text:
Vishwaroopam II, or Vishwaroop II, is a 2018 Indian spy action thriller film written and directed by Kamal Haasan and co-written by Atul Tiwari and Chakri Toleti. It is the sequel to Vishwaroopam (2013) and features Kamal Haasan alongside Rahul Bose, Shekhar Kapur, Pooja Kumar and Andrea Jeremiah, reprising their roles. While the first film was set in the United States, Vishwaroopam II takes place in India. The film was simultaneously shot in Tamil and Hindi.

Produced by Kamal Haasan and Venu Ravichandran in Tamil, the film was promoted by Ekta Kapoor and Shobha Kapoor in Hindi as Vishwaroop II. Initially planned as a late 2013 release, the film was stuck in development hell and was revived only in April 2017 when Haasan took over as producer himself.

Plot

A prequel and sequel to Vishwaroopam,

RAW Agent Major Wisam Ahmad Kashmiri is given the task to eliminate Al-Qaeda member, Omar Qureshi, and his fellow jihadis, who plan to attack New York City.

Wisam and his team travel to United Kingdom to perform the final rites of their colleague, Dawkins. Nirupama gets jealous of Ashmita's closeness with Wisam and asks her how both are related. Wisam recalls his days in the Indian Army where he found a young, enthusiastic, and brave young lady officer, Ashmita, in a parade and award ceremony. Wisam and Colonel Jagannathan together finalize Ashmita for their project, and she is trained by Wisam. Wisam was chosen for the mission as he is an illegitimate son of a Pakistani and his mother is an Indian woman. Wisam and Ashmita created an impression that she was having an affair with Wizam, to which he was court-martialed for 10 years, for having an affair with an army woman, failing to do his duties. Wisam was imprisoned but, subsequently escapes, with the help of Jagannathan. Wisam was stamped as a wanted militant and he along with Lieutenant Roy aka Imtiaz set for their mission to join Omar's group.

In the present, Wisam, Nirupama, Ashmita and Jagannathan are received by Goswami, their fellow army officer, under the instructions of his superior, Eshwar Iyer. On their way in the car to hand over Dawkins' mortal remains to his family, they are attacked by a group of militants, and their car gets into a fatal accident. While Nirupama, Ashmita, and Jagannathan are trapped in the car, Wisam is thrown out. Wisam and his team are further tracked down by a militant, and he engages in a duel to stop him and subsequently kills him. Goswami succumbs to the accident and dies happily seeing his enemy's face, which used to be his wish as an army officer.

Wisam faints and is immersed in the flashback of his final encounter with Omar. Wizam and Imtiaz work together to take down the terrorist group. Though Jagannathan trusts Wisam completely, his senior officer, Seshadri is suspicious of Wisam, and he never conceals his suspicion of Wisam's loyalty to India. Wisam tries to place a locator to give the coordinates of the Al Qaeda's prime leader Osama to the NATO troops. Omar catches Wisam red-handed and tries to shoot him, which kills Imtiaz. Wisam escapes from Omar and signals flares for the fighter choppers. Wisam is shot heavily, and the army soldiers take him to the hospital, where he is saved eventually.

Back to the present, all the accident victims reach their hotel, where they are received by Eshwar. Eshwar and Wisam engage in an argument for killing the terrorist instead of catching him. Eshwar taunts Wisam with his religion, but Wisam says he is a proud Muslim who is willing to sacrifice his life for his country. Eshwar leaves the place angrily. Wisam and Ashmita scan the room for bugging devices and eventually break one into pieces and ask Eshwar not to try anything with them. Wisam and Nirupama get into a romantic scene but are interrupted by a call for a meeting with Jagannathan, Eshwar, and an ISI agent, who turns out to be Munnavar. Wisam suspects Eshwar's hand in the terrorist attack on them. Wisam, Ashmita, and Nirupama leave to scan a building, along with an ISI agent, in a place outside the city. Wisam and Eshwar engage in a phone call during Wisam's visit to the building to search for bombs. Eshwar insists Wisam to leave the building, but Wisam disobeys. Wisam actually left the building but pretends to have got trapped inside the building. Eshwar triggers a bomb from his cell phone and is quite happy that Wisam is killed. However, Munnavar informs that Wisam is alive, and he escapes the scene. Wisam threatens Eshwar that he has revealed himself and he would hunt him down at any cost. Eshwar kills himself to avoid getting caught by Jagannathan.

The team finds a book on tides receding time, and they trace a 65-year flashback of how 1500 tons of weapons sent by America during the Second World War helped British troops getting sunk under the sea near the London shore and how militants are trying to blow the cesium weapons underwater, creating a tsunami, which can immerse London 10 to 16 meters below sea level. Nirupama, who owns a deep sea diving certificate, dives underwater to examine the intensity of the tsunami triggered. Wisam's acquaintance, Jim, is killed by a militant, and Wisam tracks him down, who tries to blast the 1500-ton bomb underwater. Wisam succeeds in stopping him by killing him.

The team reaches Delhi, and Jagannathan is called alone by his senior for debriefing. Wisam, Nirupama, and Ashmita reach their rooms to have rest. Nirupama gets angry on seeing Ashmita and Wisam sleeping in the same bed. She angrily leaves to a nearby coffee shop, where she overhears Jagannathan's debriefing session with his senior Seshadri. Jagannathan and Seshadri argue on Nirupama's role in the team. Seshadri asks him to get rid of her, but Jagannathan refuses as Wisam loves her more than anything and that they are supposed to respect his feelings. Nirupama is moved by Wisam's love. She returns to him, and the two make love. Nirupama and Wisam later go to meet Wizam's mother, who is an Alzheimer's patient living in a care home. Wisam's mother could not recognize her own son but is always lamenting looking at the childhood photos of her son. Wisam feels very sad and happy simultaneously and gets emotional with his mother. He recalls his days with her during their Kathak sessions.

Ashmita and Nirupama are kidnapped by Salim. Ashmita tries to knock him down, but he kills her and sends her chopped body to Wisam. Wisam goes to Omar to free Nirupama, who is imprisoned with her hands behind her back in handcuffs as Omar's hostage. Wisam begs Omar to spare Nirupama, but Omar refuses as Wisam was the reason behind the death of his family members during the NATO attack. Wisam tells that his wife, son, and nephew are alive. He explains to him that he handed over them to NATO to be taken to a safe haven. Omar insists on killing Wisam but pretends to spare Nirupama. Nirupama is taken to Daryaganj, where Wisam's mother too has been kept as a hostage. Omar plants a bomb on Wisam's neck with a 40-second timer. Nirupama gets devastated on seeing the bomb blast from Daryaganj. Salim tells her that to celebrate 64 years of Indian Independence, they have planted 64 bombs which would blast by an SMS. Wisam shatters the window from outside and kills Salim. A short flashback shows how Wisam escaped the bombing in a few seconds knocking everyone down and how the bomb blast burnt Omar alive. Wisam instructs his men to stop the mass bombing.

In the hospital, Wisam shows Omar's sons to Omar stating that Jalal (the elder son) has finished engineering and moving to London for MTech and Nasser (the younger son) is in his first year of med-school. Omar, on his death bed, realizes his mistake and dies. Wisam, along with Nirupama, quietly walk out of the hospital.

Cast

 Kamal Haasan as Vishwanathan / R&AW Agent Major Wisam Ahmad Kashmiri, a former soldier of the Indian army and agent of R&AW who has infiltrated the Al Qaeda during America's War on Terror; his past comes back to haunt his present
 Rahul Bose as Omar Qureshi, the main antagonist and Wisam's former friend whom he sees as a traitor and villain for the supposed death of his family
 Pooja Kumar as Dr. Nirupama Vishwanathan / Nirupama Kashmiri, Wisam's wife with knowledge of scuba diving and a doctorate in nuclear oncology
 Andrea Jeremiah as Ashmita Subramaniam, Wisam's friend and member of the army whom Nirupama gets jealous of because of her and Wisam's history.
 Shekhar Kapur as Colonel Jagannathan (Jaganath in Hindi), one of Wisam's mentors and higher orders during his infiltration and best friend of deceased John Dawkins and Gowswami
 Jaideep Ahlawat as Salim, secondary antagonist, and Omar's deputy
 Russell Geoffrey Banks as Jim, Wisam's assistant with his operations
 Waheeda Rehman as Kashmiri's mother, who's also an Alzheimer's victim
 Anant Mahadevan as Eshwar Iyer, Jagannathan/Jaganath and Goswami's boss who is revealed to be a traitor working for Omar
 Mir Sarwar
 Deepak Jethi
 Jude S. Walko as Captain Joe Black

Production

Development
The film is set in India. Kamal has earlier stated that the second part will be laced with a strong mother-son sentiment. He stated that 40% of the film was shot during the making of part one itself. While part 1 was a little over 2 hours 20 minutes, the sequel's length will be less than 2 hours.

Cast and crew
Due to personal reasons, cinematographer Sanu Varghese opted out of Vishwaroopam II and was replaced by Shamdat Sainuddeen. Ghibran was signed up to compose the music. Haasan has added Waheeda Rahman and  Anant Mahadevan for the sequel in addition to the main cast.

Filming
The shooting of Vishwaroopam II started in Thailand and the team shot scenes at the Bangkok airbase.  There was a sequence where actors defuse bombs and engage enemies in a gun battle underwater which was done by the actors themselves after training in scuba diving. The Chennai schedule of the film started on 13 June 2013. It was also reported that an underwater fight sequence involving Kamal Haasan has been shot by stunt director Ramesh in the second week of October 2016. The film started its final schedule of shooting from 27 November 2017 at Chennai. Some promotion of the film took place on the television series Bigg Boss Tamil 2, which is hosted by Hasaan, as well as on Bigg Boss Malayalam and also in Indian Idol and Dus Ka Dum hosted by Salman Khan.

Music
The soundtrack album and background score for Vishwaroopam II is composed by Ghibran, replacing Shankar-Ehsaan-Loy who composed for the first part. The lyrics for the songs of the Tamil version were written by Vairamuthu and Kamal Haasan, and the songs of the Hindi version were written by Prasoon Joshi and Sandeep Srivastava. The soundtrack rights for the Tamil version were acquired by Lahari Music, whereas the Hindi version was acquired by T-Series.

Kamal Hassan, penned and sung the song "Naanagiya Nadhimoolame" along with Kaushiki Chakraborty and Karthik Suresh Iyer, which was released as a single track by composer Ghibran, via his official Twitter account on 29 June 2018. The partial audio launch of the film was held on 30 June 2018 during the ongoing Bigg Boss Tamil 2 season. The event was telecasted on 1 July 2018, through Star Vijay. Kamal Hassan and Shruti Hassan performed the songs "Naanagiya Nadhimoolame" and "Gnyabagam Varugiradha" at the event. The latter was released a single track on 2 July 2018. While Shruti Hassan performed the song at the event, the original version of the song which was written by Vairamuthu, were rendered by Aravind Srinivas and Sarath Santhosh. The third single track "Saadhi Madham" was released on 27 July 2018. Written by Kamal Haasan and sung by Sathyaprakash and Andrea Jeremiah, the song was performed at the ongoing Bigg Boss Tamil 2 season. The full album, which consists of an EDM version of "Gnyabagam Varugiradha" and the karaoke versions of all the songs, were launched on 5 August 2018.

The title track of Vishwaroop II, which was written by Prasoon Joshi and Sandeep Shrivatsava and sung by Aravind Srinivas and Sarath Santhosh, was launched on 25 July 2018. The full soundtrack album was released on 3 August 2018.

Marketing
The first look poster of the film was released on 2 May 2017, the theatrical trailer was released on 11 June 2018.

Release

Theatrical
Vishwaroopam II was scheduled to release in February 2014, but due to delays, Hassan postponed the release. Haasan also stated initially that the film would be released as a direct to home feature in the United States. However, he later stated that he would not, as "there seems to be a lot of reticence in that." In November, Haasan stated that he hopes to release the film by 2016, but the release of the film is on hold due to the financial crisis of the producer even though some combination scenes, as well as VFX, are pending. It was also reported that Haasan will complete his portions only after his remaining salary of Rs.10 crores is settled and Lyca Productions might take over the project from Aascar Ravichandran.

In February 2017, Kamal Haasan revealed that the film was on hold as Aascar Ravichandran was not able to pay the technical crew of the film. He added that over six months' worth of post-production remained pending on the project.

The film was released theatrically on 10 August 2018.

Home media
The digital rights of the film were sold to Amazon Prime Video, while the satellite rights of the Tamil version, Vishwaroopam 2 were sold to STAR Vijay.

Reception

Box office
Vishwaroopam 2 grossed  from the Tamil Nadu box office in its first 7 days. The film collected around  at worldwide box office in first weekend. The film collected  in Andhra Pradesh and Telangana,  in Karnataka and  in rest of India. The film is believed to be earned  in overseas to  at worldwide box office. Despite this, the film grossed over  and became one of the highest grossing Tamil films of the year.

Critical response
The film received mixed reviews from critics. Rachit Gupta of The Times of India has given 2.5 stars and summarized "Despite having some genuinely good moments, the film tries to put forth a little too much, a little too quickly.". Behindwoods given 3 stars out of 5 and said "Vishwaroopam 2 is a fitting sequel to the masterpiece". Baradwaj Rangan of Film Companion said "A not-exactly-needed sequel, but a solid delivery mechanism for Kamal-isms". Raisa Nasreen of DGZ Media summarized "Kamal Haasan is heroic as the one man army". Sowmya Rajendran of The News Minute said "A predictable sequel short of fresh ideas". Vishal Menon of The Hindu wrote "Add that to particularly tacky production design and uninspiring visuals, you realise that it's easy for someone to mistake the first part as the newer film. By the end of Vishwaroopam 2 , we've seen so many bombs being planted, only for them to be defused. What's another 100 more?" Ashameera Aiyappan of The Indian Express gave the film 3.5 stars and wrote "Do catch Vishwaroopam if possible before watching the second part. You won't lose anything if you don't, but there is a lot to gain if you do." Subra Gupta of The Indian Express gave the film 1.5 out of 5 stars and stated that "That leaves Kamal Haasan, who has co-written, directed and produced the film, as the patriotic RAW agent, to do all the heavy lifting. In the first edition, the veteran star-actor led from the front: here the 'dirty soldier turned bloody espionage agent' is all thundery lines like 'main mazhab nahin mulk ke liye khoon bahaata hoon'. But even he can't rise above the shockingly inept script, which he rescues only in a few places, when his trademark intelligent, wry self-awareness manages to kick in. The rest can be safely ignored." Saibal Chatterjee of NDTV gave 1.5 out of 5 stars and wrote "That pretty much sums up Vishwaroop 2. It repeatedly fires blanks - noisy but of no use. Has a movie sequel ever been so pointless?. Janani K of India Today gave 2 out of 5 stars and wrote "On the whole, Vishwaroopam 2 is a story that is let down by the shoddy screenplay. Even a star performer like Kamal Haasan couldn't save the audience from this mess." Karishma Shetty of Pinkvilla wrote "Is Vishwaroopam 2 worth a watch? I would recommend you to see Mission Impossible - Fallout instead.  If it is of any interest, the first part of the franchise was honoured with Best Production at the National Film Awards. We rate it a 35% in the movie meter. Priyanka Sundar of Hindustan Times gave 2.5 out of 5 stars and wrote "That, however, is not the biggest problem of Vishwaroopam. That indubitably is Kamal directing the film as a director and not filmmaker. We can see the savvy politician Kamal is doing great onscreen. Wisam, unfortunately, is lost in the back ground." Shreedhar Pillai of Firstpost gave the film 2.5 out of 5 stars and stated that "The USP of the film is Kamal Haasan, who doesn't look his age (63) and does action, romance and dialogue-delivery with consummate ease and style. Some of the action scenes are well choreographed. Ghibran's music is a major plus point. But the sequel is nowhere near the first part and looks stretched (2 hours 21 minutes). Though part two lacks the finesse of part one, Vishwaroopam 2 is still an entertainer." Bollywood Hungama gave the film's rating 1.5 out of 5 and wrote "On the whole, Vishwaroop II is a highly avoidable flick. The film has too many tracks and the narration and execution is flawed and weak. Also at the box office, the film will have a tough time."

Sudhir Srinivasan of The New Indian Express gave the film 3.5 out of 5 and wrote "It's such small, effective moments that make me feel quite fondly about the Vishwaroopam films, and despondently about Tamil cinema without Kamal Haasan, the director." Reshe Manglik of India TV gave 2 out of 5 stars and wrote "In short, it's only Kamal Haasan doing all the work in the film. He delivers the thundery lines like 'Main mazhab ke liye nahi mulk ke liye khoon bahata hu' with immense grit. However, even a spectacular actor like Haasan can't rise above the script that was clearly sinking." Anupama Subramanian of Deccan Chronicle gave 3.5 out of 5 stars and wrote "To be fair, most of these stunts and action sequences are riveting and enjoyable. Kamal is ever present with his dry wit and self-aware punch dialogues, which also brings his political aspirations to the fore. He is relentless and although he looks burnt out due to Vis's heavy workload, his brains never tire and is always a step ahead of his numerous halfwit villains. Enjoyable in parts!" Moviecrow gave the film's rating 3 out of 5 stating that "An engaging sequel despite the inevitable comparison with the prequel and a few misfires. Watch it for Kamal Haasan's unsatiating desire to push the bar higher.  " Umesh Punwani of Koimoi rated the film 1.5 out of 5 stars and wrote "All said and done, fans of Vishwaroop should forget there was a sequel planned for it. They should again watch the part 1 instead and keep its memories intact. It's like one of those many Hollywood actioners which are all glam and no content." DNA India gave the film's rating 2 out of 5 and wrote "Vishwaroop 2 boasts of some great cameos by Shekhar Kapur, Waheeda Rahman and Jaideep Ahlawat but they are so good in whatever little screen-time they have got that sometimes you feel they are wasted in the film. The spy-thriller doesn't rely much on jingoism, though, which is a relief." Ananda Vikatan rated the film 42 out of 100.

Notes

References

External links

2018 films
2010s Hindi-language films
2018 action thriller films
Films directed by Kamal Haasan
Indian action thriller films
Indian sequel films
Films about the Research and Analysis Wing
2010s Tamil-language films
Indian multilingual films
Reliance Entertainment films
Films scored by Mohamaad Ghibran
War in Afghanistan (2001–2021) films
Films set in Delhi
Films about terrorism in India
Films set in Chennai
Indian Army in films
Films set in Kent
2018 multilingual films

redpajama_set_name: RedPajamaWikipedia


yes, exactly what you said. I saw this post on Tumblr and reblogged it.
Damn, Jared was unbelivable in this episode.
Thanks for the reblog on tumblr, I think this was my most reblogged post ever., it's nice to hear other fans agree!
You are so right, Jared really was unbelievable, I watched it a third time and was just blown away by how deep into character he got.


Shredding cans, otherwise known as crushing them, can be accomplished with a little knee grease. But, after time, stepping on cans to crush them can wear on your knees, and cause messy, sticky stains. Particularly if you have many cans to crush, you may want to create your own homemade aluminum .
While stomping on cans is good exercise, there is an easier and quicker way to crush them: Make your own aluminium . An easy project, it will save time and also encourage your family to crush and recycle cans.
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Whether you host the occasional sorority gathering or just drink up your own six-pack of soda pop, you'll find that empty aluminum beverage cans take up a lot of space.
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1) Retinal thickening at or within 500 microns or 1/3 disc diameter of center of macula.
2) Hard exudates at or within 500 microns of the center of the macula with adjacent retinal thickening.
3) Retinal thickening GREATER than 1 disc diameter in size which is within 1 disc diameter from the center of the macula.


text:
HUGO Gene Nomenclature
: HUGO Gene Nomenclature - XML Representation
<id value="v3-hgnc"/>

This code system http://www.genenames.org defines many codes, but they are not represented here

<url value="http://www.genenames.org"/>
<value value="urn:oid:2.16.840.1.113883.6.281"/>
<name value="Hgnc"/>
<title value="HUGO Gene Nomenclature"/>
<publisher value="HUGO Gene Nomenclature Committee (HGNC)"/>
value="HUGO Gene Nomenclature Committee (HGNC) is committee jointly funded by the US National Human Genome Research Institute and the Wellcome Trust (UK). It operates under the auspices of HUGO, with key policy advice from an International Advisory Committee, and is responsible for approving gene names and symbols. All approved symbols are stored in the HGNC database. Each symbol is unique and each gene is only given one approved gene symbol. More than 28,000 human gene symbols and names have been approved so far. The vast majority of these is for protein-coding genes, but also includes symbols for peudogenes, non-coding RNAs, phenotypes and genomic features.
HGNC gene symbols can be used with the HL7 coded data type. For example, in the HL7 messages specified according to the HL7 V2 Clinical Genomics Fully LOINC-Qualified Genetic Variation Model Implementation Guide, HGNC gene symbols can be used to as the observation values for gene identifiers. For example, OBX|1|CWE|48018-6^Gene identifier^||BRCA1^HGNC|
Versioning Information: The version of the HGNC database is reported using the last updated date and timestamp. The last updated date and timestamp is posted on the main HGNC Search screen in the format like "Monday March 30 23:00:56 2009".
HGNC is updated daily.
HGNC is a free database for the public."/>

redpajama_set_name: RedPajamaCommonCrawl


There are unwritten rules about drinking coffee in Italy. You might just get a very funny look in cafés throughout Italy if you don't learn these few simple rules.
– Beverages with milk are to be drunk only in the morning. Cappuccino, latte, macchiato late and all other favorite beverages in the world with milk are drunk in Italy only in the morning before 11 am. Also never order beverages containing milk after meals.
– Do not experiment. Don't ask for mint frapuccino or similar craziness because the Italians hold to simple recipes. The only exception is Naples – where "caffè alla nocciola" (espresso with hazelnut cream) is served and in Milan, where you can drink "marocchino" – kind of 'reverse' cappuccino (the bottom of the cup should be sprinkled with cocoa, then the gradient sparkling milk and at the top is coffee).
– Do not look for "espresso". Do not use the word "espresso" because in Italy it represents only a technical term – not the one that is in daily use. Simply order "un caffè". You can order "caffè doppio" (or double espresso) if you want, but keep in mind that the Italians do not have the habit of drinking this type of coffee. It is true that Italians drink a lot of coffee, but they always make small and strong doses.
– Be clear and loud. Italian bars are often packed and understaffed, so customers will not be able to count on fast service. Just pay a visit to the bar and call the bartender even if he is not facing you. Italians tend to drink coffee at the bar quickly, standing up. Coffee that is served at tables can be several times more expensive. If you want your coffee hot, order "caffè bollente". Apart from cappuccino, latte and macchiato coffee / late macchiato, they also drink "caffè corretto" (with the addition of brandy), "caffè freddo" / "cappuccino freddo" (iced espresso or cappuccino ice, commonly used sweetened), and "un caffè lungo" or "un caffè ristretto", depending on the amount of water in your espresso.
-Payment. In most cafés you will first have to pay for the order, then show your bill to the bartender who will serve you the drink. If you are not sure what rule applies in the restaurant, it would be best to wait first for a few minutes on the side, and observe the situation.


Honeywell today announced that its UOP Russell business has introduced a new high recovery cryogenic gas processing plant that is 50 percent larger than other pre-engineered plants currently available. With a capacity of 300 million standard cubic feet per day, the new SC-30 gas plant employs Ortloff Recycle Split Vapor technology that can recover more than 99 percent of high-value ethane from natural gas.
Due to its larger size, the SC-30 plant is about 20 percent more energy efficient on a per-cubic foot basis, lowering gas processors' capital and operating costs. The plant is prefabricated and delivered in modular form, and uses an advanced cycle process to extract high-value ethane from residue gas. Ethane is used almost exclusively as a petrochemical feedstock to produce ethylene, which is a primary component of plastics.
To make the SC-30 possible, UOP Russell created a new engineering design that included larger pumps, coolers, tanks and piping contained on modular skids. Each of these skids are manufactured in factory conditions, ensuring high manufacturing quality and on-time delivery. The entire plant can be transported on truck beds and assembled on site, anywhere natural gas is being produced.
Moreover, the SC-30's RSV technology recovers more than 99 percent of the high-value ethane present in natural gas, far more than standard Gas Subcooled Process plants that are limited to about 90 percent recovery.
According to the U.S. Energy Information Administration (EIA), U.S. production of ethane is forecast to increase from an average of 1.25 million barrels per day (b/d) in 2016 to 1.7 million b/d in 2018. The EIA predicts that increased ethane production will be consumed in the petrochemical industry domestically, in addition to exports to other countries, particularly in Europe and Asia.
UOP Russell, a subsidiary of Honeywell UOP, provides skid-mounted, pre-engineered and customized modular equipment. The equipment is easily delivered to remote locations and then assembled on-site, significantly reducing construction time and expense and improving operational reliability. This modular approach enables customers to begin processing gas and earning revenue more quickly versus conventional field-erected execution methods.
Last week, Honeywell UOP announced that it has acquired the business of Ortloff Engineers, Ltd., the industry's leading developer of specialized technologies that drive high returns in natural gas processing and sulfur recovery. Ortloff technologies are specialized to maximize gas separation and compression in modular cryogenic gas plants, providing customers with high operational flexibility and greater returns on their plant investments.


text:
The accuracy of human population maps for public health application.
Hay SI., Noor AM., Nelson A., Tatem AJ.
Human population totals are used for generating burden of disease estimates at global, continental and national scales to help guide priority setting in international health financing. These exercises should be aware of the accuracy of the demographic information used.The analysis presented in this paper tests the accuracy of five large-area, public-domain human population distribution data maps against high spatial resolution population census data enumerated in Kenya in 1999. We illustrate the epidemiological significance, by assessing the impact of using these different human population surfaces in determining populations at risk of various levels of climate suitability for malaria transmission. We also describe how areal weighting, pycnophylactic interpolation and accessibility potential interpolation techniques can be used to generate novel human population distribution surfaces from local census information and evaluate to what accuracy this can be achieved.We demonstrate which human population distribution surface performed best and which population interpolation techniques generated the most accurate bespoke distributions. Despite various levels of modelling complexity, the accuracy achieved by the different surfaces was primarily determined by the spatial resolution of the input population data. The simplest technique of areal weighting performed best.Differences in estimates of populations at risk of malaria in Kenya of over 1 million persons can be generated by the choice of surface, highlighting the importance of these considerations in deriving per capita health metrics in public health. Despite focussing on Kenya the results of these analyses have general application and are discussed in this wider context.
Tropical medicine & international health : TM & IH
TALA Research Group, Department of Zoology, University of Oxford, Oxford, UK. simon.hay@zoo.ox.ac.uk
Humans, Malaria, Risk Factors, Public Health, Climate, Endemic Diseases, Demography, Population Density, Topography, Medical, Kenya


text:
\section{Introduction}
The recent discovery of topological insulators and other topological phases~\cite{Molenkamp} has revitalized the venerable subject of band theory. In addition to the explosion of understanding of  different  forms of band topology and how symmetries protect or prevent them, rapid progress is being made on several other fronts. For example, fundamental questions such as the connection between electron count and insulating behavior~\cite{Zak1999,Zak2000,Zak2001,Parameswaran, Roy,PNAS, SciAdv,PRL}, as well as the constraints imposed by crystal symmetries on the connectivity of bands \cite{Wigner,Herring,Zak1999,Zak2000,Zak2001,PRL,KaneDirac,Combinatorics, NC, Bernevig1, Graph}, have been resolved. New information has also been gleaned by contrasting  real-space and momentum-space descriptions in all 230 crystal space groups (SGs) of nonmagnetic materials~\cite{Zak1999,Zak2000,Zak2001,SciAdv,PNAS,PRL,NC,Bernevig1,Bernevig2}.

Increasingly, attention is turning to electronic systems that combine magnetism with band topology. Here, a further panoply of novel topological phenomena is anticipated. Examples that have already been realized, to name just a few, include the quantized anomalous Hall effect in magnetic topological insulators~\cite{SCZhang,QiKunXue} and the topological Hall effect in skyrmion crystals~\cite{KimMillis,Tokura, Ong, BinzVishwanath,Pfliederer}. Promising magnetic Weyl semimetals candidates are also now being intensively studied~\cite{Nakatsuji,Felser,Yan}.  In addition to the fundamental  physics interest of these novel phases,  current-driven magnetic textures such as domain walls or skyrmions could be harnessed in technological applications, say in the development of energy-efficient memory devices~\cite{Fert}.

Despite these strong motivations, the pace of discovery of magnetic topological materials has been relatively slow compared to their nonmagnetic counterparts. There are at least two reasons for this: First, magnetic materials are necessarily correlated, making the  prediction of the magnetic structure and properties  more challenging. Consequently the set of well characterized magnetic materials is relatively small. Second, the sheer complexity of combining magnetic structures with SGs makes an exhaustive study of the theoretical possibilities daunting. Indeed, there are a total of 1651 different magnetic space groups (MSGs), which were only tabulated in the 1950s. Also, unlike for the 230 nonmagnetic SGs ~\cite{Bradley,Bilbao},  relevant  group-representation information is not always readily available.

Here, we will tackle the second problem by providing a systematic understanding of electronic band structures in the 1651 MSGs.  Inspired by the recent synthesis of atomically thin magnets ~\cite{Gong,Huang}, we also discuss the 528 magnetic layer groups (MLGs), relevant to two-dimensional magnetic materials.
We report  results for three key properties, tabulated in Appendix \ref{app:tables} and described below.

First, we determine electron fillings that can be compatible with insulating behavior in all MSGs.  The electron count is a fundamental characteristic of an electronic crystal. The presence of nonsymmorphic symmetries, such as glides and screws, enforces connectivity of bands that raises the required fillings for realizing band insulators~\cite{Herring,Zak1999,Zak2000,Zak2001}.
These conditions may be useful in the search for magnetic Weyl semimetals, as one can target fillings that, while forbidding band insulators, are nonetheless consistent with nodal-point Fermi surfaces~\cite{KaneDirac,RuChen,YoungWieder}.

Next, it has long been known that representations of energy levels at high-symmetry points must connect in specific ways in obtaining a set of isolated bands~\cite{Wigner}. This can be viewed as a refinement of the filling condition, which  imposes  additional constraints on band structures (BSs)~\cite{Combinatorics,NC,Bernevig1,Graph}. The solutions to these constraints are most conveniently represented as a vector space (but with integer coefficients, so more accurately a `lattice' in mathematical terminology)~\cite{Ari,Combinatorics,NC}, and are described using only a handful of basis vectors whose precise number $d_{\rm BS}$ depends on the MSG.

Lastly, we contrast the general BSs defined in the momentum space with the subset of those obtained from `atomic insulators' (AI), in which electrons are tightly bound to sites in the lattice furnishing orbitals with different symmetries. Significantly, we find that this produces a vector space of the {\em same} dimension, which in mathematical terms is summarized by the equality $d_{\rm AI}=d_{\rm BS}$. However, the bases for BSs and AIs do not generally coincide, and we determine the classes of band structures that cannot be reduced to any atomic limit.
This leads to an obstruction to finding symmetric localized Wannier states ~\cite{Thouless, Vanderbilt}, and corresponds to band topology that can be diagnosed without detailed information about the electronic wave-functions. We give an example of how this can be used to diagnose topological semimetals, and also an example of a nodal-line semimetal diagnosed through its filling. The latter consists of hedgehog-antihedgehog magnetic order with a lone electron at the core of these defects, leading to gapless behavior.

We note that much progress has recently been made on related problems for nonmagnetic SGs~\cite{SciAdv,PRL,NC,Bernevig1,Bernevig2}. However, in the case of MSGs these problems have only been attacked in certain restricted settings, for example in generalizing the parity criterion~\cite{FuKane} to identify Weyl semimetals, Chern insulators, and axion insulators~\cite{Ari10, Ari, PhysRevB.83.205101, Bernevig, Chern_Rotation}.
Here, we furnish a systematic study of all MSGs.
In particular, it is important to emphasize that, although our approach has several features in common with K-theory-based classifications~\cite{MooreFreed, Ktheory, Combinatorics, Ken2017}, we do not seek to fully classify crystalline electronic phases here. As discussed in a recent insightful work~\cite{Combinatorics} for two dimensional wallpaper groups, part of such a general classification includes band-structure connectivities, a problem that has been posed since the early days of band theory~\cite{Wigner}. Utilizing a result we prove below, we construct a framework that greatly simplifies such computations, which enables an extension of previous results \cite{SciAdv,PRL,NC} to the physically important case of MSGs.

In addition, topological distinctions revealed using our symmetry-based indicators fit naturally into well-established frameworks~\cite{MooreFreed,Combinatorics,Ken2017,NC}, and remain stable upon addition of trivial degrees of freedom.
Yet, our approach is inherently symmetry-based, and so for crystals with low symmetry, say with only lattice translations, topological insulators cannot be diagnosed without further knowledge of the wavefunctions.
A similar caveat pertains to the complementary quantum chemistry approach of Ref.\ \onlinecite{Bernevig1}, which elaborates on the theory developed in Ref.\ \onlinecite{Zak1999}, and takes as input a specific set of orbitals at fixed locations in the crystal. While convenient for representing quantum chemistry information, it is less suited to capturing stable topological distinctions that survive the inclusion of additional trivial bands.
 Also, one forgoes the simplification arising from the vector space like representation of energy bands that allowed us to generate results for all MSGs, while Ref. \onlinecite{Bernevig1} is currently confined to just the 230 SGs.

\section{Magnetic Band structures}
Let us begin by reviewing some background materials concerning electronic band structures arising from a magnetic material. There are in total 1651 MSGs and 528 MLGs~\cite{Bradley, Bilbao, Litvin, IUC, BNSdatabase}.
Among the 1651 MSGs, 230 of them are identical to SGs in which only unitary spatial symmetries are considered (type I MSGs).
All other MSGs have an equal number of unitary and anti-unitary elements: $\mathcal{M}=\mathcal{G}+\mathcal{A}$. The unitary part $\mathcal{G}$ is identical to one of the 230 SGs and the anti-unitary part $\mathcal{A}$ can be generally written in the form $\tilde{\mathcal{T}}\mathcal{G}$, where $\tilde{\mathcal{T}}\equiv\mathcal{T}g_0$ is the product of a spatial operation $g_0$ and time-reversal (TR) $\mathcal{T}$.  When $g_0$ belongs to $\mathcal{G}$, the MSG is simply the direct product of a SG $\mathcal{G}$ and $\mathbb Z_2^{\mathcal{T}}$.
This leads to another 230 MSGs, one for each SG, that are called type II.
When $g_0$ is not an element of $\mathcal{G}$, there are two types further differentiated by whether $g_0$ is a pure translation (type IV) or not (type III).
For type II-IV MSGs, the little group of $\vec{k}$ and the site-symmetry group of $\vec{x}$ may also have an anti-unitary part, in addition to the usual unitary part.   Note that, in the literature (e.g.\ \cite{PennState}), our definition of type III and IV MSGs are sometimes called type IIIa and IIIb respectively.
In addition, there are two common labeling schemes for MSGs: Opechowski-Guccione (OG) and Belov-Neronova-Smirnova (BNS). In this work we follow the BNS notation, where an MSG is labeled by a pair of integers written as $S.L$, with $S$ corresponding to one of the 230 SG and $L$ an extra label to differentiate between different magnetic descendants. (Please refer to, e.g., Ref.\ \onlinecite{BilbaoMagnData}, for the precise meaning of these numbers.)

Next, we introduce a general formalism for the efficient analysis of band structure properties based on representations of the little group~\cite{Ari,Combinatorics,NC}.
 Our main focus here is to address how the formalism developed for SGs can be readily applied to MSGs.
We define a band structure (BS) as a set of bands isolated from others by band gaps above and below at all high-symmetry momenta, where by a high-symmetry momentum we refer to a $\vec{k}$ in the Brillouin zone at which the \emph{unitary} part of the little group, $\mathcal{G}_{\vec{k}}$, is strictly larger than the translation subgroup $T$. In 3D, they can be high-symmetry points, lines or planes.
A BS can be characterized by the set of nonnegative integers $\vec{n}=\{n_{\vec{k}}^\alpha\}$ that count the number of times an irreducible representation (irrep) $u_{\vec{k}}^\alpha$ of $\mathcal{G}_{\vec{k}}$ appears.  As we are interested in systems of spinful electrons, the irreps $u_{\vec{k}}^\alpha$ here are generally projective due to the spin-1/2 nature of electrons (also called `double-valued'). Since $\{n_{\vec{k}}^\alpha\}$ cannot be changed smoothly without gap closing or symmetry breaking, $\vec{n}=\{n_{\vec{k}}^\alpha\}$ serves as `topological invariants' defined for each BS.

The integers $\vec{n}=\{n_{\vec{k}}^\alpha\}$ cannot be chosen freely, since symmetries demand that they satisfy a collection of compatibility relations~\cite{Wigner, Bradley}.  Since $\mathcal{G}$ is a subgroup of $\mathcal{M}=\mathcal{G}+\mathcal{A}$, the full list of compatibility relation $\mathcal{C}$ imposed on an $\mathcal{M}$-symmetric BS can be split into two sets, $\mathcal{C}_{\mathcal{G}}$ arising from $\mathcal G$ and $\tilde{\mathcal{C}}_{\mathcal{A}}$ from $\mathcal A$.
Let us denote by $\{{\rm BS}\}_{\rm phys}^{\mathcal{G}}$ and $\{{\rm BS}\}_{\rm phys}$ the set of all $\vec{n}$'s satisfying $\mathcal{C}_{\mathcal{G}}$ and $\mathcal{C}$, respectively.
Here, the subscript `phys' indicates that all $n_{\vec{k}}^\alpha$'s in $\vec{n}=\{n_{\vec{k}}^\alpha\}$ are nonnegative, which is required for interpreting them as the multiplicities of irreps in a physical band structure.
We will introduce another set $\{{\rm BS}\}^{(\mathcal{G})}$ that relaxes this nonnegative condition later.
Note that $\{{\rm BS}\}_{\rm phys}^{\mathcal{G}}$ and $\{{\rm BS}\}_{\rm phys}$ differ only in the imposition of $\tilde{\mathcal{C}}_{\mathcal{A}}$.
In general, the anti-unitary part $\mathcal{A}$ requires a pairing of $\vec{b}\in\{{\rm BS}\}_{\rm phys}^{\mathcal{G}}$ with another $\vec{b}'\in\{{\rm BS}\}_{\rm phys}^{\mathcal{G}}$, unless $\vec{b}$ itself is already symmetric under $\tilde{\mathcal{T}}$.  The pairing type can be easily determined using the Herring rule~\cite{Bradley}.
(In Appendix~\ref{app:compatibility}, we provide a more elaborated review on the compatibility relations and the Herring rule.)

\section{Band Topology}
Having described some generalities about BSs, we now review how knowledge about the real space can inform band topology \cite{SciAdv,NC}.
We define the trivial class of BSs by the AIs, which are band insulators that are smoothly connected to a limit of vanishing hopping, and hence are deformable to product states in real space. Equivalently, an AI admits symmetric, exponentially localized Wannier functions.

To specify an AI, one should choose a position $\vec{x}$ in real space at which electrons are localized, and the type of the orbital put on that site.
All inequivalent choices of the position $\vec{x}$ are classified by Wyckoff positions~\cite{ITC}. The orbital can be chosen from the (co-)irreps of the site-symmetry group of $\vec{x}$ (Appendix~\ref{app:AI}). Given these choices, an $\mathcal{M}$-invariant AI can be constructed by placing a symmetry-related orbital on each site of the $\mathcal{M}$-symmetric lattice and filling them by electrons.  The AI has a specific combination of irreps in the momentum space, which automatically satisfies $\mathcal{C}=\mathcal{C}_{\mathcal{G}}+\tilde{\mathcal{C}}_{\mathcal{A}}$.  We list up all distinct $\vec{n}$'s corresponding to an AI by varying $\vec{x}$ and the orbital type.  We list up all distinct $\vec{n}$'s corresponding to an AI by varying the position $\vec{x}$ and the orbital type and we obtain $\{{\rm AI}\}_{\rm phys}$, a subset of $\{{\rm BS}\}_{\rm phys}$.   If one replaces $\mathcal{M}$ above with $\mathcal{G}$, one gets the set of $\mathcal{G}$-symmetric AIs, $\{{\rm AI}\}_{\rm phys}^{\mathcal{G}}$.

Now we are ready to tell which elements of $\{{\rm BS}\}_{\rm phys}$ must be topologically nontrivial and which elements can be trivial.  This can be judged by contrasting the elements of $\{{\rm BS}\}_{\rm phys}$ with those in $\{{\rm AI}\}_{\rm phys}$. Namely, any $\vec{b}\in\{{\rm BS}\}_{\rm phys}$ not belonging to $\{{\rm AI}\}_{\rm phys}$ necessarily features nontrivial band topology, because, by definition, there does not exist any atomic limit of the BS with the same combinations of irreps.  This is a sufficient (but not necessary) condition to be topologically nontrivial --- here we exclusively focus on the band topology that can be diagnosed by the set of irreps at high-symmetry momenta.

The simplest way in exploring the nontrivial elements of $\{{\rm BS}\}_{\rm phys.}$ is thus to consider the complement of $\{{\rm AI}\}_{\rm phys.}$ in $\{{\rm BS}\}_{\rm phys.}$, as in Refs.~\onlinecite{NC, fragile}.
However, this set has a complicated mathematical structure. To simplify the analysis, we allow for the formal \emph{subtraction} of bands, and extend the values of $n_{\vec{k}}^\alpha$ to any integer, including the negative ones, {\it {\` a} la} a K-theory analysis.  $\{{\rm BS}\}_{\rm phys}$ then becomes an abelian group $\{{\rm BS}\}=\mathbb{Z}^{d_{{\rm BS}}}$ (known as a `lattice' in the mathematical nomenclature)~\cite{Ari,Combinatorics,NC}.   In other words, there are $d_{{\rm BS}}$ basis `vectors' $\{\vec{b}_i\}_{i=1}^{d_{{\rm BS}}}$, and $\{{\rm BS}\}$ can be expressed as $ \left \{\sum_{i=1}^{d_{{\rm BS}}}m_i\vec{b}_i\,|\,m_i\in\mathbb{Z} \right\}$.  Similarly, by allowing negative integers when taking superposition of AIs, we get another abelian group $\{{\rm AI}\}=\mathbb{Z}^{d_{{\rm AI}}}$, which is a subgroup of $\{{\rm BS}\}$. The band topology we are interested in is now encoded in the quotient group
\begin{equation}
X_{\rm BS} \equiv \{ {\rm BS}\}/\{{\rm AI} \},
\end{equation}
dubbed the symmetry-based indicator of band topology~\cite{NC}.  As we will see shortly, the quotient group is always a finite abelian group and hence must be a product of the form $\prod_i {\mathbb Z}_{n_i}$.

\section{Constructing band structures from atomic insulators}
To compute $X_{\rm BS}$, the natural first step is to identify $\tilde{\mathcal C}_{\mathcal A}$, the extra compatibility relations enforced by the anti-unitary symmetries. Contrary to this expectation, we now show that, based on our earlier results on SGs, one can directly compute $\{{\rm BS}\}$ and $X_{\rm BS}$ for any MSG $\mathcal M$ without deriving $\tilde {\mathcal C}_{\mathcal A}$. This serves to demonstrate the power of the present approach: symmetry content and connectivity of BSs can be readily extracted without the large overheads mandated by the conventional approach.

To this end, we first revisit the relevant aspects of the theory for an SG $\mathcal G$.
By definition, $\{{\rm AI}\}^{\mathcal{G}}$ is a subgroup of $\{{\rm BS}\}^{\mathcal{G}}$, and therefore, a priori, it could be the case that $d_{{\rm AI}}^{\mathcal{G}}<d_{{\rm BS}}^{\mathcal{G}}$ strictly.  However, by an explicit computation for all the 230 SGs, Ref.\ \onlinecite{NC} found that $d_{{\rm BS}}^{\mathcal{G}}=d_{{\rm AI}}^{\mathcal{G}}$ always holds.  This statement has an important implication --- every $\vec{b}\in\{ {\rm BS} \}^{\mathcal{G}}$ can, in fact, be expanded as $\vec{b} = \sum_iq_i \, \vec{a}_i$ with \emph{rational} coefficients $q_i\in\mathbb{Q}$, using a basis $\{\vec{a}_i\}_{i=1}^{d_{{\rm AI}}^{\mathcal{G}}}$ of $\{ {\rm AI} \}^{\mathcal{G}}$.
In other words, full knowledge on the group $\{ {\rm BS} \}^{\mathcal{G}}$ can be obtained from that on $\{ {\rm AI} \}^{\mathcal{G}}$.

Based on this result for SGs, we will now prove the same statement, namely $d_{{\rm BS}}=d_{{\rm AI}}$, for any MSG $\mathcal M$.
This result enables us to efficiently compute $\{{\rm BS}\}$ and $X_{\rm BS}$ for all MSGs and MLGs using only information contained in $\{{\rm AI}\}$, which can be readily extracted from the tabulated Wyckoff positions~\cite{Litvin, IUC, BNSdatabase}.

Our proof is centered on the following observation: Recall $\mathcal M = \mathcal G + \mathcal A$, where $\mathcal G$ is unitary and $\mathcal A=\tilde{\mathcal{T}}\mathcal G$ is the anti-unitary part generated by $\tilde{\mathcal{T}} = \mathcal T g_0$ for some spatial symmetry $g_0$. Note that for type II MSGs, $g_0 \in \mathcal G$ and can be chosen to be the identity, whereas for types III and IV, $g_0 \not \in \mathcal G$.
Now consider a $\mathcal G$-symmetric collection of fully-filled local orbitals in real space, which defines an AI $\vec{a} \in \{ {\rm AI}\}_{\text{phys}}^{\mathcal{G}}$. This AI is not generally symmetric under $\mathcal M$, i.e., it may not be invariant under the action of $\tilde{\mathcal{T}}$. However, if we stack it together with its $\tilde{\mathcal{T}}$-transformed copy, we will arrive at an $\mathcal M$-symmetric AI. Algebraically, this means $\vec a + \tilde{\mathcal{T}}\vec a\in \{ {\rm AI}\}_{\text{phys}}$.

In the momentum space, a similar symmetrization procedure can be performed on the representation content.
Suppose that $\{|\vec{k},i\rangle\}$ is a basis of an irrep $u_{\vec{k}}^\alpha$ of $\mathcal{G}_{\vec{k}}$. Then the $\tilde{\mathcal{T}}$-transformed copy, $\{\tilde{\mathcal{T}}|\vec{k},i\rangle\}$, forms a basis of an irrep $u_{\vec{k}'}^{\alpha'}$ of $\mathcal{G}_{\vec{k}'}$ (Appendix~\ref{app:compatibility}).  When $\vec{b}$ represents a BS that contains $u_{\vec{k}}^{\alpha}$ $n_{\vec{k}}^\alpha$-times, we denote by $\tilde{\mathcal{T}}\vec{b}$ a BS that contains $u_{\vec{k}'}^{\alpha'}$ the same number of times.   For any $\vec b \in \{ {\rm BS}\}^{\mathcal{G}}$, $\vec{b}+\tilde{\mathcal{T}}\vec{b}$ satisfies the compatibility conditions $\tilde{\mathcal{C}}_{\mathcal{A}}$, and hence belongs to $\{{\rm BS}\}$.   We present the explicit form of $\tilde{\mathcal{T}}\vec{b}$ in Appendix~\ref{app:themapf}.

\begin{table}
\begin{center}
\caption{\textbf{Characterization of band structures (BSs) in a magnetic space group (MSG); excerpt from Tables~\ref{tab:spinful_Tri}--\ref{tab:SpinfulCubic}.}
\label{tab:ex1}}
\begin{tabular}{lc|ccc}
\hline \hline
\multicolumn{2}{c|}{MSG\footnote{MSG number in the Belov-Neronova-Smirnova notation, followed by a Roman numeral ${\rm I}, \ldots, {\rm IV}$ indicating its type.}} &
$d$\footnote{Number of linearly independent BSs.}
&
$X_{\rm BS}$\footnote{Symmetry-based indicator of band topology, which takes the form $\prod_i {\mathbb Z}_{n_i}$; denoted by the collection of positive integers $(n_1,n_2, \cdots)$.}
&
$\nu_{\rm BS}$\footnote{For most of the MSGs, the set of physical BS fillings $\{\nu\}_{\rm BS}$ and the set of AI fillings $\{\nu\}_{\rm AI}$ agree with each other, and they take the form $\{\nu\}_{\rm BS}=\{\nu\}_{\rm AI}=\nu_{\rm BS} \,\mathbb{N}$.  The asterisk indicates violation to this rule, detailed in Table~\ref{tab:spinfulnuEx}.}
\\ \hline
2.4 & I & $9$ & $(2, 2, 2, 4)$ & $1$\\
2.7 & IV & $5$ & $(2)$ & $2$\\
3.4 & IV & $3$ & $(2)$ & $2$ \\
209.51 & IV & $3$ & $(1)$ & $2^*$ \\
\hline \hline
\end{tabular}
\end{center}
\end{table}

We are now ready to prove the statement. Observe any $\vec{B}\in\{{\rm BS}\}$ also belongs to $\{{\rm BS}\}^{\mathcal{G}}$, and as $d_{{\rm BS}}^{\mathcal{G}}=d_{{\rm AI}}^{\mathcal{G}}$, it can be expanded in the basis of $\{{\rm AI}\}^{\mathcal{G}}$:
\begin{equation}
\vec{B}=\textstyle\sum_{i}q_i\vec{a}_i,\quad q_i\in\mathbb{Q}.\label{eq:expand}
\end{equation}
Now we symmetrize both sides of Eq.~\eqref{eq:expand} by $\tilde{\mathcal{T}}$.  Since $\vec{B}$ is $\mathcal{M}$-invariant, $\vec{B}+\tilde{\mathcal{T}}\vec{B}=2\vec{B}$, so
\begin{equation}
\vec{B}=\textstyle\frac{1}{2}(\vec{B}+\tilde{\mathcal{T}}\vec{B})=\sum_{i}\frac{1}{2}q_i\big(\vec{a}_i+\tilde{\mathcal{T}}\vec{a}_i\big).
\end{equation}
As argued, $\vec{a}_i+\tilde{\mathcal{T}}\vec{a}_i \in \{ {\rm AI}\}$.  (Note that linearity was invoked when we extend the argument form $\{ {\rm AI}\}_{\text{phys}}^{\mathcal{G}}$ to a general element of $\{ {\rm AI}\}^{\mathcal{G}}$.) This proves that any $\vec{B}\in\{{\rm BS}\}$ can be expanded in terms of $\{{\rm AI}\}$ (using rational coefficients), an equivalent statement of $d_{{\rm AI}}=d_{{\rm BS}}$.  Furthermore, it implies that the quotient group $X_{\rm BS}=\{{\rm BS}\}/\{{\rm AI}\}$ does not contain any $\mathbb{Z}$-factor, and hence is a finite abelian group of the form $\prod_i {\mathbb Z}_{n_i}$.

To summarize, we have shown that the set of AIs and BSs are identical as far as their dimensionality goes.  This is a powerful statement, since it means we could simply focus on AIs, study their symmetry representations in momentum space, and then take rational combinations to generate all BSs. We caution, however, that one has to properly rescale the entries of $\vec n$ when an irrep is paired with another copy of itself according to the Herring rule. For full details of the treatment, we refer the interested readers to Ref.\ \onlinecite{NC}.

Following this strategy, we perform the first calculation of  $d_{\rm BS}$, $X_{\rm BS}$, and $\nu_{\rm BS}$ for all of the 1651 MSGs and 528 MLGs. The full list of the computation results are tabulated in Tables~\ref{tab:spinful_Tri}--\ref{tab:spinful_MLG} in Appendix~\ref{app:tables}.
For readers' convenience, we reproduce a few examples from these tables in Tables~\ref{tab:ex1} and \ref{tab:ex2}.

\begin{table}
\begin{center}
\caption{\textbf{Characterization of magnetic layer groups through the corresponding magnetic space group (MSG); excerpt from Table~\ref{tab:spinful_MLG}.}
\label{tab:ex2}
}
\begin{tabular}{cc|ccc}
\hline \hline
\multicolumn{2}{c|}{MSG\footnote{The numbers in parentheses label the different ways to project the MSG down to 2D planes (Appendix~\ref{app:MLG}).}} & $d$\footnote{\label{FootTableI} Defined as in Table \ref{tab:ex1}.
}
& $X_{\rm BS}$\footref{FootTableI} & $\nu_{\rm BS}$\footref{FootTableI}
\\\hline
2.5 (1) & II & $5$ & (2) & $2$ \\
2.5 (2) & II & $5$ & (2) & $2$ \\
2.5 (3) & II & $5$ & (2) & $2$ \\
3.4 (1) & IV & $3$ & $(2)$ & $2$ \\
3.4 (2) & IV & $2$ & $(1)$ & $2$ \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

\section{Representation-enforced semimetals}
As an application of our results, here we introduce new classes of representation-enforced semimetals (reSM).  A BS is said to be a reSM when (i) there are gap closings at some generic (i.e., not high-symmetry) momenta in the Brillouin zone, and (ii) the gaplessness is mandated by the combination of irreps at high-symmetry momenta.
Note that only unitary symmetries are incorporated in defining a high-symmetry momentum, and the notion of a BS only requires a continuous gap at all high-symmetry points, lines and planes. ReSMs arise when the topological band-gap closings are buried in the low-symmetry regions of the Brillouin zone, rendering them hard to diagnose in conventional electronic structure calculations, but are nonetheless readily detectable and are robust against numerical uncertainty due to their representation-enforced nature.
All previously known examples of reSMs concern TR-broken systems in 3D with either the inversion~\cite{Ari, Bernevig} or $\bar{4}$ rotation symmetry~\cite{Chern_Rotation}.  The new classes we propose here are realized in spin-orbit coupled systems in 2D or 3D, and are associated to type III or IV MSGs.

Our example is the type IV MSG 3.4 ($P_a112$) is generated by $C_{2z}$ (the $\pi$ rotation about $z$-axis) and $\tilde{\mathcal{T}}\equiv\mathcal{T}T_{\frac{1}{2}\vec{a}_1}$ (half translation by $\frac{1}{2}\vec{a}_1$ followed by TR) apart from lattice translations.  (The type III MSG 13.69 can also host a Dirac semimetal through a nearly identical mechanism. See Appendix \ref{app:tbmodel}.) According to Table~\ref{tab:ex1}, this MSG has two classes of combinations of $C_{2z}$ eigenvalues, as indicated by $X_{\rm BS}=\mathbb{Z}_2$.  As we shall see now, the 3D band structure belonging to this nontrivial class hosts at least four Weyl points, where the Weyl points with opposite charities are maximally separated in the $k_3$ direction.

To understand this band topology step by step, let us first remove the lattice translation in $z$ and take a single layer parallel to the $xy$ plane.
The MSG then reduces to the MLG $p_a112$, which corresponds to the entry 3.4 (1) in Table~\ref{tab:ex2} with $X_{\rm BS}=\mathbb{Z}_2$.  The rotation $C_{2z}$ is a symmetry at the four TR-invariant momenta of the 2D BZ, whose eigenvalues are either $\pm i$ for spinful electrons. When $k_1=0$, the anti-unitary symmetry $\tilde{\mathcal{T}}$ with $\tilde{\mathcal{T}}^2=(-1)^{\hat{F}}\hat{T}_{\vec{a}_1}=-1$ demands the pairing of the two eigenvalues of $C_{2z}$.  In contrast, when $k_1=\pi$, $\tilde{\mathcal{T}}^2=+1$ and no pairing is required.  For this MLG, $X_{\rm BS}=\mathbb{Z}_2$ and the topological index distinguishing the two class is given by
\begin{equation}
\label{eq:topoinv}
\eta=\prod_{n: \text{occupied}}\prod_{\vec{k}=(\pi,0),(\pi,\pi)}\,\eta_{n, \vec{k}},
\end{equation}
where $\eta_{n,\vec{k}}$ is the $C_{2z}$-eigenvalue of the $n$-th band.
$\eta=+1$ for any AI, while $\eta=-1$ for nontrivial BSs.
Note that this invariant is similar to those identified in some earlier works \cite{Ari, Bernevig, Chern_Rotation, FuKane,Rappe}.

\begin{figure*}[tbh!!]
\begin{center}
{\includegraphics[width=0.8 \textwidth]{reSM.pdf}}
\caption{
{\bf Magnetic representation-enforced semimetal.}
(a) An example tight-binding model for representation-enforced semimetal.  There are two sub-lattices per unit cell (shaded) due to an anti-ferromagnetic order $\vec{m}_{\vec{x}}$, producing in total four bands.  On-site potential $J$ stands for the exchange coupling $-J\vec{m}_{\vec{x}}\cdot c_{\vec{x}}^\dagger\vec{\sigma}c_{\vec{x}}$.  In addition to the standard nearest-neighbor (NN) hopping $t$, a spin-dependent hopping $\pm t_J\sigma_x$ is included, which can be viewed as originating from an exchange coupling to a magnetic moment in the middle of the NN bonds (Appendix~\ref{app:tbmodel}). (b) The dispersion relation at $J/t=1$ and $t_J/t=1/4$. In this case, $\eta$ in Eq.~\eqref{eq:topoinv} is $+1$ and the dispersion is gapped between the second and the third band. (c) The dispersion relation at $J/t=1$ and $t_J/t=3/4$. Now $\eta=-1$ and a pair of Dirac nodes exists as predicted. (d) The Fermi surface of the 3D version of the representation-enforced semimetal. The two Weyl points on the $k_3=0$ ($k_3=\pi$) plane have the chirality $+1$ ($-1$), indicating a huge Fermi arc on some 2D surfaces.  The signs on the TRIMs indicate the product of the $C_{2z}$ rotation eigenvalues of occupied bands in this model.}
\label{fig:reSM}
\end{center}
\end{figure*}

Next, we show that the nontrivial BS with $\eta=-1$ corresponds to 2D magnetic Dirac reSMs (two-fold degeneracy at the gapless point) enabled by a strong spin-rotation symmetry breaking, as illustrated in Fig.\ \ref{fig:reSM}.
Denoting the standard Pauli matrices by $\sigma_{0,\dots,3}$, the combined symmetry $\tilde{\mathcal{T}}C_{2z}$ dictates that the local $2$ by $2$ Hamiltonian $h(\vec{k})$ near a  gapless point takes the form
\begin{equation}
h(\vec{k}) = \sum_{i=0}^{2} g_i(\vec{k})\sigma_i,
\end{equation}
i.e., the $\sigma_3$ term is missing.
Since there are the same number of the tunable parameters $k_1$ and $k_2$ as the number of the relevant coefficients ($g_1(\vec{k})$ and $g_2(\vec{k})$), the gaplessness is stable against perturbations respecting the symmetries~\cite{BJY}.
To see this more explicitly, we note that the symmetry $\tilde{\mathcal{T}}C_{2z}$ quantizes the Berry phase $B(k_1)=\int_{-\pi}^{\pi} dk_2 A_2(k_1)$ to be $0$ or $\pi$ at any $k_1$.  The topological index $\eta=-1$ dictates that $B(0)\neq B(\pi)$, implying the presence of a gap closing somewhere in $0<k_1<\pi$.
Using a $\vec k \cdot \vec p$ analysis, one can confirm that the bands disperse linearly, leading to a Dirac node.
We note that, compared to the protection of Dirac node by an eigenvalue exchange of nonsymmorphic symmetries~\cite{YoungKane,YoungWieder,JingWang,Vanderbilt2}, the present Dirac node has a more topological character.

\begin{figure*}[tbh!!]
\begin{center}
{\includegraphics[width=0.98 \textwidth]{feSM.pdf}}
\caption{
{\bf Magnetic filling-enforced semimetals.}
(a,b) Symmetries of a magnetic order can prohibit atomic insulators (AIs) at odd site fillings.
(a) The magnetic-point-group symmetries of a ferromagnetic arrangement is compatible with nondegenerate local energy levels. (b) Those of the depicted hedgehog defect, however, force all the energy levels to exhibit even degeneracies, which forbids AIs when a lone electron is localized to the purple site.
(c) When hedgehog and anti-hedgehog defects are arranged into a diamond lattice, the previous argument suggests that no AI is allowed whenever the site fillings at the defect cores are odd.
(d) The hypothetical magnetic structure in (c) could be realizable in spinel structures if the diamond sites are occupied by atoms with odd atomic numbers (purple), and the magnetic atoms (blue) at the pyrochlore sites exhibit  an all-in-all-out magnetic order.
(e--h) Filling-enforced semimetals arising form the magnetically-ordered Fu-Kane-Mele model, Eq.\ \eqref{eq:JFKM}.
(e) When the magnetically modulated hopping $t_J$ is weak compared to the spin-orbit coupling $\lambda$, the fermiology is governed by rings of gap-closing (circled in red), growing out from the original Dirac points at $X$ when $t_J=0$. (f) The positions of the rings in the Brillouin zone are shown in red.
(g,h) For $t_J \geq 2\sqrt{2} \lambda$, the nodal rings become connected at the momentum W. (Thin lines indicate copies of the gapless momenta in the repeated zone scheme, included to illustrate the connectivity of the rings.)
}
\label{fig:feSM}
\end{center}
\end{figure*}

Given this understanding of the 2D layer with MLG 3.4, let us now recover the third dimension by stacking the copies of 2D layers with generic inter-layer couplings.   The resulting band structure, fully symmetric under MSG 3.4, host in total four Weyl points; two on the $k_3=0$ plane and the other two on the $k_3=\pi$ plane, as depicted in Fig.~\ref{fig:reSM}d.  Note that the two Weyl points on the $k_3=0$ plane are related by $C_{2z}$ rotation (or, equivalently in this setup, $\tilde{\mathcal T}$) symmetry, and so must have the same chirality.  Consequently, the two Weyl points on the $k_3=\pi$ plane have the opposite chirality. This large separation of the Weyl points with opposite chiralities dictates the presence of long Fermi arcs on the surface.  Let us stress again that this nontrivial band structure is indicated by the rotation eigenvalues at TRIMs, via the formula in Eq.~\eqref{eq:topoinv} applied to the $k_3=0$ plane of the 3D band structure.This illustrates how the ideas  described in the present work can guide the search for ideal magnetic Weyl semimetals.
We leave a full analysis of magnetic reSMs to future works.

\section{Filling-enforced semimetals}
As another application of our theory, we attack the following problem: Given an MSG $\mathcal M$, what are the electron fillings $\nu$ at which band insulators are allowed? One of the primary interests in solving this problem lies in the search of topological semimetals --- if a material has an electron filling incompatible with any band insulator, it must possess symmetry-protected gaplessness near the Fermi energy, provided a band-theory description is applicable. We refer to such systems as filling-enforced (semi-)metals.
Once the MSG describing the magnetic order is identified, such systems can be readily predicted using our results on $\nu_{\rm BS}$. For stoichiometric compounds with perfect crystalline order and commensurate magnetism, one can further show that spatial symmetries quantize the physically allowed fillings, which in turns allows one to further diagnose the fermiology of the filling-enforced (semi-)metals. We relegate a detailed discussion to Appendix \ref{app:feSM}, and summarize the results on the predicted fermiology in Tables \ref{tab:feSM-p}--\ref{tab:feSMSOC}.

While a systematic analysis of filling-enforced semimetals (feSMs) can be performed using the tables mentioned above, it is instructive to discuss a physical picture linking a real-space description of the magnetic ordering to the momentum-space obstruction to realizing any band insulator. To this end, we first contrast the effect of topologically trivial and nontrivial magnetic orders on local electronic energy levels.
When the surrounding magnetic moments order ferromagnetically, as in Fig.\ \ref{fig:feSM}a, the electronic levels are Zeeman-split and nondegenerate. In contrast, when the moments are arranged into the hedgehog defect shown in Fig.\ \ref{fig:feSM}b, the magnetic-point-group symmetries at the defect core demand that all single-particle local energy levels exhibit even degeneracies, i.e., such a magnetic order forbids a gapped state when a lone electron is localized to the defect core.

To be compatible with lattice translations, however, there must be a balance between defects and anti-defects in each unit cell. An interesting situation arises when the positions of the defects and anti-defects are further related by symmetries prohibiting them from annihilation.
As an example, consider the diamond lattice with one sublattice occupied by hedgehog defects, and the other sublattice, symmetry-related to the first by a glide mirror, occupied by anti-hedgehogs (Fig.\ \ref{fig:feSM}c). The symmetries of this system are described by MSG 227.131 (type III).
Suppose the defect core (i.e., the diamond sites) corresponds to an atom with an odd atomic number, which leads to an electron filling of  $\nu = 2 \mod 4$ per primitive unit cell. By the previous argument, it should be impossible to obtain an AI by localizing all the electrons. This suggests the electron filling might lead to an obstruction to forming a band insulator and hence enforcing a (semi-)metallic behavior.
Indeed, our result of $\nu_{\rm BS} = 4$ for this MSG (Table \ref{tab:SpinfulCubic}) implies all band insulators are ruled out at the specified filling of $\nu = 2 \mod 4$. From Table \ref{tab:feSM-m},  we further see that this corresponds to a feSM with movable nodal features.

Curiously, the hypothetical structure depicted in Fig.\ \ref{fig:feSM}c could be relevant for materials with the chemical formula $A B_2 O_4$ taking the spinel structure, where the $A$ atoms, occupying the diamond sites, are surrounded by the $B$ atoms sitting at the pyrochlore positions. If the $B$ atoms are magnetically ordered into the all-in-all-out configuration (Fig.\ \ref{fig:feSM}d), it can be viewed as a lattice realization of the described hedgehog-anti-hedgehog lattice in MSG 227.131.
This suggests the following tight-binding model as an example of the feSM described:
\begin{equation}\begin{split}\label{eq:JFKM}
\hat H = & \sum_{\langle ij\rangle} \hat c_{i}^\dagger \left( t \, \sigma_0 + 4\, t_J\, \vec d_{ij} \cdot \vec \sigma \right) \hat c_j\\
&~~~~~~~~~~~~ + i\, 8 \lambda \sum_{\langle \langle ij \rangle \rangle} \hat c_{i}^\dagger \left(\vec d_{li}  \times \vec d_{lj}\right)\cdot \vec \sigma\,\hat c_j,
\end{split}\end{equation}
where $\hat c_i$ and $\hat c_i^\dagger$ represent the fermion operators corresponding to an $s$-orbital localized to the diamond site $i$ (spin indices are suppressed), $\langle i j \rangle$ and $\langle \langle ij \rangle \rangle$ respectively denote first- and second-nearest-neighbor bonds, and $\vec d_{ij}$ is the vector connecting the site $j$ to $i$ (the lattice constant of the conventional cell is set to $1$).
In the term $\propto \lambda$, we let $l$ denote the common nearest neighbor of $i$ and $j$.
Note that $t_J$ can be physically interpreted as a modulation of the nearest-neighbor hopping by exchange coupling with the magnetic moment at pyrochlore sites, and $\lambda$ parameterizes the symmetry-allowed spin-orbit coupling between the second-nearest-neighbor sites.

When $t_J=0$, Eq.~\eqref{eq:JFKM} is TR-symmetric (corresponding to the type II MSG 227.129), and reduces to the undistorted Fu-Kane-Mele model defined on the diamond lattice hosting three symmetry-related Dirac points (Table \ref{tab:feSM-p}) \cite{Z2_3D}. Unlike lattice distortions, which lead to topological insulators~\cite{Z2_3D}, TR-symmetry breaking described by $t_J\neq 0$ leads to a feSM.
Interestingly, the Dirac points in the original Fu-Kane-Mele model are unstable towards such perturbation, and our results indicate that the semimetallic behavior is enforced by a more delicate band connectivity (Table \ref{tab:feSM-m}).
This is explicitly verified in the example band structures plotted in Figs.\ \ref{fig:feSM}e-h, showing how the Dirac points split into (doubly degenerate) nodal rings as $t_J$ increases from $0$.

This example serves only as a particular instance of the 421 MSGs we identified to be compatible with feSMs (Appendix \ref{app:feSM}). A full list of such MSGs is presented in Tables \ref{tab:feSM-p} and \ref{tab:feSM-m}.
Armed with this list, we studied the magnetic structures listed on the Bilbao Crystallographic Server \cite{BilbaoMagnData}, and found that the experimentally-characterized magnetic materials YFe$_4$Ge$_2$ and  LuFe$_4$Ge$_2$ (and related compounds) in the type III MSG 58.399 are realistic magnetic feSM candidates \cite{YFG, LFG}.
For these compounds, a nonmagnetic atom with an odd atomic number (Lu or Y) sits at a maximal-symmetry site of multiplicity 2, whereas their  surrounding magnetic moments (Fe) ordered into a pattern symmetric under the combined symmetry of spatial inversion and TR.
Such a magnetic ordering falls into our broad description of nontrivial defect lattices, and these compounds are expected to feature Dirac points, pinned to high-symmetry momenta, near the Fermi energy (Table \ref{tab:feSM-p}).

We remark that the method described here can be readily applied to any other commensurate magnetic crystals where both the chemical formula and the MSG describing the magnetic order have been identified. In particular, a local-moment description is inessential --- the filling criterion established based on our computation of $\nu_{\rm BS}$ applies equally well to systems exhibiting itinerant magnetism, as long as a band-theory description remains applicable.

\begin{figure*}[tbh!!]
\begin{center}
{\includegraphics[width=0.60 \textwidth]{inversion.pdf}}
\caption{
{\bf Symmetry indicators for anti-ferromagnetic topological insulators.}
(a) An example of the inversion parity combination of valence bands in the $\mathbb{Z}_2$ nontrivial phase of MSG 2.7. (b) The realization of the $\mathbb{Z}_2$ nontrivial phase by staggered stacking of Chern insulators. The red (blue) disks represent a Chern insulator with $C=+1$ $(C=-1)$. (c) Breaking the $\tilde{T}$-symmetry (the half translation in $z$ followed by the time-reversal) leads to a `higher-order' state with a 1D equatorial chiral mode on the surface \cite{SitteRoschAltmanFritz, ZhangKaneMele,Teo,  Benalcazar17,  Hashimoto2017, Song17, Schindler17,  Benalcazar17Aug, Fang17, Langbehn17}.
}
\label{fig:AFTI}
\end{center}
\end{figure*}

\section{Indicator for Antiferromagnetic Topological Insulator}
As the last application of our results, let us make a connection of symmetry-based indicators of MSGs, fully computed in this work, to some previously studied topological insulators in TR broken settings.

As a canonical example, we discuss the type IV MSG 2.7 generated by the inversion symmetry $I$ and the half translation followed by TR, $\tilde{\mathcal{T}}\equiv\mathcal{T}T_{\frac{1}{2}\vec{a}_z}$.  This MSG has two classes of the combination of inversion parities as indicated by $X_{\rm BS}=\mathbb{Z}_2$ in Table~\ref{tab:ex1}.  The topological index distinguishing the two classes is given by
\begin{equation}
\label{eq:topoinv2}
\xi=(-1)^{\frac{1}{4}\sum_{n: \text{occupied}}\sum_{\text{TRIMs}}\,\xi_{n, \vec{k}}}=\pm1
\end{equation}
Here, $\xi_{n, \vec{k}}$ is the parity of the $n$-th occupied band.  To understand this formula, note first that even- and odd-parity bands form a Kramers pair at the TRIMs with $k_z=\pi$. This is because $I$ and $\tilde{\mathcal{T}}$ anti-commute when $k_z=\pi$. Thus, the four TRIMs with $k_z=\pi$ never contribute to the index. On the other hand, when $k_z=0$, $I$ and $\tilde{\mathcal{T}}$ commute, and $\tilde{\mathcal{T}}^2=-1$. This means that the same parity eigenvalue appears twice at TRIMs with $k_z=0$. In Fig.~\ref{fig:AFTI}a, we illustrated an example of the parity combination with $\xi=-1$.

As demonstrated in Ref.~\cite{MongEssinMoore}, the index $\xi=-1$ implies a nontrivial band topology, the so-called `antiferromagnetic topological insulator' that supports gapless surface Dirac modes. The anti-ferromagnetic topological insulator can be understood as a staggered stacking of Chern insulators, in which a Chern insulator with $C=+1$ (described by red disks in Fig.~\ref{fig:AFTI}b) is on every $z=n$ ($n\in\mathbb{Z}$) plane and the one with $C=-1$ (blue disks) is on $z=n+\frac{1}{2}$ ($n\in\mathbb{Z}$) plane. All of these Chern insulators are related by the $\tilde{ \mathcal T}$ symmetry.

If we introduce a perturbation that breaks the $\tilde{\mathcal T}$ symmetry, the $C=+1$ Chern insulator at $z=n$ ($n\neq0$) and the $C=-1$  Chern insulator at $z=n\mp\text{sign}(n)\frac{1}{2}$ can pair annihilate with each other in an inversion symmetric manner, leaving only the $C=+1$ Chern insulator on the $z=0$ plane as shown in Fig.~\ref{fig:AFTI}c. Such one-dimensional chiral edge state on the surface of 3D topological insulators has been studied in Refs.\ \onlinecite{ZhangKaneMele,SitteRoschAltmanFritz}, and their counterparts in many other symmetry settings have been discovered recently \cite{Teo,  Benalcazar17,  Hashimoto2017, Song17, Schindler17,  Benalcazar17Aug, Fang17, Langbehn17}.
Assuming that the inversion symmetry is unbroken, we see that this phase can be diagnosed by another symmetry-based indicator for the type-I MSG 2.4, which is simply the SG with only the inversion $I$ apart from the lattice translation. According to the Table~\ref{tab:ex1}, this MSG has $\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_4$ classification, and the insulator hosting the 1D chiral edge mode constructed here belongs to the class $(0,0,0,2)$ in this classification.

\section{Conclusion}
In this work, we revisit the old problem of assigning a global irrep label to each connected branch of a band structure~\cite{Wigner}, generalizing our recent results on time-reversal symmetric systems \cite{NC} to all 1651 magnetic space groups.
Our central results are the computation of three fundamental quantities associated with magnetic band structures, which are tabulated in Appendix \ref{app:tables}: (i) $d_{\rm BS}$, which characterizes the number of independent building blocks of energy bands; (ii) $X_{\rm BS}$, which, akin to the Fu-Kane parity criterion \cite{FuKane}, serves as a symmetry-based indicator of band topology; and (iii) $\nu_{\rm BS}$, which dictates the electron fillings at which band insulators are possible. We further demonstrate the utility of these results by applying them to the study of topological semimetals, focusing on cases where the absence of a band gap is either diagnosed through the symmetry representations at isolated high-symmetry points, or is mandated by the electron filling. In particular, we identify the exhaustive list of magnetic space groups capable of hosting filling-enforced semimetals (Appendix \ref{app:feSM}).

Although a full database on our computation of the basis `vectors' of the groups $\{{\rm BS}\}$ and $\{{\rm AI}\}$ is not included, we note that such a database can be readily generated \cite{NC}: Our proof of $d_{\rm BS} = d_{\rm AI}$ for all magnetic space groups implies full knowledge on $\{{\rm BS}\}$ is encoded in that of $\{{\rm AI} \}$, and the latter can be computed by analyzing, for instance, hypothetical structures constructed by combining irreps of the site-symmetry groups to each of the Wyckoff positions. As long as all high-symmetry momenta are included in the analysis and the doubling of any irrep (according to the Herring rule) is incorporated, $\{{\rm BS}\}$ can be recovered by forming linear superpositions of the entries in $\{ {\rm AI} \}$ using rational coefficients, subjected only to the constraint that the resultant irrep multiplicities must all be integer-valued.
Such calculations can be performed without deriving the compatibility relations, and this circumvents the careful convention-fixing required in both tabulating and utilizing such results.  In view of this, we refrain from providing a database on them.

We close by highlighting several interesting open questions that we leave for future studies: (i) performing a comprehensive study on the identification and characterization of representation-enforced semimetals in all 2D and 3D magnetic space groups; (ii) developing a theory of quantized physical responses, or a proof of the absence thereof, for each nontrivial class in the exhaustive list of $X_{\rm BS}$ we computed; and (iii) discovering realistic topological materials based on systematic searches using the diagnostics our theory provides. Finally, we remark that some aspects of our filling criterion generalize to the interacting setting \cite{MSGInt}, similar to the corresponding relation between the results for time-reversal symmetric crystals \cite{PRL, PNAS}.

\begin{acknowledgments}
HW thank Shuichi Murakami and Chen Fang for useful discussion. AV and HCP were supported by NSF DMR-1411343. AV acknowledges support from a Simons Investigator Award.
HW acknowledges support from JSPS KAKENHI Grant Number JP17K17678.
\end{acknowledgments}

\clearpage

redpajama_set_name: RedPajamaArXiv


text:
Un número primo triádico (en inglés 3-way prime o triadic prime) es un número primo, capicúa en algún sistema de numeración y que además es simétrico a lo largo de la línea horizontal en la que se escriben sus dígitos (razón por la cual sus dígitos sólo pueden ser 0, 1, 3 y 8 en la base decimal).

C. W. Trigg les puso el nombre de palíndromos reflectables.

En el sistema binario, todos los números primos capicúas son triádicos, mientras que en sistemas de numeración con base mayor no es así.

Ejemplo
Un ejemplo de este tipo de números es:
10...0111013101110...01'donde la cadena 0...0'' contiene 2509 ceros.

Enlaces externos

Números primos


text:
Q: how to get different digits starting from the most significant digit in a number in c? I am solving a problem in which a positive integer is given and i have to display it in words.
For example if a number is 2134 the output should be "two one three four" .
If i use the modulus operator and use the recursion technique, i get the digits starting from the least significant digit i.e "four three one two"
I can also reverse the number and then use modulus operator but i am looking for a better method.
What can be a better approach of solving this problem?.What concept am i missing?

A: My simple answer:
void printNum(int x)
{
    static const char * const num[] = {
        "zero ", "one ", "two "  , "three ", "four ",
        "five ", "six ", "seven ", "eight ", "nine "
    };

    if (x < 10) {
        printf(num[x]);
        return;
    }
    printNum(x / 10);
    printNum(x % 10);
}

edit
After some more experimenting and tweaking, I came up with this version.
I think this is the about the most "pure" recursive function I can make.
void printNum(int x)
{
    static const char * const num[] = {"zero ",  "one ", "two ", "three ",
                                       "four ",  "five ", "six ", "seven ",
                                       "eight ", "nine "};
    (x < 10)? printf(num[x]) : (printNum(x / 10), printNum(x % 10));
}

A: In case you're looking for recursive solution:
void num2word(int n) {
    if (n / 10 == 0) {
        // print the word for n
    }
    else {
        num2word(n / 10);
        // print the word for n % 10
    }
}

A: Going for a pure math answer, because I think that's what you're looking for:
#include <math.h>
#include <stdio.h>

main() {
    long long mynum = 2987612345;
    long long firstDigitValue;
    int       firstDigit;
    char *names[] = { "zero", "one", "two", "three", "four", "five",
                      "six", "seven", "eight", "nine" };

    while (mynum > 0) {
        firstDigit = (int) (mynum/pow(10,(int)log10(mynum)));
        firstDigitValue = firstDigit * pow(10,(int)log10(mynum));
        mynum -= firstDigitValue;
        printf("%s ", names[firstDigit]);
    }
    printf("\n");
}

And then running it produces:
two nine eight seven six one two three four five

It should work with any size number that a long long can handle (in other languages, or with a big number library in C, it can handle something arbitrarily large).
Now...  I'm not sure the use of log10 and pow is the fastest.  But it's the most fun :-)

A: If it is a null terminated char *, then why not just do this:
int i = 0;
while(char[i] != null){
    switch(char[i]):

         case '1': printf("One "); break;
         ....
         i++;

}

if its not a char *, turn it into one with char * temp = itoa(number);

A: 1234 / 1000 => 1
1234 % 1000 => 234     234 / 100 => 2
1234 % 100  => 34      34  / 10  => 3
1234 % 10   => 4

You can pretty easely build a loop around that, but you may go even easer with itoa, sprintf, and std::to_string.

redpajama_set_name: RedPajamaStackExchange


Congress president Rahul Gandhi has jumped into the issue of detention of Economist and activist Jean Dreze and questioned the Jharkhand government. He has called it an attack on those who are working for the poor. Watch to know more!


text:
The bidirectional gut-brain-microbiota axis as a potential nexus between traumatic brain injury, inflammation, and disease
Mark H. Sundman, Nan kuei Chen, Vignesh Subbian, Ying hui Chou
As head injuries and their sequelae have become an increasingly salient matter of public health, experts in the field have made great progress elucidating the biological processes occurring within the brain at the moment of injury and throughout the recovery thereafter. Given the extraordinary rate at which our collective knowledge of neurotrauma has grown, new insights may be revealed by examining the existing literature across disciplines with a new perspective. This article will aim to expand the scope of this rapidly evolving field of research beyond the confines of the central nervous system (CNS). Specifically, we will examine the extent to which the bidirectional influence of the gut-brain axis modulates the complex biological processes occurring at the time of traumatic brain injury (TBI) and over the days, months, and years that follow. In addition to local enteric signals originating in the gut, it is well accepted that gastrointestinal (GI) physiology is highly regulated by innervation from the CNS. Conversely, emerging data suggests that the function and health of the CNS is modulated by the interaction between 1) neurotransmitters, immune signaling, hormones, and neuropeptides produced in the gut, 2) the composition of the gut microbiota, and 3) integrity of the intestinal wall serving as a barrier to the external environment. Specific to TBI, existing pre-clinical data indicates that head injuries can cause structural and functional damage to the GI tract, but research directly investigating the neuronal consequences of this intestinal damage is lacking. Despite this void, the proposed mechanisms emanating from a damaged gut are closely implicated in the inflammatory processes known to promote neuropathology in the brain following TBI, which suggests the gut-brain axis may be a therapeutic target to reduce the risk of Chronic Traumatic Encephalopathy and other neurodegenerative diseases following TBI. To better appreciate how various peripheral influences are implicated in the health of the CNS following TBI, this paper will also review the secondary biological injury mechanisms and the dynamic pathophysiological response to neurotrauma. Together, this review article will attempt to connect the dots to reveal novel insights into the bidirectional influence of the gut-brain axis and propose a conceptual model relevant to the recovery from TBI and subsequent risk for future neurological conditions.
Biological Phenomena
Gastrointestinal Microbiome
Gut-brain axis
Intestinal dysfunction
Microglia
Sundman, M. H., Chen, N. K., Subbian, V., & Chou, Y. H. (2017). The bidirectional gut-brain-microbiota axis as a potential nexus between traumatic brain injury, inflammation, and disease. Brain, Behavior, and Immunity, 66, 31-44. https://doi.org/10.1016/j.bbi.2017.05.009
The bidirectional gut-brain-microbiota axis as a potential nexus between traumatic brain injury, inflammation, and disease. / Sundman, Mark H.; Chen, Nan kuei; Subbian, Vignesh; Chou, Ying hui.
In: Brain, Behavior, and Immunity, Vol. 66, 11.2017, p. 31-44.
Sundman, MH, Chen, NK, Subbian, V & Chou, YH 2017, 'The bidirectional gut-brain-microbiota axis as a potential nexus between traumatic brain injury, inflammation, and disease', Brain, Behavior, and Immunity, vol. 66, pp. 31-44. https://doi.org/10.1016/j.bbi.2017.05.009
Sundman MH, Chen NK, Subbian V, Chou YH. The bidirectional gut-brain-microbiota axis as a potential nexus between traumatic brain injury, inflammation, and disease. Brain, Behavior, and Immunity. 2017 Nov;66:31-44. https://doi.org/10.1016/j.bbi.2017.05.009
Sundman, Mark H. ; Chen, Nan kuei ; Subbian, Vignesh ; Chou, Ying hui. / The bidirectional gut-brain-microbiota axis as a potential nexus between traumatic brain injury, inflammation, and disease. In: Brain, Behavior, and Immunity. 2017 ; Vol. 66. pp. 31-44.
@article{3f433be4d6264bdcb4c60baaa20a91d2,
title = "The bidirectional gut-brain-microbiota axis as a potential nexus between traumatic brain injury, inflammation, and disease",
abstract = "As head injuries and their sequelae have become an increasingly salient matter of public health, experts in the field have made great progress elucidating the biological processes occurring within the brain at the moment of injury and throughout the recovery thereafter. Given the extraordinary rate at which our collective knowledge of neurotrauma has grown, new insights may be revealed by examining the existing literature across disciplines with a new perspective. This article will aim to expand the scope of this rapidly evolving field of research beyond the confines of the central nervous system (CNS). Specifically, we will examine the extent to which the bidirectional influence of the gut-brain axis modulates the complex biological processes occurring at the time of traumatic brain injury (TBI) and over the days, months, and years that follow. In addition to local enteric signals originating in the gut, it is well accepted that gastrointestinal (GI) physiology is highly regulated by innervation from the CNS. Conversely, emerging data suggests that the function and health of the CNS is modulated by the interaction between 1) neurotransmitters, immune signaling, hormones, and neuropeptides produced in the gut, 2) the composition of the gut microbiota, and 3) integrity of the intestinal wall serving as a barrier to the external environment. Specific to TBI, existing pre-clinical data indicates that head injuries can cause structural and functional damage to the GI tract, but research directly investigating the neuronal consequences of this intestinal damage is lacking. Despite this void, the proposed mechanisms emanating from a damaged gut are closely implicated in the inflammatory processes known to promote neuropathology in the brain following TBI, which suggests the gut-brain axis may be a therapeutic target to reduce the risk of Chronic Traumatic Encephalopathy and other neurodegenerative diseases following TBI. To better appreciate how various peripheral influences are implicated in the health of the CNS following TBI, this paper will also review the secondary biological injury mechanisms and the dynamic pathophysiological response to neurotrauma. Together, this review article will attempt to connect the dots to reveal novel insights into the bidirectional influence of the gut-brain axis and propose a conceptual model relevant to the recovery from TBI and subsequent risk for future neurological conditions.",
keywords = "Chronic Traumatic Encephalopathy, Concussion, Gut, Gut-brain axis, Intestinal dysfunction, Microbiota, Microglia, Neurodegenerative disease, Neuroinflammation, Traumatic Brain Injury",
author = "Sundman, {Mark H.} and Chen, {Nan kuei} and Vignesh Subbian and Chou, {Ying hui}",
T1 - The bidirectional gut-brain-microbiota axis as a potential nexus between traumatic brain injury, inflammation, and disease
AU - Sundman, Mark H.
AU - Chen, Nan kuei
AU - Subbian, Vignesh
AU - Chou, Ying hui
N2 - As head injuries and their sequelae have become an increasingly salient matter of public health, experts in the field have made great progress elucidating the biological processes occurring within the brain at the moment of injury and throughout the recovery thereafter. Given the extraordinary rate at which our collective knowledge of neurotrauma has grown, new insights may be revealed by examining the existing literature across disciplines with a new perspective. This article will aim to expand the scope of this rapidly evolving field of research beyond the confines of the central nervous system (CNS). Specifically, we will examine the extent to which the bidirectional influence of the gut-brain axis modulates the complex biological processes occurring at the time of traumatic brain injury (TBI) and over the days, months, and years that follow. In addition to local enteric signals originating in the gut, it is well accepted that gastrointestinal (GI) physiology is highly regulated by innervation from the CNS. Conversely, emerging data suggests that the function and health of the CNS is modulated by the interaction between 1) neurotransmitters, immune signaling, hormones, and neuropeptides produced in the gut, 2) the composition of the gut microbiota, and 3) integrity of the intestinal wall serving as a barrier to the external environment. Specific to TBI, existing pre-clinical data indicates that head injuries can cause structural and functional damage to the GI tract, but research directly investigating the neuronal consequences of this intestinal damage is lacking. Despite this void, the proposed mechanisms emanating from a damaged gut are closely implicated in the inflammatory processes known to promote neuropathology in the brain following TBI, which suggests the gut-brain axis may be a therapeutic target to reduce the risk of Chronic Traumatic Encephalopathy and other neurodegenerative diseases following TBI. To better appreciate how various peripheral influences are implicated in the health of the CNS following TBI, this paper will also review the secondary biological injury mechanisms and the dynamic pathophysiological response to neurotrauma. Together, this review article will attempt to connect the dots to reveal novel insights into the bidirectional influence of the gut-brain axis and propose a conceptual model relevant to the recovery from TBI and subsequent risk for future neurological conditions.
AB - As head injuries and their sequelae have become an increasingly salient matter of public health, experts in the field have made great progress elucidating the biological processes occurring within the brain at the moment of injury and throughout the recovery thereafter. Given the extraordinary rate at which our collective knowledge of neurotrauma has grown, new insights may be revealed by examining the existing literature across disciplines with a new perspective. This article will aim to expand the scope of this rapidly evolving field of research beyond the confines of the central nervous system (CNS). Specifically, we will examine the extent to which the bidirectional influence of the gut-brain axis modulates the complex biological processes occurring at the time of traumatic brain injury (TBI) and over the days, months, and years that follow. In addition to local enteric signals originating in the gut, it is well accepted that gastrointestinal (GI) physiology is highly regulated by innervation from the CNS. Conversely, emerging data suggests that the function and health of the CNS is modulated by the interaction between 1) neurotransmitters, immune signaling, hormones, and neuropeptides produced in the gut, 2) the composition of the gut microbiota, and 3) integrity of the intestinal wall serving as a barrier to the external environment. Specific to TBI, existing pre-clinical data indicates that head injuries can cause structural and functional damage to the GI tract, but research directly investigating the neuronal consequences of this intestinal damage is lacking. Despite this void, the proposed mechanisms emanating from a damaged gut are closely implicated in the inflammatory processes known to promote neuropathology in the brain following TBI, which suggests the gut-brain axis may be a therapeutic target to reduce the risk of Chronic Traumatic Encephalopathy and other neurodegenerative diseases following TBI. To better appreciate how various peripheral influences are implicated in the health of the CNS following TBI, this paper will also review the secondary biological injury mechanisms and the dynamic pathophysiological response to neurotrauma. Together, this review article will attempt to connect the dots to reveal novel insights into the bidirectional influence of the gut-brain axis and propose a conceptual model relevant to the recovery from TBI and subsequent risk for future neurological conditions.
KW - Chronic Traumatic Encephalopathy
KW - Concussion
KW - Gut
KW - Gut-brain axis
KW - Intestinal dysfunction
KW - Microbiota
KW - Microglia
KW - Neurodegenerative disease
KW - Neuroinflammation
KW - Traumatic Brain Injury


text:
Q: How do i make a custom Kubuntu distro? I am trying to make a custom kubuntu 20.04 distro. Are there any good tools I can use?

A: I would first try to package your config changes and added dependencies into one or multiple .deb files.
once they are installable easily, you can think of how yo are going to distribute it.
the suggestions posted as comments to your questions sounded like good suggestions.
have fun


Welcome back to SOS – How can we help you?
At SOS our goal is to be an ongoing resource for our patients and a partner in the team that provides you with excellent healthcare. If you have questions about your therapy, concerns with your equipment or just need a little help deciphering how various insurance companies pay for treatment – we are here to help.
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The Boundary 500 Motorcycle Group were formed in June 2006 to raise funds for The Great North Air Ambulance Service and Zoes Baby Hospice. As of December 2010 the amount raised stands at over three hundred thousand pounds.
The Group hold a major event in June for the Air Ambulance and December for the Baby Hospice.
On the weekend on 10th, 11th and 12 June 2011 we will be holding the Boundary 500 mile Challenge and the Boundary Classic a 125 ride around the Yorkshire Moors. Both events are open to any age of motorcycle.
On Saturday 9th April 2011 we are holding our anual Easter party at the Blackwell Grange Hotel Darlington, Co Durham. The bar opens at 2.30pm and closes at 1am. There is an evening meal, free use of the leisure facilities including the swimming pool. There is accommodation and breakfast. The inclusive price is £42 per person.
For further details contact the hotel direct on 01325 509955 and ask for functions.
If you would like any further details about anything we are involved with please contact the Group Secretary Karen at sec.b500@yahoo.co.uk or phone 01642 343479.
Created new bike club "The Boundary 500 Motorcycle Group"