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List of jugglers A juggler is a person who practices object manipulation for entertainment, sport or recreation. Famous professional jugglers have come from many countries and have performed their skills live in circuses, variety theatres, casinos, cruise ships, festivals, street venues and on television. This list shows notable jugglers for reasons of success as a professional performer, world record holder, official competition winner or inventor of patterns or tricks. Australia Space Cowboy Kenny Cheung Austria Christoph and Manuel Mitasch - hold multiple club passing numbers juggling world records Canada Wilfred Dubois (b. 1892) Czech Republic Alan Šulc - holds multiple bounce numbers juggling world records Germany Francis Brunn - numerous appearances on stage and television. Paul Cinquevalli Kris Kremo - numerous appearances on stage and television. Holder of cigar box pirouette world record. Son of Béla Kremo. Thomas Dietz Hungary Trixie - known as 'the first lady of juggling'. Appearance on stage and film. Italy Enrico Rastelli - a former holder of multiple world records. Netherlands Niels Duinker - comedy juggler and holder of multiple Guinness World Records. Russia (or the former USSR) Vova and Olga Galchenko Sergej Ignatov - former holder of various world records. Many appearances in circuses and on stage. Anatoli Miagkostoupov and Viktor Pilipovich United Kingdom Alex Barron - holds multiple ball numbers juggling world records Sean Gandini - holds multiple ring passing numbers juggling world records Rod Laver - notable for ping pong ball juggling routine. Numerous appearances on stage and television worldwide. Haggis and Charlie - juggling performers, teachers and authors. Appearances at festivals, stage and television. Steve Rawlings - many appearances on stage and television. United States Jack Bremlov - reputedly the first person to juggle seven clubs. Part of a juggling troupe 'The Bremlovs' W.C. Fields - Actor and vaudeville performer. Dave Finnegan - author of The Complete Juggler, one of the most widely read self teach juggling books Dr Hot & Neon - numerous performances on stage and television. The Flying Karamazov Brothers - juggling and comedy troupe who have been performing since 1973. Jason Garfield - president of the World Juggling Federation Anthony Gatto - holds various number juggling world records, considered by many to be the world's greatest juggler. Michael Goudeau - ex-circus clown and frequent collaborator of Penn Jillette. Penn Jillette - half of the duo Penn & Teller, who incorporates his juggling skills into their magic act. Albert Lucas - holds some number juggling world records. Club juggling trick 'Alberts' and by association 'Treblas' named after him. Marcus Monroe - Comedian and Juggler, Winner of the Andy Kaufman Award Bobby May - vaudeville performer Steve Mills - inventor of the juggling pattern Mills Mess Michael Moschen - popularised contact juggling in 1990s Frank Olivier - numerous performances on stage and television. Kit Summers - author of Juggling with Finesse Paul Klimek - invented siteswap References Further reading Ziethen, Karl-Heinz & Serena, Alessandro 2003 Virtuosos of Juggling, Renegade Juggling, Santa Cruz Ziethen, Karl-Heinz & Allen, Andrew 1985 Juggling: The Art and its Artists, Werner Rausch & Werner Luft Inc, Berlin Summers, Kit 1987 Juggling with Finesse, Finesse Press, San Diego Finnigan, Dave 1987 The Complete Juggler, Vintage Books, New York Dancey, Charlie 2001 Encyclopedia of Ball Juggling, Butterfingers, Devon Dancey, Charlie 1995 Compendium of Club Juggling Butterfingers, Bath Category:Jugglers

Drospirenone, a new progestogen, for postmenopausal women with hypertension. The prevalence of hypertension increases in women after the menopause. Associated with the rise in postmenopausal blood pressure (BP) are increased salt sensitivity and imbalance between the renin-angiotensin-aldosterone system and nitric oxide pathways that lead to sodium and water retention. Drospirenone is the first synthetic progestogen with antialdosterone activity similar to natural progesterone. Drospirenone counteracts the salt- and water-retaining effects of estrogen and causes natriuresis, which leads to a reduction in BP. In preclinical studies as well as early efficacy studies (for menopausal symptoms), drospirenone exhibited antihypertensive and natriuretic effects. Subsequent clinical trials in postmenopausal women proved that drospirenone with 17beta-estradiol has a significant BP-lowering effect in untreated hypertension and has additive effects when coadministered with ACE inhibitors, angiotensin II type 1 receptor antagonists and thiazide diuretics. The lowest effective dose of drospirenone for reduction in BP is 2mg, a dose that is also protective of the uterus in women treated with estrogen therapy. Additionally, clinical trials have shown that drospirenone up to 3 mg/day has an acceptable safety profile with no clinically significant elevations in plasma potassium in patients with concomitant NSAID use, diabetes mellitus or mild to moderate renal insufficiency. In addition to effectively relieving menopausal symptoms and lowering BP, drospirenone reduces bodyweight and lipoprotein concentrations. Thus, drospirenone is a unique progestogen that confers the additional benefit of BP reduction, an effect that could lead to potential benefit with respect to some cardiovascular risk concerns in women taking hormone therapy.

SUBJECT: Chhattisgarh Governments Plan to Impose a Ban on PUCL and Ekta Parishad Dear Mr Shekhar Dutt, We are concerned at the news appearing in a section of the media that the Chhattisgarh Government was contemplating imposing a ban on the Peoples Union for Civil Liberties (PUCL) & Ekta Parishad under the Chhattisgarh Special Public Security Act 2005. Any such an action by the Government will only provide further support to our contention that the CSPSA 2005 is an un-democratic law, promulgated in violation of the fundamental rights guaranteed under the Constitution of India; that its sole purpose is to suppress dissent and true democracy, and it is primarily meant to be used against genuine human rights activists and social movements working for the right to life and livelihood of vulnerable sections of society like tribals, dalits, women, workers, child and bonded labourers etc. In fact, the Peoples Union for Civil Liberties Chhattisgarh Unit has challenged the constitutional validity of the CSPSA 2005 in the Chhattisgarh High Court at Bilaspur, and the Writ Petition No. WP(C) 2163/2009 titled Peoples Union for Civil Liberties v. Union of India & Anr is pending in the High Court, which too has been unduly delayed. We are convinced that the attempts by the Chhattisgarh Government to impose ban on PUCL & Ekta Parishad are politically motivated and are examples of state repression. We would like to draw your attention to the fact that PUCL was founded by Lok Nayak Jai Prakash Narain, along with Mr. V M Tarkunde, Former Judge of Bombay High Court, during the Emergency Rule in India in 1975-76, to protect the Rights of the Citizens guaranteed under Constitution of India. Dr. Rajni Kothari, renowned social scientist, Mr. Arun Shourie, senior journalist, Mr. Rajendar Sachar, Former Chief Justice of Delhi High Court, and a number of legal luminaries, writers, journalists, social and political workers are associated with it. We wish to reiterate that PUCL has never advocated violence, and that it believes in peaceful and democratic means to work for the preservation and promotion of civil liberties & democratic rights, as is clearly declared in the PUCL constitution. That PUCL has adhered to such principles and policies is evident from its various actions, which include the filing of a Writ Petition in the Supreme Court of India on the Right to Food, that resulted in the mid-day meal scheme to be implemented all over the country; its intervention through the Supreme Court resulting in a landmark order making it necessary for all electoral candidates to declare their financial assets and criminal records, etc. Similarly, the Chhattisgarh Unit of PUCL (which was originally part of the Madhya Pradesh unit) has made significant contributions in promoting and preserving civil liberties & democratic rights in Chhattisgarh since the days of Emergency. It is responsible for the release of some 4000 bonded labourers in Raipur district, through the intervention of the Supreme Court of India in April 1988. It obtained a landmark judgment from the MP High Court at Jabalpur in the case of slum-dwellers in Raipur city, because of which even today thousands of slum-dwellers cannot be evicted and displaced, and their right to shelter and work are still protected. It has successfully fought the cases of arbitrary and preventive detentions under various draconian laws like the NSA & MISA, externment proceedings against trade union and political leaders; including those of the legendary labour leader Com. Shankar Guha Niyogi, peasant leader Kanker Munjare (who later was elected MLA in Balaghat district), Ram Ghulam Thakur (Baghbehra), Sahdeo Sahu of Chhattisgarh Mukti Morcha (Dalli-Rajhara). PUCL also carried out fact-finding missions and published reports on various police firings in CG such as the firing on mine-workers at Dalli-Rajhara on June 2-3, 1977, on mine workers at Bailladilla in 1978, on Industrial workers at Abhanpur on May 30, 1990, on workers at Bhilai on July 1, 1992, etc. In addition, PUCL has exposed crimes committed by the State, especially extra-judicial killings, starvation deaths, tribal deaths due to blood dysentery in Bastar, excesses committed by Salwa-Judum and security forces, displacement due to development, etc. It is a matter of record that PUCL has meticulously and tirelessly carried on its campaign for protecting and promoting constitutional rights of all citizens, especially when they are violated by the powerful machinery of the State. It is also a matter of record that each political party in power has resorted to cheap tactics to discredit PUCL in the public eye. Since the State is the biggest violator of human rights, PUCLs expositions of its violations such as extra-judicial killings, fake encounters, custodial deaths, crimes committed by the Security Forces and Political Groups, violations of peoples rights such as on forest, land and water, labour laws, pollution etc. have earned it the wrath of every government. It also bears mentioning that many political leaders, cutting across political parties, have also been associated with the PUCL at one or the other time. We also note that Ekta Parishad is a Gandhian organization, which believes in using non-violence as a tool of conflict resolution and amplifying the voice of the marginalized classes of the society, and has been working on land and forest rights at a national level. It has been built up over twenty years growing from the local, to the state, to the national and increasingly, to the international level. The two main activities of Ekta Parishad are dialoguing with the government at the state and national level and mobilizing the villagers for struggle at the grassroots level, both of which are interlinked: people struggling at the bottom level and their struggle is supported by a formation of institutions giving them the tools to fight for their rights at the top level, through dialogue. Vice versa, supporters dialoguing at the top level give space for political action or struggle at the bottom level. Founded in 1991, Ekta has revitalised Mahatma Gandhi's message of non-violent action to deal with the problem of the widening gap between the rich and poor in India today. By organising marches and peaceful protests Ekta Parishad brings marginalised groups together and supports India's rural communities' in their demands for justice. Eminent Gandhians, and social workers like Dr. S N Subba Rao, Shri Bal Vijay Ji, Shri Rajiv Vora, Shri Rajendra Singh, Shri Yogendra Parekh, Yathish Mehta and many more are associated closely with Ekta Parishad. In Chhattisgarh today, corporate houses, including those of some newspapers, exercise tremendous influence over parliamentary institutions and media, which are being systematically deployed to repress democratic voices and movements, especially those opposing the usurpation of peoples resources. It is in this context that the Chhattisgarh Government is leading a vilification campaign against PUCL and Ekta Parishad, as evident from the recent defamatory and irresponsible statement of the Home Minister Sri Nanki Ram Kanwar that PUCL was supplying arms to naxalites. Thus, we would like to APPEAL to your Constitutional Office: 1. To immediately intervene in taking steps to stop such an un-democratic and un-constitutional act of the Chhattisgarh Government in imposing a ban on the Peoples Union for Civil Liberties (PUCL) & Ekta Parishad under the Special Public Security Act 2005; 2. To take appropriate action in protecting the constitutional and democratic rights of organisations like PUCL & Ekta Parishad, and also other such NGOs and social and human rights workers who are facing repressive measures at the hands of the State Government and its various wings like the police; 3. To take adequate steps to stop the vilification campaign carried out by some of the ministers and officials of the Chhattisgarh Government against prestigious organisations and individuals who have been known for protecting and preserving the right to life and livelihood of the people; 4. To intervene to create a conducive climate for resolving some of the pressing problems through dialogue and reconciliation. We are confident that you would definitely take concrete and immediate steps to stop the Chhattisgarh Government from taking such un-constitutional and un-democratic action. Sign The Petition If you already have an account please sign in, otherwise register an account for free then sign the petition filling the fields below.Email and the password will be your account data, you will be able to sign other petitions after logging in. Name (mandatory) Surname (mandatory) Email (mandatory) Choose a password (mandatory) Privacy in the search engines? You can use a nickname: City (optional) Nickname (optional) Comment Attention, the email address you supply must be valid in order to validate the signature, otherwise it will be deleted. Donate Help keep Online Petitionfree! Please consider making a small donation today. This will allow us to keep Oppose CG government's plans to ban PUCL and Ekta Parishad alive and available to a wide audience. One-time donation Every cent you give goes toward maintaining the website and the petitions. Even a small contributions helps!Support the Cause

The anti-Muslim film, “The Innocence of Muslims,” that caused controversy - and even violence - in portions of the Middle East now has an appeals court ordering its removal from YouTube. The decision made over the anti-Muslim film was the result of a divided three-judge panel of San Francisco’s 9th United States Circuit Court of Appeals, according to a Fox News report on Wednesday. The appeals court’s decision reinstated the lawsuit that was filed against YouTube by one of the actresses who appeared in the film, Cindy Lee Garcia. The movie, being realized on YouTube in 2012, negatively depicted the Prophet Muhammad. Previously, YouTube didn’t take the movie down after world leaders – including President Barack Obama – had said they wanted the video taken off the web site. YouTube claimed that taking the video off of the web site was equated to unwarranted government censorship which would have been in violation of the firm’s free speech protections. Additionally, YouTube stated that the makers of the video – not the actors – owned the copyright. Therefore, only the filmmakers could remove it from the YouTube web site. However, the court ruled that this court case is atypical and that Cindy Lee Garcia has retained a copyright claim that YouTube must honor. Of concern was that Garcia thought she was acting in a different movie - than the one that she ended up being part of - online. Cris Armenta, the actress’ lawyer, asserted that Garcia would never have agreed to appear in the movie if she had known the true nature of the film. The court said that the ruling on this film is not to be rubber-stamped, giving all actors copyright protection of the movies in which they appear. It was said that this case was different – and, in this case, Garcia’s acting was worthy of copyrighted protection. Garcia had been paid $500 to perform for five seconds in the movie. She was told that the movie was called “Desert Warrior” and she believed it was not about radical Islam or religion. After Garcia performed in the movie, her lines were switched – via dubbing – to her asking Muhammad if he was a child molester. Google, which owns YouTube, is going to appeal the decision – possibly appealing to the United States Supreme Court. At this time, YouTube has removed the video.

Social comparisons and temporal comparisons after coronary artery surgery. To examine the use and efficacy of social and temporal comparisons used before and after coronary artery surgery. Secondary analysis of data from a prospective study designed to examine social support. 141 subjects undergoing surgery. One third of subjects made spontaneous social comparisons. Most who made a social comparison before and 1 month after surgery viewed themselves as similar to others. Social comparisons were related to mood states only before surgery. The majority of subjects made temporal comparisons 1 year after surgery. Subjects generally saw themselves as the same or better than they were before surgery. Temporal comparisons were related to both emotional and functional outcomes. Social comparisons were not consistently related to emotional and functional status; thus whether they can be used to formulate interventions needs further exploration. On the other hand, use of temporal comparisons was related to better mood state and functional status. Enhancing an individual's ability to view self as stable or improved compared with before surgery may be beneficial. Results are discussed in terms of how social comparison theory fits within the overall context of coping with physical illness.

Q: Shared codebase Azure cloud service we want to deploy a webrole on the Azure cloud service (PaaS) we have multiple Virtual Applications that have exactly the same codebase (dlls) but different web.configs this leads to multiple time the same dll's in the package to upload; resulting in a very large package file is there any way to share the bin folder for these 'same' Virtual Applications to minimize the size of the package? greetings, Tim A: Create a cloud application with one web role containing the codebase and additional web roles containing only the Web.config for the multiple virtual applications. On ServiceDefinition.csdef, define the virtual applications for the roles as required. Set a startup task to copy the contents from the complete web role to each of the additional virtual applications. This will be executed at role initialization time, with administrator privileges. This way your deployment package won't need to include multiple copies of the same artifacts, and the virtual applications will be set up when the role instance is initialized.

Q: How to use one 12V power supply to supply two Op/amp with 5V and 12V voltage supply? I am trying to design a circuit and i have two Op-Amp with different supply voltage with -+12V for LT1220 and +-5V for LT1193. i want to use just one power supply with +-12V. in that case for Lt1193 i would like to make a local power supply for that. using Zener diode with zener voltage regulator is good approach or is there any high efficient way in practice for this purpose? A: Depending on your current requirements you basically have two approaches: low currents just use a linear regulator, something like the (somewhat) good, (definitely) old 7805. A linear regulator dissipates a lot of power, the current through it being constant. For a 7V drop at 100mA you get 700mW, it is not a lot but it can be too much. I guess your current needs are much lower, so a linear regulator might suit you. high currents you can use a monolithic DCDC step down regulator to get some 6V from your 12V, then use an LDO (Low Drop Out) regulator to get to 5V, nice and smooth. You need the LDO because usually the output voltage of a DCDC regulator is quite messy and noisy, something you normally cannot accept in an opamp circuit. The second solution is actually the best in almost any case, unless your current requirements are very low (less than 100mA let's say), but your BOM gets bigger and more costly. If you need low currents the linear only solution is the best, mainly because you only need one chip and a couple of capacitors. If you opt for the linear path please search a suitable part for your application, don't just throw in a 7805, there are cheaper and better solutions on the market.

Measles-mumps-rubella vaccine and autistic spectrum disorder: report from the New Challenges in Childhood Immunizations Conference convened in Oak Brook, Illinois, June 12-13, 2000. Parents and physicians are understandably concerned about the causes and treatment of autism, a devastating disease that affects the entire family. Although much has been learned about autism, there are many gaps in our knowledge about what causes the disorder and how it can be prevented. Autistic symptoms occur along a spectrum, often referred to as autistic spectrum disorder (ASD). Concern has been raised about a possible association between measles-mumps-rubella (MMR) vaccine and inflammatory bowel disease (IBD) and ASD, especially autism with regression. Also, increased requests for educational services related to ASD have raised concerns about possible increases in the incidence of ASD. On June 12-13, 2000, the American Academy of Pediatrics (AAP) convened a conference titled "New Challenges in Childhood Immunizations" in Oak Brook, Illinois. At this conference, parents, practitioners, and scientists presented information and research on MMR vaccine and ASD. Attendees included representatives from select AAP committees and sections as well as federal and other organizations that address related issues. The multidisciplinary panel of experts reviewed data on what is known about the pathogenesis, epidemiology, and genetics of ASD and the available data on hypothesized associations with IBD, measles, and MMR vaccine. Supplemental information was requested from authors who have proposed the hypotheses and other experts in relevant areas. Autism is a complex disorder of uncertain and probably multiple etiologies. Genetic predisposition to ASD may involve as many as 10 genes. Many experts believe that the abnormal brain development in autism occurs before 30 weeks' gestation in most instances. In utero rubella is a known cause of autism. Animal model data support the biologic plausibility that exposure to yet unrecognized infectious or other environmental agents could cause ASD. Several factors may contribute to apparent increases in incidence of ASD in recent years. Most data indicate increased recognition and reporting as primary factors, but the epidemiologic data are insufficient to determine if there has been a true increase in the incidence of ASD. Increased reporting of ASD in recent years has occurred long after the introduction of MMR vaccine in the United States in 1971 and widespread use of this vaccine in the 1970s for routine immunization of children at 12 to 15 months of age. Appropriate detailed studies are needed to define the true incidence and prevalence of ASD. Epidemiologic studies in Europe indicate no association between MMR vaccine and ASD. Some children with ASD have gastrointestinal symptoms, but an increased rate of any specific gastrointestinal disorder in children with ASD has not been established. Studies to detect evidence of measles virus in intestinal tissue specimens from patients with IBD or autism with gastrointestinal symptoms have not used uniform techniques. Several laboratories have found no evidence of measles viruses in tissue specimens from patients with IBD, but 2 groups have found evidence of measles virus using different techniques. A group that found evidence of measles virus in affected tissue specimens from patients with IBD has also reported detecting portions of measles virus in peripheral blood lymphocytes and intestinal tissue specimens from patients with autism and gastrointestinal disorders. Finding a portion of a virus using molecular techniques does not constitute evidence for a causal relationship, because some viruses persist in unaffected hosts. Additional controlled studies in several laboratories are needed to determine if portions of measles virus persist in intestinal and other tissues of people with and without gastrointestinal disease and/or ASD. Although the possible association with MMR vaccine has received much public and political attention and there are many who have derived their own conclusions based on personal experiences, the available evidence does not support the hypothesis that MMR vaccine causes autism or associated disorders or IBD. Separate administration of measles, mumps, and rubella vaccines to children provides no benefit over administration of the combination MMR vaccine and would result in delayed or missed immunizations. Pediatricians need to work with families to ensure that children are protected early in the second year of life from these preventable diseases. Continued scientific efforts need to be directed to the identification of the causes of ASD.

Q: How to turn a list of objects into a keyed array/object? I'm trying to write code using Ramda to produce a new data structure, using only the id and comment keys of the original objects. I'm new to Ramda and it's giving me some fits, although I have experience with what I think is similar coding with Python. Given the following initial data structure… const commentData = { '30': {'id': 6, 'comment': 'fubar', 'other': 7}, '34': {'id': 8, 'comment': 'snafu', 'other': 6}, '37': {'id': 9, 'comment': 'tarfu', 'other': 42} }; I want to transform it into this… { '6': 'fubar', '8': 'snafu', '9': 'tarfu' } I found the following example in the Ramda cookbook that comes close… const objFromListWith = R.curry((fn, list) => R.chain(R.zipObj, R.map(fn))(list)); objFromListWith(R.prop('id'), R.values(commentData)); But values it returns includes the whole original object as the values… { 6: {id: 6, comment: "fubar", other: 7}, 8: {id: 8, comment: "snafu", other: 6}, 9: {id: 9, comment: "tarfu", other: 42} } How can I reduce the values down to the value of their comment key only? I don't need to use the code I got from the cookbook. If anyone can suggest some code that will give the results I'm looking for that's also better (simpler, shorter, or more efficient) than the example here, I'll be happy to use that instead. A: If you don't mind, you don't need to use Ramda for that, pure JS can handle it nicely: You can use a combination of Object.values(), to get all values of your first object (commentData) and .forEach() (or even .map(), but slower), in the array that results from Object.values to insert values into a new object dynamically. const commentData = { '30': {'id': 6, 'comment': 'fubar', 'other': 7}, '34': {'id': 8, 'comment': 'snafu', 'other': 6}, '37': {'id': 9, 'comment': 'tarfu', 'other': 42} }; let values = Object.values(commentData) let finalObj = {}; values.forEach(x => finalObj[x.id] = x.comment) console.log(finalObj) But, if you want an one-liner, you can go with Object.fromEntries() after returning arrays of key/value from .map() based on id and comment, like below: const commentData = { '30': {'id': 6, 'comment': 'fubar', 'other': 7}, '34': {'id': 8, 'comment': 'snafu', 'other': 6}, '37': {'id': 9, 'comment': 'tarfu', 'other': 42} }; console.log(Object.fromEntries(Object.values(commentData).map(x => [x.id, x.comment]))) A: A Ramda one-liner would be const foo = compose(fromPairs, map(props(['id', 'comment'])), values) const commentData = { '30': {'id': 6, 'comment': 'fubar', 'other': 7}, '34': {'id': 8, 'comment': 'snafu', 'other': 6}, '37': {'id': 9, 'comment': 'tarfu', 'other': 42} } console.log( foo(commentData) ) <script src="//cdnjs.cloudflare.com/ajax/libs/ramda/0.26.1/ramda.js"></script> <script>const {compose, fromPairs, map, props, values} = R </script> But that doesn't seem a lot cleaner than (a slight tweak to) the suggestion in epascarello's comment: const foo = (obj) => Object.values(obj).reduce((o, v) => (o[v.id] = v.comment) && o, {}) or a similar version I would write, if it didn't cause any performance issues: const foo = (obj) => Object.values(obj).reduce((o, {id, comment}) => ({...o, [id]: comment}), {})

Q: Scala: type mismatch in function composition, found (Int, Int) => Seq[Int] require ? => Seq[Int] I'm trying to compose 2 functions in a script and getting a type mismatch I can not solve. Here is the sample code: def generate(start: Int, end: Int): Seq[Int] = (start until end).toSeq def restrain(seq: Seq[Int]) = seq.dropWhile(_ < 20).takeWhile(_ < 60) val com: (Int, Int) => Seq[Int] = (restrain _ compose generate) By loading this in the REPL with: :load test.sc I get the following error: val com: (Int, Int) => Seq[Int] = (restrain _ compose generate) ^ test.sc:1: error: type mismatch; found : (Int, Int) => Seq[Int] required: ? => Seq[Int] What am I doing wrong? A: The types Function2[Int, Int, Seq[Int]] and Function1[(Int, Int), Seq[Int]] are not the same. The (generate _) produces the former, whereas for this composition, you need the latter. Try: restrain _ compose (generate _).tupled

Q: Darkcoin nomp pool payment processing error and missing ui We are setting up nomp https://github.com/zone117x/node-open-mining-portal When we run node init.js This happens: root@drk:~/node-open-mining-portal# node init.js 2014-05-17 18:51:17 [POSIX] [Connection Limit] (Safe to ignore) POSIX module not installed and resource (connection) limit was not raised 2014-05-17 18:51:17 [Master] [CLI] CLI listening on port 7777 2014-05-17 18:51:17 [Master] [PoolSpawner] Spawned 1 pool(s) on 1 thread(s) 2014-05-17 18:51:17 [Profit] [Config] No alternative coins to switch to in current config, switching disabled. 2014-05-17 18:51:17 [Switching] [Setup] (Thread 1) Loading last proxy state from redis 2014-05-17 18:51:17 [Pool] [darkcoin] (Thread 1) Share processing setup with redis (127.0.0.1:6379) 2014-05-17 18:51:17 [Switching] [Setup] (Thread 1) Switching "switch1" listening for x11 on port 3333 into darkcoin 2014-05-17 18:51:23 [Pool] [darkcoin] (Thread 1) Stratum Pool Server Started for darkcoin [DRK] {x11} Network Connected: Mainnet Detected Reward Type: POW Current Block Height: 70293 Current Connect Peers: 7 Current Block Diff: 2816.770470913 Network Difficulty: 2755.22977025 Network Hash Rate: 55.22 GH Stratum Port(s): 3331 Pool Fee Percent: 1% Block polling every: 1000 ms 2014-05-17 18:52:18 [Pool] [darkcoin] (Thread 1) No new blocks for 55 seconds - updating transactions & rebroadcasting work 2014-05-17 18:53:13 [Pool] [darkcoin] (Thread 1) No new blocks for 55 seconds - updating transactions & rebroadcasting work Then we navigate to our pool @ http://ourpoolexample.com/MPOS/public but we see a blank page. Where is the UI? Here are our settings (user and password have been omitted): ~/node-open-mining-portal/config.json { "logLevel": "debug", "logColors": true, "cliPort": 7777, "clustering": {"enabled":true,"forks":"auto"}, "defaultPoolConfigs":{ "blockRefreshInterval":1000, "jobRebroadcastTimeout":55, "connectionTimeout":600, "emitInvalidBlockHashes":true, "validateWorkerUsername":true, "tcpProxyProtocol":false, "banning":{"enabled":true,"time":600,"invalidPercent":50,"checkThreshold":500,"purgeInterval":300}, "redis":{"host":"127.0.0.1","port":6379}}, "website":{ "enabled":false, "host":"0.0.0.0", "port":80, "stratumHost":"cryppit.com", "stats":{"updateInterval":60,"historicalRetention":43200,"hashrateWindow":300}, "adminCenter":{"enabled":false,"password":"password"}}, "redis": {"host":"127.0.0.1","port":6379}, "switching":{ "switch1":{"enabled":true,"algorithm":"x11","ports":{ "3333":{"diff":32,"varDiff":{"minDiff":8,"maxDiff":512,"targetTime":15,"retargetTime":90,"variancePercent":30}} }} }, "profitSwitch": {"enabled":true,"updateInterval":600,"depth":0.90,"usePoloniex":true,"useCryptsy":true,"useMintpal":true} } ~/node-open-mining-portal/pool_configs/darkcoin.json { "enabled": true, "coin": "darkcoin.json", "address": "XgZLPCQkGvvpK42jAAtgRHvs8J25xKn1XS", "rewardRecipients": {"XgZLPCQkGvvpK42jAAtgRHvs8J25xKn1XS":1.0}, "paymentProcessing":{ "enabled":false, "paymentInterval":20, "minimumPayment":70, "daemon":{"host":"127.0.0.1","port":19332,"user":"user","password":"password"}}, "ports":{ "3331":{"diff":32,"varDiff":{"minDiff":8,"maxDiff":512,"targetTime":15,"retargetTime":90,"variancePercent":30}}}, "daemons":[ {"host":"127.0.0.1","port":8332,"user":"user","password":"password"}], "p2p": {"enabled":false,"host":"127.0.0.1","port":19333,"disableTransactions":true}, "mposMode":{ "enabled":false, "host":"127.0.0.1", "port":3306, "user":"user", "password":"password", "database":"drk", "checkPassword":true, "autoCreateWorker":false} } ~/node-open-mining-portal/coins/darkcoin.json { "name": "Darkcoin", "symbol": "DRK", "algorithm": "x11", "mposDiffMultiplier": 256 } ~/.darkcoin/darkcoin.conf server=1 gen=0 rpcport=8332 rpcallowip=127.0.0.1 rpcuser=user rpcpassword=password Notice we have disabled payment processing in pool_configs/darkcoin.json as we always get the error: 2014-05-17 19:21:19 [Payments] [darkcoin] Error with payment processing daemon {"type":"offline","message":"connect ECONNREFUSED"} I am assuming this may be a problem with ports conflicting but there is a line on the official page that confuses me on the payments processing main daemon: /* This daemon is used to send out payments. It MUST be for the daemon that owns the configured 'address' that receives the block rewards, otherwise the daemon will not be able to confirm blocks or send out payments. */ It uses port 19332 twice in the example (matching), but when we match them the pool wont even start (another ECONNREFUSED this time for [pool]) What is the problem? We are very close.. A: In your darkcoin.json, the ports for the daemons must match the rpcport in the respective daemons data directory conf file. Since 8332 is the rpc port in your darkcoin.conf in the data directory, this line: "daemon":{"host":"127.0.0.1","port":19332,"user":"user","password":"password"}}, should be "daemon":{"host":"127.0.0.1","port":8332,"user":"user","password":"password"}},

Prognostic validity of the Rorschach. Fifty adult stutterers entering therapy at the UCLA Psychology Clinic were administered the Rorschach, with a Klopfer method inquiry, scoring, form level rating, and calculation of scores on the Rorschach Prognostic Rating Scale (RPRS). On the basis of independent clinician ratings of attitudinal or psychotherapeutic improvement, subjects were divided into groups of Improved Most (n = 21) and Improved Least (n = 29). Subjects were also divided into Continued (n = 43) and Dropped (n = 7). Logistic regression was employed to compare groups on the following Rorschach dimensions: Prognostic Score (RPRS); Human Movement; Animal Movement; Inanimate Movement; Shading; Color; Form Level. The Improved Most group was significantly higher in M, FM, Shading, and Productivity. The finding that M and FM discriminates between improvement groups corroborates results obtained in a previous study (Sheehan et al., 1954). The Rorschach movement variables, particularly M and FM, seem to be stable indicators of capacity for improvement in psychotherapy.

Diversity of Pestalotiopsis-Like Species Causing Gray Blight Disease of Tea Plants (Camellia sinensis) in China, Including two Novel Pestalotiopsis Species, and Analysis of Their Pathogenicity. Several Pestalotiopsis-like species cause gray blight disease in tea plants, resulting in severe tea production losses. However, systematic and comprehensive research on the diversity, geographical distribution, and pathogenicity of pathogenic species associated with tea plants in China is limited. In this study, 168 Pestalotiopsis-like isolates were obtained from diseased tea plant leaves from 13 primary tea-producing provinces and cities in China. Based on a multilocus (internal transcribed spacer, translation elongation factor 1-α, and β-tubulin gene region) phylogenetic analysis coupled with an assessment of conidial characteristics, 20 Neopestalotiopsis unclassified isolates, seven Pestalotiopsis species, including two novel (Pestalotiopsis menhaiensis and Pestalotiopsis sichuanensis), four known (Pestalotiopsis camelliae, Pestalotiopsis chamaeropis, Pestalotiopsis kenyana, and Pestalotiopsis rhodomyrtus) and one indistinguishable species, and three Pseudopestalotiopsis species, including two known (Pseudopestalotiopsis camelliae-sinensis and Pseudopestalotiopsis chinensis) and one indistinguishable species, were identified. This study is the first to evaluate Pestalotiopsis chamaeropis on tea plants in China. The geographical distribution and pathogenicity tests showed Pseudopestalotiopsis camelliae-sinensis to be the dominant cause of gray blight of tea plants in China. In vitro antifungal assays demonstrated that theobromine not only derepressed mycelial growth of the 29 representative isolates but also increased their growth. Correlation analysis revealed a linear positive relationship between the mycelial growth rate and pathogenicity (P = 0.0148).

Q: Show that the space is not locally Euclidean (Topology) Definition: A topological space $X$ is locally Euclidean if there exists $n \in \mathbb{N}$ so that for all $x \in X$ there is a neighborhood $U$ of $x$ which is homeomorphic to an open subset $V \subset \mathbb{R} ^n$. I want to show that $$X = \{ (x,y) \in \mathbb{R^2} : x^2+y^2 = 1 \} \cup \{ (x,0) : -1 \leq x \leq 1 \}$$ is not locally Euclidean. I proved that $X$ is not homeomorphic to the circle $S^1 \subset \mathbb{R^2}$, is this enough? If not, how could we proceed/conclude? I am using Introduction to Topology by Munkres. A: Points $(-1,0)$ and $(1,0)$ do not have any euclidean neighborhood. If there was an homeomorphism from an open set of $\mathbb{R}^n$ to one of these neighborhoods $U$, what would the counterimage of $U-\{(1,0)\}$ be? Certainly an open set of $\mathbb{R}^n$ consisting in not more than two connected components, while $U-\{(1,0)\}$ will have three distinct connected components (unless of course you choose $U$ to cointain $(-1,0)$, but you don't have to).

Rolling Balls When you will tilting your phone – balls begin to roll under gravity. To win the game – you need to throw all the white balls into the hole. Except white balls there are still balls of other colors and sizes. Red balls will add white balls to level, it don’t need to roll. Wooden boxes will disturb you to throw up white balls into the hole.

Languages USA: Alert as USA hints at attacks on other countries After three days of the U.S. bombing campaign in Afghanistan, the United States has informed the Security Council that attacks against other organisations and other countries may be necessary, a move that would be a drastic stretch of the international law concept of self-defence and would serve to further intensify and escalate the ongoing violence. The Women's Caucus for Gender Justice suggests actions and gives contact information for your governments in the capitals and at the permanent missions to the United Nations in New York. The statement was made in a letter from the United States' UN ambassador, John Negroponte, to the Security Council. According to different news reports, the letter from Negroponte stated: "Our inquiry is in the early stages. We may find that our self-defense requires further actions with respect to other organizations and other countries." "One sentence which has caused some anxiety amongst the membership which I've also asked about was the question that they may find it necessary to go after other organizations and other States. The US has indicated that this is not a predictor of any intention that it intends to take but basically it is a statement that they are at the early stages and keeping their options open." The UN Charter allows for the exercise of the right of self-defense until the Security Council takes action to restore international peace and security. Self-defense can be exercised when the need is immediate and overwhelming and when there are no other options available. In the absence of Security Council action, countries which have been threatened or attacked can make the claim of self-defense in determining the nature and scope of their military response, which must still be within the parameters of humanitarian law. Since the adoption of Security Council resolution 1373 on September 28th, the Security Council has appeared content to let the U.S. and any other countries willing to opt into its multilateral campaign against terrorism determine the means necessary to respond to the September 11th attacks. However, a response that is United Nations-led and adheres to international law is even more urgent given the rapid escalation of the cycle of violence and retaliation and the possible spread of this violence into other countries and regions. Three days into the bombing campaign by British and U.S. forces, UN aid workers and civilians have fallen victim while the alleged perpetrators of the September 11th attack are still in hiding. It is imperative at this moment that the Security Council assert its authority in this matter, as the U.S. has indicated the possibility of widening the scope of its military action to other organizations and other countries which could further threaten international peace and security.

A DFT and TD-DFT approach to the understanding of statistical kinetics in substitution reactions of M3Q4 (M = Mo, W; Q = S, Se) cuboidal clusters. For many years it has been known that the nine water molecules in [M(3)Q(4)(H(2)O)(9)](4+) cuboidal clusters (M = Mo, W; Q = S, Se) can be replaced by entering ligands, such as chloride or thiocyanate, and kinetic studies carried out mainly on the substitution of the first water molecule at each metal centre reveal that the reaction at the three metal centres occurs with statistical kinetics; that is, a single exponential with a rate constant corresponding to the reaction at the third centre is observed instead of the expected three-exponential kinetic trace. Such simplification of the kinetic equations requires the simultaneous fulfilment of two conditions: first that the three consecutive rate constants are in statistical ratio, and second that the metal centres behave as independent chromophores. The validity of those simplifications has been checked for the case of the reaction of [Mo(3)S(4)(H(2)O)(9)](4+) with Cl(-) by using DFT and TD-DFT theoretical calculations. The results of those calculations are in agreement with the available experimental information, which indicates that the H(2)O ligands trans to the μ-S undergo substitution much faster than those trans to the μ(3)-S. Moreover, the energy barriers for the substitution of the first water molecule at the three metal centres are close to each other, the differences being compatible with the small changes in the numerical values of the rate constants required for observation of statistical kinetics. TD-DFT calculations lead to calculated electronic spectra, which are in reasonable agreement with those experimentally measured, but the calculations do not indicate that the three metal centres behave as independent chromophores, although the mathematical conditions required for simplification of the kinetic traces to a single exponential are reasonably well fulfilled at certain wavelengths. A re-examination of the kinetics of the reaction by using global fitting procedures yields results, which are compatible with statistical kinetics, although an alternative interpretation in which substitution only occurs at a single metal centre under reversible conditions is also possible.

Q: Why does the inner div not expand to the set height/min-height? Why does height: 100% have no effect on #baz in the following code? How could you fix this when min-height on (some of) the ancestor element(s) is required? HTML: <div id="foo"> <div id="bar"> <div id="baz"> foo bar baz </div> </div> </div> CSS: div { border: 3px solid red; padding: 5px; } #foo { height: 300px; } #bar { min-height: 100%; } #baz { height: 100%; } See example at http://jsfiddle.net/pmmyP/ Tested with Chrome 12 and Firefox 4. A: Using the following kind of works: #bar { min-height: 100%; position: relative; } #baz { position: absolute; top: 0; bottom: 0; left: 0; right: 0; } But is there another (or better) way? Example at http://jsfiddle.net/pmmyP/1/

--- author: - 'Madeline Brandt, DJ Bruce, Taylor Brysiewicz, Robert Krone, Elina Robeva' bibliography: - 'ref.bib' title: 'The degree of $\operatorname{SO}(n)$' --- Introduction {#introsect} ============ The *special orthogonal group* $\operatorname{SO}(n,\mathbb{R})$ is the group of automorphisms of $\mathbb{R}^n$ which preserve the standard inner product and have determinant equal to one. The complex special orthogonal group is the complexification of the special orthogonal group and can be thought of more explicitly as the group of matrices $$\operatorname{SO}(n):=\operatorname{SO}(n,{\mathbb{C}})=\left\{M\in \text{Mat}_{n,n}({\mathbb{C}}) \; | \; \det M =1, \quad M^{t}M=\text{Id}\right\}.$$ As these conditions are polynomials in the entries of such a matrix, we view $\operatorname{SO}(n)$ as a complex variety. Recall that the degree of a complex variety $X$ is the generic number of intersection points of $X$ with a linear space of complementary dimension. Problem $4$ on Grassmannians in [@fitness] asks for a formula for the degree of the of $\operatorname{SO}(n)$. Our primary result is the following theorem, which answers this question completely. [theorem]{}[maintheorem]{} \[degson\] The degree of $\operatorname{SO}(n)$ is given by $$\deg \operatorname{SO}(n)=2^{n-1}\det \left( {2n-2i-2j}\choose{n-2i} \right)_{1\leq i , j \leq \lfloor{\frac n 2}\rfloor}.$$ Our proof of Theorem \[degson\] uses a formula of Kazarnovskij [@kazarnovskii] (see also Theorem \[Kazarnovskij\]) for the degree of the image of a representation of a connected, reductive, algebraic group over an algebraically closed field. By applying this formula to the case of the standard representation of $\operatorname{SO}(n)$ we are able to express the degree of $\operatorname{SO}(n)$ in terms of its root data and other invariants. In addition to this result, Theorem \[non-intersecting\] provides a combinatorial interpretation of this degree in terms of non-intersecting lattice paths. In contrast to Theorem \[degson\], the combinatorial statement has the immediate benefit of being obviously non-negative. Let ${\Bbbk}$ be a field of characteristic zero. We can define $\operatorname{SO}(n,{\Bbbk})$ using the same system of equations since they are defined over the prime field ${\mathbb{Q}}$. For ${\Bbbk}$ that is not algebraically closed, the degree of a variety can be defined in terms of the Hilbert series of its coordinate ring. Since the Hilbert series does not depend on the choice of ${\Bbbk}$, the degree does not either. We choose to work over ${\mathbb{C}}$ not only for simplicity, but also so that we may use the above definition of degree. Our methods are not specific to $\operatorname{SO}(n)$. The same approach can be used to compute the degree of other algebraic groups. For example, toward the end of Section \[sec:degson\] we provide a similar closed formula for the degree of the symplectic group. This formula is also interpreted combinatorially in Section \[NILP\]. In order to verify Theorem \[degson\], as well as explore the structure of $\operatorname{SO}(n)$ in further depth, it is useful to compute this degree explicitly. We were able to do this for small $n$ using symbolic and numerical computations. A comparison of the success of these two approaches, together with our formula from Theorem \[degson\], is illustrated by the following table. \[fig:SO\] $\mathbf{n}$ **Symbolic** **Numerical** **Formula** -------------- -------------- --------------- ------------- 2 2 2 2 3 8 8 8 4 40 40 40 5 384 384 384 6 - 4768 4768 7 - 111616 111616 8 - - 3433600 9 - - 196968448 : Degree of $\operatorname{SO}(n)$ computed in various ways This project started in the spring of 2014, when Benjamin Recht asked the fifth author to describe the geometry of low-rank semidefinite programming (see Section \[SDP\]). In particular, he asked why the augmented Lagrangian algorithm for solving this problem [@BM] almost always recovers the correct optimum despite the existence of multiple local minima. It quickly became clear that to even compute the number of local extrema, one needs to know the degree of the orthogonal group. In Section \[SDP\] we find a formula for the number of critical points of low-rank semidefinite programming (see Theorem \[thm:LRSDP\]). The rest of this article is organized as follows. In Section \[backgroundsect\] we give the reader a brief introduction to algebraic groups and state Kazarnovskij’s Theorem. Section \[sec:degson\] proves Theorem \[degson\] by applying Kazarnovskij’s Theorem and simplifying the resulting expressions. After simplification, we are left with a determinant of binomial coefficients which can be interpreted combinatorially using the celebrated Gessel-Viennot lemma which we describe in Section \[NILP\]. The relationship between the degree of $\operatorname{SO}(n)$ and the degree of low-rank semidefinite programming is elaborated upon in Section \[SDP\]. Section \[Computational\] contains descriptions of the symbolic and numerical techniques involved in the explicit computation of $\deg \operatorname{SO}(n)$. Finally, in Section \[reality\] we explore questions involving the real points on $\operatorname{SO}(n)$. Background {#backgroundsect} ========== In this section we provide the reader with the necessary language to understand the statement of Kazarnovskij’s Theorem (see Theorem \[Kazarnovskij\]), our main tool for determining the degree of $\operatorname{SO}(n)$. We invite those who already are familiar with Lie theory to skip to the statement of Theorem \[Kazarnovskij\] and continue to Section \[sec:degson\] for our main result. We note, that aside from applying Theorem \[Kazarnovskij\], no understanding of the material in this section is necessary for understanding the remainder of the proof of Theorem \[degson\]. A more thorough treatment of the theory of algebraic groups can be found in [@derksen; @fulton; @humphreys]. An *algebraic group* $G$ is a variety equipped with a group structure such that multiplication and inversion are both regular maps on $G$. When the unipotent radical of $G$ is trivial and $G$ is over an algebraically closed field, we say that $G$ is a *reductive group*. Throughout this section, we let $G$ denote a connected reductive algebraic group over an algebraically closed field ${\Bbbk}$. Let ${{\mathbb{G}}_{\mathrm{m}}}$ denote the multiplicative group of ${\Bbbk}$, so as a set, ${{\mathbb{G}}_{\mathrm{m}}}= {\Bbbk}\setminus \{0\}$. Let $T$ denote a fixed maximal torus of $G$. By *maximal torus*, we mean a subgroup of $G$ isomorphic to ${{\mathbb{G}}_{\mathrm{m}}}^r$ and which is maximal with respect to inclusion. The number $r$ is well-defined and is called the *rank* of $G$. After fixing $T$, we define the *Weyl group* of $G$, denoted $W(G)$, to be the quotient of the normalizer of $T$ by its centralizer, $W(G) = N_G(T)/Z_G(T)$. Like $r$, $W(G)$ does not depend on the choice of $T$ up to isomorphism. \[maxTori\] We can parametrize $\operatorname{SO}(2,{\mathbb{C}})$ by ${{\mathbb{G}}_{\mathrm{m}}}$ via the map $$\mathbf{R}(t) := \frac{1}{2}\begin{pmatrix} t+t^{-1} & -i(t-t^{-1}) \\ i(t-t^{-1}) & t+t^{-1} \end{pmatrix},$$ which is in fact a group isomorphism. (Note that $\mathbf{R}(e^{i\theta})$ is the rotation matrix by angle $\theta$.) Therefore $\operatorname{SO}(2)$ has rank 1. Fix $r \in \mathbb{N}$. Then $$\begin{aligned} T_{2r}&:=\left\{ \begin{pmatrix} \mathbf{R}(t_1) & 0 & 0 & \cdots & 0\\ 0 & \mathbf{R}(t_2) & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \mathbf{R}(t_r) \end{pmatrix} \Bigg| t_i \in {{\mathbb{G}}_{\mathrm{m}}}\right\}\cong \operatorname{SO}(2)^{r}\subset \operatorname{SO}(2r)\\ T_{2r+1}&:=\left\{ \begin{pmatrix} \mathbf{R}(t_1) & 0 & 0 & \cdots & 0 & 0\\ 0 & \mathbf{R}(t_2) & 0 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & \cdots & \mathbf{R}(t_r) & 0\\ 0 & 0 & 0 & \cdots & 0 & 1 \end{pmatrix}\; \Bigg| \; t_i \in {{\mathbb{G}}_{\mathrm{m}}}\right\}\cong \operatorname{SO}(2)^{r}\subset \operatorname{SO}(2r+1)\end{aligned}$$ are maximal tori of rank $r$ of their respective groups. Therefore, $\text{rank}(\operatorname{SO}(2r))=\text{rank}(\operatorname{SO}(2r+1))=r$ and we see that the rank of $\operatorname{SO}(n)$ depends fundamentally on the parity of $n$. The *character group* $M(T)$ is the set of algebraic group homomorphisms from $T$ to ${{\mathbb{G}}_{\mathrm{m}}}$, i.e. group homomorphisms defined by polynomial maps, $$M(T):=\operatorname{Hom}_{{\mathbf{AlgGrp}}}(T,{{\mathbb{G}}_{\mathrm{m}}}).$$ Since $T$ is isomorphic to ${{\mathbb{G}}_{\mathrm{m}}}^r$, all such homomorphisms must be of the form $$(t_1,\ldots,t_r) \mapsto t_1^{a_1}\cdots t_r^{a_r}$$ with $a_1,\ldots,a_r$ integers. This character group is isomorphic to ${\mathbb{Z}}^r$ and for this reason it is often called the character lattice. Dual to this is the *group of 1-parameter subgroups* $$N(T):=\operatorname{Hom}_{{\mathbf{AlgGrp}}}({{\mathbb{G}}_{\mathrm{m}}},T),$$ which is also isomorphic to ${\mathbb{Z}}^r$. Indeed, each 1-parameter subgroup is of the form $t \mapsto (t^{b_1},\ldots,t^{b_r})$ for integers $b_1,\ldots,b_r$. There exists a natural bilinear pairing between $N(T)$ and $M(T)$, given by $$\begin{aligned} M(T)\times N(T)&\rightarrow \operatorname{Hom}_{{\mathbf{AlgGrp}}}({{\mathbb{G}}_{\mathrm{m}}},{{\mathbb{G}}_{\mathrm{m}}})\cong{\mathbb{Z}}\\ \langle \chi, \sigma \rangle &\mapsto \chi \circ \sigma.\end{aligned}$$ Now if $\rho:G\rightarrow{}\operatorname{GL}(V)$ is a representation of $G$ we attach to it special characters called weights. A *weight* of the representation $\rho$ is a character $\chi\in M(T)$ such that the set $$V_\chi:=\bigcap_{s\in T}\ker(\rho(s)-\chi(s)\operatorname{Id}_V)$$ is non-trivial. This condition is equivalent to saying that all of the matrices in $\{\rho(s) \; | \; s\in T\}$ have a simultaneous eigenvector $v\in V$ such that the associated eigenvalue for $\rho(s)$ is $\chi(s)$. We will use $C_V$ to denote the convex hull of the weights of the representation $\rho$. \[ex:weight\] An example that will be important for us later will be the standard representation coming from the natural embedding $\rho:\operatorname{SO}(n) \to \operatorname{GL}({\mathbb{C}}^n)$. For any $t \in {{\mathbb{G}}_{\mathrm{m}}}$, the matrix $\mathbf{R}(t) \in \operatorname{SO}(2)$ has eigenvectors $e_1 + ie_2$ and $e_1 - ie_2$ with eigenvalues $t$ and $t^{-1}$ respectively. From the explicit description of $T$ in Example \[maxTori\] we see that the eigenvectors of $\rho(t_1,\ldots,t_r)$ are all vectors of the form $e_{2j-1} \pm ie_{2j}$ with $1\leq j \leq r$ and the eigenvalues are $t_1^{\pm 1},\ldots,t_r^{\pm 1}$. These eigenvalues, viewed as characters, are the weights of $\rho$. Additionally when $n = 2r+1$, we have that $e_{2r+1}$ is an eigenvector with eigenvalue 1, corresponding to the trivial character. Another representation of a matrix group $G \subseteq \operatorname{End}(V)$ is the *adjoint representation*, $\operatorname{Ad}:G \to \operatorname{GL}(\operatorname{End}(V))$, with $\operatorname{Ad}(g)$ the linear map defined by $A \mapsto gAg^{-1}$. The *roots* of $G$ are the weights of the adjoint representation. Given a linear functional $\ell$ on $M(T)$, we define the *positive roots* of $G$ with respect to $\ell$ to be the roots $\chi$ such that $\ell(\chi)>0$. We denote the positive roots of $G$ by $\alpha_1,\ldots,\alpha_l$. For the algebraic groups in this paper, we can choose $\ell$ to be the inner product with the vector $(r,r-1,\ldots,1)$ so that a root of the form $e_j-e_k$ is positive if and only if $j<k$. To each root $\alpha$, we associate a *coroot* $\check \alpha$, defined to be the linear function $\check \alpha (\vec{x}):= \frac{2\langle\vec{x},\alpha\rangle}{\langle\alpha,\alpha\rangle}$ where $\langle , \rangle$ must be $W(G)$-invariant. Throughout this paper, we fix this to be the standard inner product. We now compute the roots of $\operatorname{SO}(n)$, starting with $n$ even. It can be shown that the simultaneous eigenvectors of $\operatorname{Ad}(s)$ over all $s \in T$ are matrices $A$ with the following structure. These matrices are zero outside a $2\times 2$ block $B$ in rows $2j-1,2j$ and columns $2k-1,2k$ for some $1\leq j,k \leq r$. Furthermore, $B = v_1v_2^T$ with each $v_k$ equal to one of the eigenvectors of $\mathbf{R}(t)$, $e_1 \pm ie_2$. Indeed, suppose $s \in T$ has blocks along the diagonal $\mathbf{R}(t_j)$ with $t_1,\ldots,t_r \in {{\mathbb{G}}_{\mathrm{m}}}$. Then $\operatorname{Ad}(s)(A)$ will also be zero except in the same $2\times 2$ block, and that block will be $$\mathbf{R}(t_j)B\mathbf{R}(t_k)^T = t_j^{\pm 1}t_k^{\pm 1}B,$$ where the signs in the exponents depend on the choices of $v_1$ and $v_2$. Thus the roots of $\operatorname{SO}(2r)$ are the characters of the form $t_j^{\pm 1}t_k^{\pm 1}$ for $1\leq j,k \leq r$. In the case that $n$ is odd, $A$ has an extra row and column. Consider $A$ with support only in the last column. Then for $s \in T$, $\operatorname{Ad}(s)(A) = sAs^{-1}$ but $s^{-1}$ acts trivially on the left, while $s$ acts on the last column as an element of $\operatorname{GL}({\mathbb{C}}^n)$ as in the standard representation. As in Example \[ex:weight\] we get weights $t_1^{\pm 1},\ldots,t_r^{\pm 1},1$. The same weights appear for $A$ with support in the last row. Associated to $G$ is a Lie algebra $\mathfrak{g}$, which comes equipped with a Lie bracket $[\ ,\ ]$. A *Cartan subalgebra* $\mathfrak{h}$ is a nilpotent subalgebra of $\mathfrak{g}$ that is self-normalizing, meaning if $[x,y] \in \mathfrak{h}$ for all $x \in \mathfrak{h}$, then $y \in \mathfrak{h}$. Let $S(\mathfrak{h}^*)$ be the ring of polynomial functions on $\mathfrak{h}$. The Weyl group $W(G)$ acts on $\mathfrak{h}$, and this extends to an action of $W(G)$ on $S(\mathfrak{h}^*)$. The space $S(\mathfrak{h}^*)^{W(G)}$ of polynomials which are invariant up to the action of $W(G)$ is generated by $r$ homogeneous polynomials whose degrees, $c_1+1,\ldots,c_r+1$, are uniquely determined. The values $c_1, \ldots, c_r$ are called *Coxeter exponents*. We are now prepared to state Kazarnovskij’s theorem. \[Kazarnovskij\] Let $G$ be a connected reductive group of dimension $m$ and rank $r$ over an algebraically closed field. If $\rho:G\rightarrow{}\operatorname{GL}(V)$ is a representation with finite kernel then, $$\deg\overline{\rho\left(G\right)}=\frac{m!}{|W(G)|(c_1!c_2!\cdots c_r!)^2|\ker(\rho)|}\int_{C_V}(\check{\alpha}_{1}\check{\alpha}_2\cdots\check{\alpha}_l)^2dv.$$ where $W(G)$ is the Weyl group, $c_i$ are Coxeter exponents, $C_V$ is the convex hull of the weights, and $\check \alpha_i$ are the coroots. If $\rho$ is the standard representation for an algebraic group $G$, then it follows that $\deg \overline{\rho(G)}=\deg G$. Therefore, in order to compute $\deg \operatorname{SO}(n)$, all we must do is apply this theorem for the standard representation of $\operatorname{SO}(n)$. The relevant data for this theorem is given in Table \[data\] below for $\operatorname{SO}(n)$ and $\operatorname{Sp}(n)$. \[data\] [ l c c c c c c]{} Group& Dimension& Rank & Positive Roots & Weights & $|W(G)|$ & Coxeter Exponents\ \ $\operatorname{SO}(2r)$& ${2r}\choose{2}$ & $r$ &$\{e_i \pm e_j\}_{ i<j }$ & $\{\pm e_i\}$ & $r!2^{r-1}$ & $1,3,\ldots,2r-3,r-1$\ \ $\operatorname{SO}(2r+1)$ & ${2r+1}\choose{2}$& $r$ & $\{e_i \pm e_{j}\}_{i<j} \cup \{e_i\}$ & $\{\pm e_i\}$ & $r! 2^r$ & $1,3, 5, \ldots, 2r-1$\ \ $\operatorname{Sp}(r)$ & ${2r}\choose{2}$ & $r$ &$\{e_i \pm e_j\}_{i<j }\cup\{2e_i\}$ & $\{\pm e_i\}$ & $r! 2^r$ & $1,3, 5, \ldots, 2r-1$\ \ Main Result: The Degree of $\operatorname{SO}(n)$ {#sec:degson} ================================================= We now prove our main result, Theorem \[degson\]. At the end of this section we use the same method to obtain a formula for the degree of the symplectic group. We begin by directly applying Theorem \[Kazarnovskij\] to $\operatorname{SO}(2r)$ and $\operatorname{SO}(2r+1)$ to obtain $$\begin{aligned} \label{eqn:even-int} \deg\operatorname{SO}(2r)&=\frac{\displaystyle \binom{2r}{2}!}{\displaystyle r!2^{r-1}(r-1)!^2\prod_{k=1}^{r-1}(2k-1)!^2} \int_{C_V} \left(\prod_{1\leq i<j\leq r}(x_i^2-x_j^2)^2\right)dv,\\ \label{eqn:odd-int} \deg\operatorname{SO}(2r +1) &= \frac{\displaystyle \binom{2r +1}{2}!}{r!2^r \displaystyle \prod_{k=1}^r(2k-1)!^2} \int_{C_V} \left(\prod_{1\leq i<j\leq r}(x_i^2-x_j^2)^2\prod_{i=1}^r(2x_i)^2\right)dv.\end{aligned}$$ Thus, to compute the degree of $\operatorname{SO}(n)$ it suffices to find formulas for the integrals above. We do this by first expanding the integrand into monomials, and then integrating the result. We use the well-known expression for the determinant of the Vandermonde matrix, $$\prod_{1\leq i < j\leq r}(y_j-y_i)=\sum_{\sigma\in S_r}\left(\operatorname{sgn}(\sigma)\prod_{i=1}^r y_i^{\sigma(i)-1}\right).$$ Substituting $y_i = x_i^2$ and squaring the entire expression yields $$\label{rewrite-integrand} \prod_{1\leq i<j\leq r}(x_i^2-x_j^2)^2 = \sum_{\sigma,\tau \in S_r}\left(\operatorname{sgn}(\sigma\tau) \prod_{i=1}^r x_i^{2\sigma(i)+2\tau(i)-4}\right).$$ Additionally, we point out that every variable in the integrand is being raised to an even power and $C_V$ is the convex hull of weights, $\{\pm e_i\}$. Because of this symmetry, the integrals over $C_V$ are $2^r$ times the same integrals over $\Delta_r$, the standard $r$-simplex. We have now reduced the computation of this integral to understanding the integral of any monomial over the standard simplex. The following proposition provides a formula for this. \[integral-monomial\] Let $\Delta_{r}\subset \mathbb{R}^{r}$ be the standard $r$-simplex. If $\mathbf{a}=(a_1,\ldots,a_r)\in \mathbb{Z}_{>0}^r$ then $$\int_{\Delta_r}\mathbf{x}^\mathbf{a}d\mathbf{x}=\int_{\Delta_r}x_1^{a_1}x_{2}^{a_2}\cdots x_{r}^{a_r}dx_1dx_2\cdots dx_r= \frac{1}{(r+\sum a_i)!}\prod_{i=1}^r a_i!.$$ We can now get expressions for the integrals in (\[eqn:even-int\]) and (\[eqn:odd-int\]) directly by applying (\[rewrite-integrand\]) and Proposition \[integral-monomial\]. \[two-integrals\] Let $I_{even}(r)$ and $I_{odd}(r)$ denote the integrals in (\[eqn:even-int\]) and (\[eqn:odd-int\]) respectively. Then, $$I_{even}(r) =\frac{r!2^r}{\binom{2r}{2}!}\det\left((2i+2j-4)!\right)_{1\leq i,j\leq r}.$$ $$I_{odd}(r) =\frac{r!2^{3r}}{\binom{2r+1}{2}!}\det\left((2i+2j-2)!\right)_{1\leq i,j\leq r}.$$ As mentioned above, we can compute $I_{odd}$ by considering the integrand only over the simplex. This, along with equation gives us that $$\begin{aligned} I_{odd}(r) &= 2^r \int_{\Delta_r} \prod_{1\leq i<j\leq r} (x_i^2-x_j^2)^2 \prod_{i=1}^r (2x_i)^2 dv\\ &= 2^r \int_{\Delta_r} \left(\sum_{\sigma,\tau \in S_r}\operatorname{sgn}(\sigma\tau) \prod_{i=1}^r x_i^{2\sigma(i)+2\tau(i)-4}\right) \prod_{i=1}^r(2x_i)^2 dv\\ &= 2^{3r} \sum_{\sigma,\tau \in S_r}\operatorname{sgn}(\sigma\tau)\int_{\Delta_r}\prod_{i=1}^r x_i^{2\sigma(i)+2\tau(i)-2} dv.\end{aligned}$$ As the integrand is homogeneous of degree $4\binom{r}{2}+2r$, applying Proposition \[integral-monomial\] and simplifying yields $$\begin{aligned} I_{odd}(r) &= \frac{2^{3r}}{\left(4\binom{r}{2}+3r\right)!} \sum_{\sigma,\tau\in S_r} \operatorname{sgn}(\sigma\tau) \prod_{i=1}^r (2\sigma(i)+2\tau(i)-2)!,\end{aligned}$$ which after replacing $i$ with $\sigma^{-1}(i)$ gives us $$\prod_{i=1}^r(2\sigma(i)+2\tau(i)-2)! = \prod_{i=1}^r(2i+2\tau\sigma^{-1}(i)-2)!.$$ Let $\rho = \tau\sigma^{-1}$. Over all pairs $\sigma,\tau \in S_r$, the permutation $\rho$ appears as each permutation in $S_r$ exactly $r!$ times, and $\operatorname{sgn}(\sigma\tau) = \operatorname{sgn}(\rho)$. Therefore, we have that $$\begin{aligned} I_{odd}(r) &= \frac{r!2^{3r}}{\left(4\binom{r}{2}+3r\right)!} \sum_{\rho\in S_r} \operatorname{sgn}(\rho) \prod_{i=1}^r (2i+2\rho(i)-2)!\\ &= \frac{r!2^{3r}}{\binom{2r+1}{2}!} \det\left((2i+2j-2)!\right)_{1\leq i,j\leq r}.\end{aligned}$$ The derivation of $I_{even}$ follows precisely the same steps. Theorem \[degson\] now follows directly from the subsequent simplification. $$\begin{aligned} \deg \operatorname{SO}(2r+1) &= \frac{2^{2r}}{(1!3!\cdots(2r-1)!)^2} \det ((2i+2j-2)!)\\ &=\frac{2^{2r}}{(1!2!\cdots (2r-1)!)} \det\left(\frac{(2i+2j-2)!}{(2i-1)!}\right)\\ &=2^{2r}\det\left(\frac{(2i+2j-2)!}{(2i-1)!(2j-1)!}\right)\\ &=2^{2r}\det\left(\binom{2i+2j-2}{2i-1}\right)_{1\leq i,j\leq r}.\end{aligned}$$ Reversing the order of the rows and columns of the final matrix and reindexing produces the formula given in Theorem \[degson\]. Similarly, for the even case, we have $$\begin{aligned} \deg \operatorname{SO}(2r) &= \frac{2}{(1!3!\cdots(2r-3)!(r-1)!)^2} \det ((2i+2j-4)!)\\ &= \frac{2 \cdot (2^{r-1})^2}{(1!3!\cdots(2r-3)!2\cdot 4 \cdots (2r-2))^2} \det ((2i+2j-4)!)\\ &= 2^{2r-1} \det \left(\frac{(2i+2j-4)!}{(2i-2)!(2j-2)!}\right)\\ &=2^{2r-1}\det\left(\binom{4r-2i-2j}{2r-2i}\right)_{1\leq i,j\leq r}.\end{aligned}$$ This finishes the proof of Theorem \[degson\]. Since the orthogonal group $\operatorname{O}(n)$ has two components that are isomorphic to $\operatorname{SO}(n)$, we immediately get a formula for the degree of $\operatorname{O}(n)$. \[degOn\] The degree of $\operatorname{O}(n)$ is given by $$\deg \operatorname{O}(n)=2^{n}\det \left( {2n-2i-2j}\choose{n-2i} \right)_{1\leq i , j \leq \lfloor{\frac n 2}\rfloor}.$$ Furthermore, as mentioned in the introduction, there is no reason, [*a priori*]{}, that the steps taken in this section are particular to $\operatorname{SO}(n)$. We now apply these methods to find the degree of $\operatorname{Sp}(r)$, the group of (complex) symplectic matrices. Recall the *symplectic group* over ${\mathbb{C}}$ is defined to be $$\operatorname{Sp}(r):=\operatorname{Sp}(r,{\mathbb{C}}) = \{ M \in \text{Mat}_{2r,2r}({\mathbb{C}}) \ |\ M^T \Omega M = \Omega\},$$ where $$\Omega = \begin{pmatrix} 0 & I_r \\ -I_r & 0 \end{pmatrix}.$$ \[thm:symplectic\] The degree of $\operatorname{Sp}(r)$ is given by $$\deg \operatorname{Sp}(r) =\det\left({2i + 2j - 2 \choose 2i-1}\right)_{1\leq i,j\leq r}.$$ For $1 \leq r \leq 5$ the values of $\deg \operatorname{Sp}(r)$ are $2,24,1744,769408,2063048448,\ldots$. This was verified using both numerical and symbolic techniques up to $r=3$. This is an application of Kazarnovskij’s result which is completely analogous to the computation for the special orthogonal group. The integral is the same as the one for $\operatorname{SO}(2r+1)$ up to factors of 2, so it is evaluated in the same way, and then the expression can be simplified $$\begin{aligned} \deg \left(\text{Sp}(r)\right) &=\frac{(r(2r+1))!}{r! 2^r (1!3!\cdots(2r-1)!)^2}\int_{C_V} \left(\prod_{1\leq i<j\leq r}(x_i-x_j)^2(x_i+x_j)^2\prod_{i=1}^r x_i^2\right)dv \\ &=\frac{1}{(1!3!\cdots(2r-1)!)^2} \det\left((2i+2j-2)!\right)_{1\leq i,j\leq r}\\ &=\det\left({2i + 2j - 2 \choose 2i-1}\right)_{1\leq i,j\leq r}.\end{aligned}$$ We remark that our formula for $\deg \text{Sp}(r)$ is particularly interesting because the determinant in Theorem \[thm:symplectic\] is the same as the determinant in Theorem \[degson\] when $n=2r+1$. \[cor:sp\] $$\deg \operatorname{SO}(2r+1)=2^{2r}\deg \text{Sp}(r)$$ Sending the $(i,j)$ entry of the matrix in Theorem \[thm:symplectic\] to the ${(r-i+1,r-j+1)}$ entry does not change the determinant and gives us that $$\deg \text{Sp}(r)=\det\left({4r+2-2i-2j \choose 2r+1-2i}\right)_{1\leq i,j\leq r}.$$ When $n=2r+1$, this is the matrix appearing in Theorem \[degson\] and all that is different is the coefficient in front. Accounting for this coefficient finishes the proof. Non-Intersecting Lattice Paths {#NILP} ============================== The formulas given in the previous section for the degrees of $\operatorname{SO}(n),\operatorname{O}(n),$ and $\text{Sp}(r)$ can be interpreted as a count of non-intersecting lattice paths via the Gessel-Viennot Lemma [@GV]. \[GV\] Let $A=\{a_1, \ldots,a_r\}$, $B=\{b_1, \ldots,b_r\}$ be collections of lattice points in $\mathbb{Z}^2$. Let $M_{i,j}$ be the number of lattice paths from $a_i$ to $b_j$ using only unit steps in either the North or East direction. If the only way that a system of these lattice paths from $A \to B$ do not cross each other is by sending $a_i\mapsto b_i$, then the determinant of $M$ equals the number of such non-intersecting lattice paths. The number of lattice paths from $(0,0)$ to $(i,j)$ is the binomial coefficient ${i+j} \choose i$. Since the matrix involved in the formulas for the degrees of $\operatorname{SO}(n),$ $\operatorname{O}(n),$ and $\text{Sp}(r)$ has binomial coefficients as entries, it is natural to search for a interpretation of its determinant via Gessel-Viennot. \[non-intersecting\] Let $N(n)$ count the number of non-intersecting lattice paths from $A(n):=\{a_i\}_{i=1}^{\lfloor \frac n 2 \rfloor}$ to $B(n):=\{b_j\}_{j=1}^{\lfloor \frac n 2 \rfloor}$ where $a_i=(2i-n,0)$ and $b_j=(0,n-2j)$. Then $$\begin{aligned} \deg\operatorname{SO}(n) &= 2^{n-1}N(n), \\ \deg\operatorname{O}(n) &= 2^{n}N(n), \\ \deg\text{Sp}(r) &= N(2r+1).\end{aligned}$$ It is enough to prove this theorem for $\operatorname{SO}(n)$ and apply Corollaries \[degOn\] and \[cor:sp\]. Noticing that the matrix appearing in Theorem \[degson\] is the minor of Pascal’s matrix which skips every other row and every other column up to $\lfloor \frac n 2 \rfloor$ shows that we have a correct point configuration for Gessel-Viennot. Figure \[fig:GVExample\] computes that $N(5)=24$ by explicitly listing all $24$ non-intersecting lattice paths from $A(5)$ to $B(5)$. Then, according to Theorem \[non-intersecting\], we see that $\deg \operatorname{SO}(5)=2^4 \cdot 24=384$, $\deg \operatorname{O}(5)=2^5 \cdot 24=768$, and $\deg \text{Sp}(2)=24$. Theorem \[non-intersecting\] suggests a relationship between these non-intersecting lattice paths and the degrees of $\operatorname{SO}(n),\operatorname{O}(n),$ and $\text{Sp}(r)$. Such a direct interpretation could be interesting, and so we pose the question: \[ourQuestion\] Does Theorem \[non-intersecting\] have a deeper combinatorial interpretation? Because the formula for the degree of the symplectic group has no coefficient in front of the lattice path count in Theorem \[non-intersecting\], studying the combinatorial meaning of the degree of $\text{Sp}(r)$ may be an ideal starting point to tackle Question \[ourQuestion\]. An Application - The Degree of Low Rank Semidefinite Programming {#SDP} ================================================================ In this section we show how knowing the degree of $\operatorname{SO}(n)$ can be used to compute the number of critical points for a certain optimization problem (cf. Theorem \[thm:LRSDP\]). Consider the standard formulation of *semidefinite programming* $$\begin{aligned} \label{primalProblem} &\text{minimize}_{X\in\mathcal S^n} \quad C\bullet X\notag\\ &\text{such that }\quad A_i\bullet X = b_i, i=1,..., m, \quad X\succeq 0.\end{aligned}$$ Here $\mathcal S^n$ is the set of $n\times n$ real symmetric matrices, $b\in\mathbb Q^m$ is a vector, $C, A_1, ..., A_m\in\mathbb Q\mathcal S^n$ are matrices, and $\bullet$ denotes the trace inner product for matrices: $U\bullet V = \text{trace}(UV)$. Semidefinite programming can be solved in polynomial time in the size $n$ of the unknown matrix $X$ and in the number of constraints $m$. It is a widely used method in practice, and many NP-hard problems possess semidefinite relaxations [@BV2; @GW2]. However, it is often the case that the size $n$ is very large, and solving exactly can be computationally prohibitive. On the other hand, the rank $r$ of the optimal solution $X^*$ is often much smaller than $n$, and in those cases we can solve  more rapidly by replacing $X$ by the low rank positive semidefinite matrix $RR^T$, where $R\in\mathbb R^{n\times r}$. This idea and an algorithm to solve the new problem are due to Burer and Monteiro [@BM]. The problem becomes $$\begin{aligned} \label{newProblem} &\text{minimize}_{R\in\mathbb R^{n\times r}} \quad C\bullet (RR^T)\notag\\ &\text{such that }\hspace{0.33cm} A_i\bullet (RR^T) = b_i, ~~i=1, ..., m.\end{aligned}$$ The constraint $X\succeq 0$ is now implicit and the number of variables has decreased from $n^2$ to $nr$. However, the objective function and the constraints are no longer linear; instead, they are quadratic and the feasible set is non-convex. In [@BM]  Burer and Monteiro propose a fast algorithm for solving . Despite the existence of multiple local minima, in practice this algorithm quickly finds the global minimum. It starts by choosing the rank $r=1$, and increments it until $C - \sum_{i=1}^my_i A_i\succeq 0$, which ensures that we have arrived at the smallest optimal $r$. For each fixed rank $r$, the optimization problem is non-convex, and its appealing behavior still remains to be examined. In Theorem \[thm:LRSDP\] we give a formula for the number of critical points of this optimization problem. We call a [*critical point*]{} of the optimization problem any point $(R, y)$ which satisfies the Lagrange multipliers equations arising from this problem. Here $y$ is a vector of size $m$, and its entries $y_1,\dots, y_m$ are the new dual variables introduced for the $m$ constraints in (see equation ). Before we state our theorem, we need the following definition. Let $$\psi_i = 2^{i-1}, \quad \psi_{i,j} = \sum_{k=i}^{j-1}\binom{i+j-2}k \text{ when } i < j,$$ and $$\psi_{i_1,\dots, i_r}= \operatorname{Pf}(\psi_{i_k, i_l})_{1\leq k < l\leq r} \text{ if } r \text{ is even},$$ $$\psi_{i_1,\dots, i_r}= \operatorname{Pf}(\psi_{i_k, i_l})_{0\leq k < l\leq r} \text{ if } r \text{ is odd}$$ where $r>2$, $\psi_{0, k } = \psi_{k}$, and $\operatorname{Pf}$ denotes the Pfaffian. Then, define $\delta(m, n, r)$ as $$\delta(m, n, r) = \sum_{I} \psi_I\psi_{I^c},$$ where the sum runs over all strictly increasing subsequences $I = \{i_1, ..., i_{n-r}\}$ of $\{1, ..., n\}$ of length $n - r$ and such that $i_1 + ... + i_{n-r} = m$. \[thm:LRSDP\] The number of critical points of the low-rank semidefinite programming algorithm is $$2 (\deg\operatorname{SO}(r)) \delta(m, n, r).$$ The number $\delta(m, n, r)$ is called the [*algebraic degree of semidefinite programming*]{} and was originally defined in [@NRS] as the number of critical points of the original semidefinite programming problem for which the matrix $X$ has rank $r$. The final formula for it was computed in [@VR]. *Proof of Theorem \[thm:LRSDP\]:* In order to analyze the optimality conditions for the program (\[newProblem\]) for a fixed $r$, consider the Lagrangian function $$\begin{aligned} \label{Lagrangian} L(R, y) = C\bullet (RR^T) - \sum_{i=1}^my_i(A_i\bullet(RR^T) - b_i).\end{aligned}$$ Taking derivatives, we find out that the critical points $(R, y)$ of this optimization problem are given by the Lagrange multipliers equations: $$\begin{aligned} \label{equations} \left(C - \sum_{i=1}^my_iA_i\right)RR^T&= 0\\ A_i\bullet(RR^T) &= b_i, i=1,2,...,m.\notag\end{aligned}$$ In addition, those critical points relevant for applications have to be real and have to satisfy $$\begin{aligned} \label{addEquation} \left(C - \sum_{i=1}^my_iA_i\right)\succeq 0,\end{aligned}$$ since this is the constraint in the dual to the optimization problem . However, in this article we are primarily concerned with counting all of the critical points. Analogously, in [@NRS] Nie, Ranestad, and Sturmfels show that the critical points of the original semidefinite programming problem satisfy $$\begin{aligned} \left(C - \sum_{i=1}^my_iA_i\right)X &= 0,\label{optimalSolutions1}\\ A_i\bullet X &= b_i, i=1,...,m.\label{optimalSolutions2}\end{aligned}$$ In addition, the critical points relevant for applications have to satisfy $$\begin{aligned} \left(C - \sum_{i=1}^my_iA_i\right)\succeq 0 \text{ and } X&\succeq 0,\label{optimalSolutions3}\end{aligned}$$ but these conditions are disregarded and the total number of critical points is counted. Nie, Ranestad, and Sturmfels show that the number of solutions $(X, y)$ to -, for which the rank of $X$ is $r$, equals $\delta(m, n, r)$ (c.f. Definition 1). Comparing our system of equations to the equations -, we see that the fiber of the map $(R, y) \mapsto (RR^T, y)$ above each point $(X, y)$, satisfying -, consists of all points $(R, y')$, satisfying , and such that $y' = y$ and $X = RR^T$. Given $X$ and one matrix $R$ such that $X = RR^T$, all other matrices $S$ such that $(S, y)$ is in the fiber above $(X, y)$ have the form $S = RU$ where $U$ runs over all orthogonal $r\times r$ matrices. In other words, this fiber is isomorphic to a copy of the orthogonal group $O(r)$. Therefore, the number of solutions to is equal to $2(\deg\operatorname{SO}(r))\delta(m, n, r)$. The number of critical points of low-rank semidefinite programming grows rapidly with the rank $r$, and the appealing behavior of the augmented Lagrangian algorithm [@BM] still needs to be explained. It would be quite interesting and relevant for applications to examine how many of the critical points computed in Theorem 5 and in [@NRS] are real, and moreover, how many of them satisfy the additional linear matrix inequality constraints and respectively. This is a real algebraic problem and would involve counting polynomial system solutions over semialgebraic sets. This question is addressed more in Section \[reality\]. Computational Methods {#Computational} ===================== Although we have already derived a formula for the degree of $\operatorname{SO}(n)$, it is natural to want to compute this degree explicitly for particular values of $n$. Aside from merely verifying the formula in Theorem \[degson\], the computation of this degree gives us access to other useful data along the way. In our case, this manifests itself as either a Gröbner basis or a witness set for $\operatorname{SO}(n)$. Once computed, either may be used in further computations (such as those done in Section \[reality\] using witness sets). Additionally, $\operatorname{SO}(n)$ serves as a prime example of when numerical algorithms are better suited for computation than other techniques. Even though our computations focus on $\operatorname{SO}(n)$, these methods are useful for studying many other varieties. In this section, we describe three techniques which can compute the degree of a variety: a Gröbner basis algorithm, polynomial homotopy continuation, and a numerical monodromy algorithm. The first is symbolic and the last two use numerical algebraic geometry. The results of our symbolic and numerical computations for $\deg \operatorname{SO}(n)$ appear in the first two columns of Table \[fig:SO\]. Code for each method is given in the appendix. Using Gröbner bases, we were able to compute the degree of $\operatorname{SO}(n)$ for $n \leq 5$. The standard algorithm computes a Gröbner basis for the ideal of $\operatorname{SO}(n)$ over $\mathbb{Q}$ and uses this to produce the Hilbert polynomial. However, since the dimension of $\operatorname{SO}(n)$ grows quadratically in $n$, this method quickly becomes computationally infeasible. Computing a Gröbner basis over a finite field can speed up the computation, but this method is still quite slow. A common numerical algorithm for computing the degree of $\operatorname{SO}(n)$ over $\mathbb{C}$ randomly chooses an affine linear space $\mathcal L$ of complementary dimension and counts the number of complex solutions $S$ to the zero-dimensional system corresponding to $\operatorname{SO}(n) \cap \mathcal L$. This data is contained in the triple $(\operatorname{SO}(n),\mathcal L, S)$ which is called a *witness set* for $\operatorname{SO}(n)$. This is the fundamental data type in numerical algebraic geometry in the sense that the computation of a witness set is often a necessary step for other numerical algorithms. Such techniques include sampling points on the variety at a rapid rate, studying its asymptotic behaviour, computing its monodromy group, or even studying its real locus, as we do in Section \[reality\]. Both numerical algorithms presented below produce a witness set for $\operatorname{SO}(n)$. *Polynomial homotopy continuation* computes a witness set by solving a system of polynomials describing these points. Briefly, this method begins with a “start” polynomial system that has similar structure to the “target” system we want to solve, but for which the solutions are obvious. The solutions of the start system are quickly tracked through a homotopy towards those of the target system [@SW]. The most basic start system one uses for this technique has a solution count equal to the product of the degrees of the polynomials in the target system. This number is called the *Bézout bound* and for our case is equal to $2^{n(n+1)/2}$ (for $n=6$, this is already $2097152$). The *polyhedral start system*, however, has a solution count equal to the mixed volume of the Newton polytopes of these polynomials. In our case, this count provides no savings as it is equal to the Bézout bound. Because of how many paths need to be tracked with this method, we were only able to compute the degree of $\operatorname{SO}(n)$ up to $n=5$ with this method, just like with Gröbner bases. The method that proved to be the most efficient takes advantage of the monodromy group of $\operatorname{SO}(n)$. The basic idea is that if we know some point on a linear cut $W=\mathcal{L} \cap \operatorname{SO}(n)$, we can track this solution from the slice $W$ along some path $\gamma$ to another slice $W'$ using homotopy methods. Tracking this solution along a different path $\gamma'$ back to $W$ then induces a permutation $\sigma_{\gamma,\gamma'}$ on the points in $W$. Therefore, applying this action to a point $x_0 \in W$ will likely produce a new point $\sigma_{\gamma,\gamma'}(x_0) \in W$. One iterates this process hoping to populate the witness set associated to $W$. Other than knowing the degree [*a priori*]{}, stopping criteria for this method tend to be heuristic in nature: one can wait until the algorithm fails to produce new points (suggesting there are no new points to be found) or one can compute a *trace test* [@tracetest] which numerically decides whether or not a witness set is complete. This monodromy method has been implemented in the package *monodromySolver* for *Macaulay2* [@M2] and is explained in much more detail in [@duff2016solving]. A major computational result arising from this project was the computation of witness sets for $\operatorname{SO}(6)$ and $\operatorname{SO}(7)$. This was done in $630$ and $42790$ seconds respectively using *monodromySolver*. The algorithm stopped when no new points were found on ten consecutive iterations. Real Points on $\operatorname{SO}(n)$ {#reality} ===================================== An interesting question pertaining to $\operatorname{SO}(n)$ is whether or not this variety always admits some witness set consisting of only real points. Since tracking points of one witness set to those of another is computationally inexpensive via homotopy continuation, we use this method to generate experimental data regarding real points on witness sets of $\operatorname{SO}(3),$ $\operatorname{SO}(4),$ and $\operatorname{SO}(5)$. The number of coefficients needed to produce a linear cut of $\operatorname{SO}(n)$ is $(n^2+1) {{n}\choose 2}$. We randomly choose these coefficients using the *random* function in *Macaulay2* in order to sample linear cuts of $\operatorname{SO}(n)$. We then use homotopy continuation to track solutions of a precomputed witness set to those lying on the randomly chosen linear cut. Finally, we determine how many solutions in the new cut are real by checking whether each solution is within a $0.001$ numerical tolerance of a real point coordinate-wise. One can certify the results using the software *alphaCertify* which implements Smale’s $\alpha$ theory [@alphaCertified]. For the sake of speed, we chose not to certify all of the results, but instead certify at least one witness set achieving the observed maximum of real points (cf. Table \[fig:SO3Real\], Table \[fig:SO4Real\], and Table \[fig:SO5Real\]). After computing $1398000$, $1004100$, and $48200$ witness sets for $\operatorname{SO}(3),\operatorname{SO}(4),$ and $\operatorname{SO}(5)$ respectively, we have summarized the number of real solutions found in each witness set in the frequency tables and histograms below. Explicit data and code used can be found in [@dataSite]. Note that very rarely, numerical failures occur because the path that homotopy continuation is being performed over is ill-conditioned (for example, almost singular). These occurrences are also tallied below under “fail”. -------------------- ------------------------------------------------------------------------ -- -- -- -- -- -- \#(Real Solutions) **[Fail]{}&**[0]{} & **[2]{} & **[4]{} & **[6]{}& **[8]{}&**[Total]{}\ Frequency &2& 285676 & 420049 & 549875 & 127699 & 14699&1398000\ ************** -------------------- ------------------------------------------------------------------------ -- -- -- -- -- -- : Number of real points on witness sets of $\operatorname{SO}(3)$[]{data-label="fig:SO3Real"} -------------------- -------------- ---------------------------------------------------------------- -- -- -- -- -- \#(Real Solutions) [**Fail**]{} **0 & **2& **4& **6& **[8]{}& **[10]{}\ Frequency &51 &183427 &108273 &132143 &156010 &159630 &124843\ ************ -------------------- -------------- ---------------------------------------------------------------- -- -- -- -- -- : Number of real points on witness sets of $\operatorname{SO}(4)$[]{data-label="fig:SO4Real"} [ |c|c|c|c|c| c | c | c | c | c | c | c | c ||c]{} **[12]{}& **[14]{}& **[16]{}& **[18]{}& **[20]{} &**[22]{} &**[24]{}&**[26]{}&**[28]{}&**[30]{}&**[32]{}&$\cdots$&**[40]{}&**[Total]{}\ ************************** 76965 &38243 &16150 &5780 &1897 &510 &145 &23 &9 &1 &0 &$\cdots$ &0 &1004100\ -------------------- -------------- ----------------------------------------------------------------------- -- -- -- -- -- -- -- -- \#(Real Solutions) [**Fail**]{} **0 & **2& **4& **6& **[8]{}& **[10]{}& **[12]{}& **[14]{}& **[16]{}\ Frequency &81 &6162 &2628 &2377 &2306 &2275 &2272 &2275 &2383 &2473\ ****************** -------------------- -------------- ----------------------------------------------------------------------- -- -- -- -- -- -- -- -- : Number of real points on witness sets of $\operatorname{SO}(5)$[]{data-label="fig:SO5Real"} [|c|c|c|c|c|c|c|c|c|c|c|c|c|c]{} **[18]{}& **[20]{}& **[22]{}& **[24]{}& **[26]{}& **[28]{}& **[30]{}& **[32]{}& **[34]{}& **[36]{}& **[38]{}& **[40]{}& **[42]{}& **[44]{}\ **************************** 2497 &2527 &2504 &2485 &2280 &2009 &1755 &1644 &1331 &1051 &802 &591 &468 &362\ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- **[46]{}& **[48]{}& **[50]{}& **[52]{}& **[54]{}& **[56]{}& **[58]{}& **[60]{}& **[62]{} &**[64]{} &**[66]{}&**[68]{}&**[70]{}&**[72]{}&**[74]{}&**[76]{}&**[78]{}&$\cdots$&**[384]{}&**[Total]{}\ 235 &150 &118 &60 &44 &21 &16 &8 &4 &3 &1 &0 &0 &0 &0 &2 &0 &$\cdots$ &0 &48200\ ************************************** ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- : Number of real points on witness sets of $\operatorname{SO}(5)$[]{data-label="fig:SO5Real"} \[fig:so4histogram\] \[fig:so5histogram\] In each case, we were able to find a witness set which failed to have any real solutions on it. This is unsurprising as $\operatorname{SO}(n)$ is compact over the real numbers. Despite the fact that all witness sets computed for $\operatorname{SO}(4)$ had fewer than $40$ solutions, and for $\operatorname{SO}(5)$, fewer than $384$, there is little evidence suggesting that a non-trivial upper bound for the number of real solutions on a witness set of $\operatorname{SO}(n)$ exists. We end with a conjecture. For any $n$, $\operatorname{SO}(n)$ admits some real witness set. Appendix: Macaulay2 Code {#appendix-macaulay2-code .unnumbered} ======================== This section contains code which computes the degree of $\operatorname{SO}(n)$ for various $n$ using Gröbner bases, polynomial homotopy continuation, *MonodromySolver*, and Theorem \[degson\] respectively. They are all done in *Macaulay2*. First, we compute the degree of $\operatorname{SO}(5)$ using Gröbner bases. The computation is done over the finite field $\mathbb{Z}_{101}$ for $\operatorname{O}(5)$ and the result is halved to give the degree of $\operatorname{SO}(5)$. n=5 R = ZZ/101[x_(1,1)..x_(n,n)] M = genericMatrix(R,n,n) J = minors(1,M*transpose(M)-id_(R^n)) degOn = degree J degSOn = degOn//2 Computing the degree of $\operatorname{O}(n)$, rather than $\operatorname{SO}(n)$ directly, is useful because it throws out the polynomial of highest degree in the system. This is especially useful in numerical methods since they perform best with polynomials of low degree. The code below uses the package *NumericalAlgebraicGeometry* to solve the zero dimensional system given by a linear slice of $\operatorname{O}(3)$. The method *solveSystem* employs the standard method of polynomial homotopy continuation. Again, the answer is halved to give $\deg \operatorname{SO}(3)$. loadPackage "NumericalAlgebraicGeometry" n = 3 L = toList apply( (0,0)..(n-1,n-1), (i,j)->"x"|toString i|toString j ) R = CC[L] M = genericMatrix(R,n,n) B = M*(transpose M) - id_(R^n) polys = flatten for i from 0 to n-1 list( for j from i to n-1 list B_(i,j) ) linearSlice = apply( binomial(n,2), i->random(1,R)-random(CC) ) S = solveSystem(polys|linearSlice); degOn = #S degSOn = degOn//2 Next, we provide code that computes the degree of $\operatorname{SO}(7)$ using the package *MonodromySolver*. We again do not include the determinant condition, but this time we do *not* need to halve the result. This is because our starting point, the identity matrix, lies on $\operatorname{SO}(7)$ and this method only discovers points on the irreducible component corresponding to our starting point. The linear slices are parametrized by the $t$ and $c$ variables which are varied within the function *monodromySolve* to create monodromy loops. The method stops when ten consecutive loops provide no new points. Although it is possible that this stopping criterion is satisfied prematurely, in our case the program stopped at the correct number, serving as a testament to the practicality of the software and also this stopping criterion. loadPackage "MonodromySolver" N=7 d=binomial(N,2) R=CC[c_1..c_d,t_(1,1,1)..t_(d,N,N)][x_(1,1)..x_(N,N)] M=genericMatrix(R,N,N) B=M*transpose(M)-id_(R^N) polys=flatten for j from 0 to N-1 list( for k from j to N-1 list B_(j,k) ); linearSlice=for i from 1 to d list( c_i+sum( flatten for j from 1 to N list( for k from 1 to N list t_(i,j,k)*x_(j,k) ) ) ); G = polySystem join (polys,linearSlice) x0coords = flatten entries id_(CC^N) setRandomSeed 0 (p0, x0) := createSeedPair(G,x0coords) elapsedTime (V,npaths) = monodromySolve(G,p0,{x0},NumberOfNodes=>2,NumberOfEdges=>4); --node1: 111616 --node2: 111616 -- 42790.9 seconds elapsed Finally, for the mathematician wanting to compute the degree of $\operatorname{SO}(n)$ quickest, we give code that evaluates the formula in Theorem \[degson\]. degSO = method() degSO(ZZ) := N ->( n := N//2; M := matrix for i from 1 to n list ( for j from 1 to n list ( binomial(2*N-2*i-2*j,N-2*i) ) ); 2^(N-1)*(det M) ) This article was initiated during the Apprenticeship Weeks (22 August-2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. The authors are very grateful to Jan Draisma for his tremendous help with understanding Kazarnovskij’s formula, and to Kristian Ranestad for many helpful discussions. The authors thank Anton Leykin for performing the computation of $\operatorname{SO}(7)$. The first three authors would also like to thank the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany for their hospitality where some of this article was completed. The motivation for computing the degree of the orthogonal group came from project that started by the fifth author at the suggestion of Benjamin Recht. The first author was supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1106400. The second author was partially supported by the NSF GRFP under Grant No. DGE-1256259 and the Graduate School and the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation.

Better versus worse family therapy sessions as reflected in clients' alliance-related behavior. To be responsive to clients' evaluations of the unfolding therapy process, therapists must first accurately "read" client behavior, a particularly challenging task in conjoint family therapy. In this study, the authors compared client behavior in 28 sessions that one family member and the therapist concurred, on the Session Evaluation Questionnaire (W. B. Stiles & J. S. Snow, 1984), were relatively better or worse than their other sessions. Client behavior was rated from videotapes using the System for Observing Family Therapy Alliances (SOFTA-o; M. L. Friedlander et al., 2006). In contrast to the worse sessions, the better sessions were characterized by significantly greater client Engagement in the Therapeutic Process and Safety within the Therapeutic System. Notably, whereas only the worse sessions had exceptionally poor within-family collaboration, 40% of the better sessions were characterized by mild family conflict. Implications are discussed for building theory on therapist responsiveness and for future research that may benefit practice, training, and supervision.

Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding. We present an inversion algorithm for the retrieval of particle size distribution parameters, i.e., mean (effective) radius, number, surface area, and volume concentration, and complex refractive index from multiwavelength lidar data. In contrast to the classical Tikhonov method, which accepts only that solution for which the discrepancy reaches its global minimum, in our algorithm we perform the averaging of solutions in the vicinity of this minimum. This averaging stabilizes the underlying ill-posed inverse problem, particularly with respect to the retrieval of number concentration. Results show that, for typical tropospheric particles and 10% error in the optical data, the mean radius could be retrieved to better than 20% from a lidar on the basis of a Nd:YAG laser, which provides a combination of backscatter coefficients at 355, 532, and 1064 nm and extinction coefficients at 355 and 532 nm. The accuracy is improved if the lidar is also equipped with a hydrogen Raman shifter. In this case two additional backscatter coefficients at 416 and 683 nm are available. The combination of two extinction coefficients and five backscatter coefficients then allows one to retrieve not only averaged aerosol parameters but also the size distribution function. There was acceptable agreement between physical particle properties obtained from the evaluation of multiwavelength lidar data taken during the Lindenberg Aerosol Characterization Experiment in 1998 (LACE 98) and in situ data, which were taken aboard aircraft.

The present invention relates to data center management, and in particular enclosures for high speed data transport fiber cabling systems. Optical fibers allow for transmission of communications over longer distances and at higher bandwidths than wire cables. Optical fibers are also advantageous for communication systems because signals suffer less loss than wire cables and are immune to electromagnetic interference. Optical fibers are therefore often used for high bandwidth, long distance applications. One of the primary functions of a data center is to provide connections between incoming and outgoing optical fiber connections. A user may desire to use different sized fiber optic modules housing optical fiber connections. At present, such a user installs and removes entire banks of patch panels or trays in order to accommodate different sized fiber optic modules because trays are currently designed to only support one size of fiber optic module. Thus, it may be advantageous to provide a tray that allows different sized fiber optic modules to be installed within the tray without replacing or removing the tray itself.

A known storage cell includes a gas discharging section (safety valve) that opens to prevent, for example, rupture of a cell case when an internal pressure of the storage cell increases due to heat generated by internal short circuiting or the like and reaches a predetermined pressure. For example, PTL 1 discloses a metal battery case including a gas discharging section provided on a bottom surface portion of the case. When the internal pressure of the battery reaches a predetermined pressure, a large opening is formed in the bottom surface portion of the metal battery case. In the case where a plurality of storage cells are connected together to form an electricity storage device (for example, a battery pack), a connecting member is attached to the bottom surface portion of each cell case. In the storage cell of the related art disclosed in PTL 1, a connecting member is welded to the gas discharging section.

Santa Barbara shooting suspect’s ‘Twisted World’ manifesto A lawyer says the family of the Woodland Hills man suspected in a shooting rampage near UC Santa Barbara called police several weeks ago after being alarmed by YouTube videos “regarding suicide and the killing of people.” Rodger is believed to have fired for 10 minutes in streets filled with university students walking, biking and skateboarding in the beach community near Santa Barbara, picking off people one by one in a deadly rampage. Seven people were killed in all, including the shooter.

369 F.2d 488 NATIONAL LABOR RELATIONS BOARD, Petitioner,v.MOORE'S SEAFOOD PRODUCTS, INC., Respondent. No. 15510. United States Court of Appeals Seventh Circuit. Oct. 28, 1966, Rehearing Denied Dec. 23, 1966, Rehearing Denied Dec. 23, 1966,(En Banc) Marcel Mallet-Prevost, Asst. Gen. Counsel, Leonard M. Wagman, Atty., N.L.R.B., Washington, D.C., Arnold Ordman, Gen. Counsel, Dominick L. Manoli, Associate Gen. Counsel, George B. Driesen, Atty., N.L.R.B., for petitioner. Myron E. Ropella, Albert H. Petajan, Roemer & Ropella, Milwaukee, Wis., for respondent, Moore's Seafood Products, Inc. Before KILEY, SWYGERT and FAIRCHILD, Circuit Judges. KILEY, Circuit Judge. 1 The Board has petitioned for enforcement of its order based upon findings of Moore's Seafood Products' unlawful coercion and interference with employee rights in violation of section 8(a)(1) and of refusing to recognize and bargain in good faith with the Union1 in violation of sections 8(a)(5) and 8(a) (1). We enforce the Board's order. 2 The Union was certified as bargaining agent of Respondent's employees October 19, 1962. On March 11, 1963, the Union and Moore's entered into a maintenance of membership agreement to run until March 10, 1964, and thereafter from year to year, unless terminated by either party on written notice not less than sixty days prior to any expiration date. Under the terms of the agreement, Respondent could, 'at least' fifteen days before expiration of the contract or any yearly renewal, inform its employees of their right to continue as Union members or withdraw from the Union. In December, 1963, the Union gave due notice of its desire to reopen the contract and negotiations followed until May 20, u964. Moore's conduct during this period is the basis for the 8(a)(1) complaint. On May 25, Moore's challenged the Union majority and refused to recognize it as bargaining agent. The general counsel's complaint alleging the 8(a)(1) and (5) violations was filed on June 19, 1964. 3 The trial examiner found that Respondent had violated section 8(a)(1) but that it did not violate section 8(a)(5). With respect to the 8(a)(1) violation, the Board sustained the trial examiner's finding based on Respondent's circulation on March 2, 3 and 4, 1964, of a petition disavowing the Union among employees, prohibition of employees' conversations about the Union on their own time, suggestion of an 'inside' union and interrogation of and threats to employees over support of the Union. The only issue presented concerning the 8(a)(1) violation is the substantiality of the evidence on the record as a whole. We have considered the objections made, and we hold that there was substantial evidence in the record as a whole to support the findings of the Board.2 4 With respect to the 8(a)(5) violation, the trial examiner found that Moore's refusal to bargain was not unlawful because the Union did not command the support of a majority of the employees on March 1.3 At the hearings Moore's contended that the 'critical date' for determining majority for purposes of the 8(a)(5) claim was March 11, 1964, the day after the contract, as renewed, expired. The examiner decided, however, that because of Respondent's unfair labor practices beginning on March 2, the 'critical date' was March 1, subject to evidence that defections after that date were not caused by the unfair labor practices. Medo Photo Supply Corp. v. NLRB, 321 U.S. 678, 687, 64 S.Ct. 830, 88 L.Ed. 1007 (1944); Joy Silk Mills, Inc. v. NLRB, 87 U.S.App.D.C. 360, 185 F.2d 732, 744 (1954), cert. denied, 341 U.S. 914, 71 S.Ct. 734, 95 L.Ed. 1350 (1951). 5 The trial examiner decided that on March 1 the bargaining unit consisted of 77 employees, finding that all but one of a group referred to as 'laid off employees' /4/ should be excluded. He found that 44 of these 77 were Union members on March 1.5 He further found, however, that in February, prior to any unfair labor practices, six of the Union members had an intention to leave the Union and that this evidence 'disentangled' their subsequent defections from other defections attributable to the unfair labor practices. This finding rested upon his inference of 'unequivocal action' by the six employees, demonstrating their desire not to be represented. Five of these employees had signed a paper repudiating the Union, and the other, as well as two of the signers, had expressed their intentions to leave the Union to the Union steward. Thus, the examiner concluded that only 38 out of 77 supported the Union. 6 The Board disagreed with the examiner and found that Moore's had unlawfully refused to bargain with the Union. The only finding of the Board we need consider is the inclusion of the six employees excluded by the examiner in determining majority.6 The Board found that the six employees took no unequivocal action to repudiate the Union until each wrote a letter of resignation to the Union and gave copies to Respondent and that since these letters were written after March 1 their resignations did not impair the Union majority. If the Board's finding is justified, the 8(a)(5) violation is established. 7 There is no question of credibility of the testimony of the six employees, even though the examiner stated 'I credit their testimony.' Their testimony is uncontroverted. 8 From the uncontroverted testimony, the Board was entitled to draw the inference that the action taken by the six employees was not unequivocal. Universal Camera Corp. v. NLRB, 340 U.S. 474, 494, 71 S.Ct. 456, 95 L.Ed. 456 (1951). It was a reasonable exercise of the Board's expertise to rule that the evidence of disaffection in February was not 'determinative' to disentangle the later unequivocal resignations from the other effects of Respondent's unfair labor practices.7 See Universal Camera Corp. v. NLRB, 340 U.S. 474, 490, 71 S.Ct. 456, 95 L.Ed. 456 (1951). It was within the province of the Board to infer that the destruction of the signed paper and the comments to the Union steward indicated merely an unsettled attitude prior to the unfair labor practices. NLRB v. National Seal Corp., 127 F.2d 776, 779 (2d Cir. 1942). 'It is for the Board, not the courts to determine how the effect of prior unfair labor practices may be expunged.' Franks Bros. Co. v. NLRB, 321 U.S. 702, 704, 64 S.Ct. 817, 818, 88 L.Ed. 1020 (1944). 9 We need not and do not reach the question of what evidence is necessary to rebut the presumption that defections subsequent to unfair labor practices will be attributed to those practices. 10 We need go no further in affirming the Board's decision, reported at 152 N.L.R.B. No. 66 (May 19, 1965). So far as needed to support our judgment and reasoning, we adopt that decision. 11 The order will be enforced. 1 Amalgamated Meat Cutters and Butcher Workmen of North America, Local No. 64 2 We need not discuss the Board's challenged disagreement with the examiner's finding against the claim that Respondent was guilty of surveillance. There is sufficient basis, without that element, for the 8(a)(1) violation 3 The defense of good faith doubt of majority status was not raised in this court 4 These employees were laid off by Moore's because of the termination of the night shift. The factual issue before the trial examiner was whether they had been terminated or merely laid off without a loss of employee rights. The examiner found that with one exception (Libakken) they were terminated prior to March 1 5 The trial examiner referred to '42 or 43' depending on the inclusion of one Schueler. This reference did not include Libakken, an admitted Union member who was one of the 'laid off' employees. The exclusion of Libakken from this total is the apparent source of confusion as to the total found by the Board to support the Union. See note 6 infra 6 In determining the size of the bargaining unit, the Board included an additional five of the 'laid off' employees, plus two others, for a total of 84. The number of Union supporters was found to be 49, on the basis of the following: the addition of the seven added to the unit total (who were admittedly Union members); the addition of the six employees excluded by the examiner because of their preunfair-labor-practice intention to leave the Union; and the subtraction of one employee included by the examiner but found by the Board to have joined the Union after March 1. Although there seems to be a disparity of one employee in the computations of the Board and the examiner (our computations based on the Board inclusions and exclusion put the number at 50), there is no need to consider it. If the Board was correct in including the six employees, the Union had a majority (at least 42 of 77, excluding Schueler and Libakken) without regard to the other findings 7 The Board's continued reference to the 'critical date' somewhat clouds the issue in this case. The rule supporting the choice of the 'critical date' is that an employer may not assert a lack of majority status as a defense to an 8(a)(5) violation when the defections are due to his previous unfair labor practices. Medo Photo Supply Co. v. NLRB, 321 U.S. 678, 687, 64 S.Ct. 830, 88 L.Ed. 1007 (1944); Joy Silk Mills, Inc. v. NLRB, 87 U.S.App.D.C. 360, 185 F.2d 732, 744 (1950), cert. denied, 341 U.S. 914, 71 S.Ct. 734, 95 L.Ed. 1350 (1951). Our holding in this case is confined to the Board's refusal to consider 'equivocal' evidence as determinative of the cause of the defections of the six employees involved

I slipped away from my normal mustang restore, because it was a really good deal, and bought this 1982 Corvette Collectors Edition needing some TLC. Now I'm done and after I enter it in the Dodge City Days Classic Car Show this Saturday I will put it up for trade on a good mustang.

Armenian Uruguayans Armenian Uruguayans number around 15,000-19,000 of the population, making Uruguay to have one of the largest Armenian populations around the world. The Armenian community in Uruguay is one of the oldest communities in South America, with most of them residing in the capital Montevideo. History One of the Armenian diaspora's smaller communities, Armenians in Uruguay are concentrated mostly in the capital city, Montevideo. Many of them are third- or even fourth-generation descendants of the first wave of immigrants coming from the Ottoman Empire between the end of the 19th century and the Armenian Genocide. The Armenian General Benevolent Union (AGBU) established a chapter in Uruguay in 1939 and inaugurated a community center complex in 1953. Recognition of the Armenian Genocide by various world parliaments was spearheaded by Uruguay's Parliament, when in 1965 it became the first country in the world to recognize the Genocide. The Parliament has subsequently consistently supported various resolutions in favor of the Armenians. Community Between 1974 and 1975, the AGBU Uruguay Chapter established an educational center which was completed in two phases: first to be completed was the Nubarian Elementary School in honor of the founder of AGBU, Boghos Nubar; then came the Alex Manoogian High School, named after the then AGBU President. The Armenians are very active in the arts. Alvaro Hagopian is the conductor of the Montevideo Philharmonic Orchestra. Also operating is the "Gayane" Dance Group, which belongs to the Armenian National Center. Two long-running radio programs, "Radio Armenia" and "Radio Gomidas," were founded in 1935. The former was closed in 2007 and replaced by "Radio Arax." "Radio Gomidas" continues broadcasting to this day. Montevideo has a public square named Armenia. The Uruguayan Government also has a minister (and former member of Parliament) of Armenian origin, Liliam Kechichian. Religion Most Armenians belong to the Armenian Apostolic Church. The main center is the Armenian Church of Montevideo, Uruguay (). This church has a memorial statue by sculptor Nerses Ounanian, dedicated to the victims of the Armenian Genocide. There is also a significant presence of Armenian Catholics and Armenian Evangelicals. The main Armenian places of worship in Montevideo are: St. Nerses Shnorhali Church (Armenian Apostolic) Cathedral of Our Lady of Bzommar (Armenian Catholic) Armenian Evangelical Church (Armenian Evangelical) Notable people Coriún Aharonián – musicologist and composer Pablo Aprahamian - judoka Garo Arakelian – Famous Uruguayan rock star, solo musician, founder and member of La Trampa Avedis Badanian – journalist Dora Bagdassarián – Dean of the Law School of the University of the Republic Nuvart Bezjián – painter Joaquín Boghossian – footballer for Club Plaza Colonia de Deportes Mauro Guevgeozián – footballer for Club Atlético Temperley Liliam Kechichián – government minister Roberto Markarián – Rector of the University of the Republic Sergio Markarián – football coach and retired footballer Vartan Matiossian – scholar of Armenian studies Gabriel Melconian Alvez – swimmer Nerses Ounanian – sculptor and artist Diego Rossi Marachlian – footballer for LAFC Adrian Sarkissian – retired footballer José Luis Satdjian – businessman and politician Asadur Vaneskahian – journalist and news anchor Christian Yeladian – footballer for Club Sportivo Cerrito Gabriel Melconian — Swimmer of the Olympics Games London of 2012 Martín Melconian — Swimmer of the Olympics Games of Rio 2016 Oriana Aharonian — Pro signer and writer of her own songs. Also plays guitar and piano See also Armenian diaspora Armenian-Uruguayan relations References External links AGBU Web site for Armenians of Uruguay Tigran Ghanalanyan, Armenian Protestant communities in South America, http://noravank.am/eng/issues/detail.php?ELEMENT_ID=5722 Armenian-Uruguayan cultural identity Uruguay Category:Asian Uruguayan Category:Ethnic groups in Uruguay Category:Immigration to Uruguay Category:Armenia–Uruguay relations

Many people are aware of the law enforcement and tactical training that Gunsite Academy is known for…but they also cater to hunters seeking to hone their skills. In this video, Gunsite instructor Il Ling New discusses some of the skills and features of this interesting course. Whether you consider yourself an expert hunter or this hunting season will be your first, it’s always important to review safety rules. Il Ling New reviews hunter safety in this week’s Tips & Tactics video. “Any team member, needs to talk to other team members so that they have a plan formulated. I doesn’t have to be a very complex plan–it can be the ‘baby-steps’ of a plan. The important thing is that everybody has a plan, a common plan in mind, and the first steps of that plan in mind, so that they can execute together. And one thing a woman has to keep in mind also is that she does not need to, nor should she, rely completely upon–say–her husband, or her son, or the man of the team to give her direction. She needs to be able to think, and react, and act on whatever information she’s getting, just as much as he does.” It’s unfortunate, but the fact is that the holster industry tends to give women short shrift when it comes to supplying gear appropriate for their concealment needs. In this Sheriff’s Tip from American Guardian Television, Il Ling and Sheriff Jim Wilson discuss the special issues women face when choosing a concealed carry rig that works for them.

Christiane Karg Christiane Karg (born 6 August 1980) is a German operatic soprano. The award-winning singer became known for performing Mozart roles at the Salzburg Festival, and made an international career. Career Born in Feuchtwangen, Bavaria, Karg studied at the Mozarteum, voice with Heiner Hopfner and Lied with Wolfgang Holzmair. She studied the Italian repertory for half a year at the conservatory of Verona. She graduated at the Mozarteum in 2008 and was awarded the Lilli Lehmann Medal. She took master classes with Grace Bumbry, Mirella Freni, Robert Holl and Ann Murray, among others. Karg made her debut at the Salzburg Festival in 2006, as Melia in Mozart's Apollo et Hyacinthus and as Weltgeist in his Die Schuldigkeit des ersten Gebots. A year later she appeared there as Madame Silberklang in his Der Schauspieldirektor and in a title role of his Bastien und Bastienne. From the 2008/09 season, Karg was a member of the Frankfurt Opera where she appeared as Susanna in Mozart's Le nozze di Figaro, as Pamina in his Die Zauberflöte, as Musetta in Puccini's La bohème, Zdenka in Arabella by Richard Strauss, and the title role in Debussy's Pelléas et Mélisande. She appeared as Sophie in Der Rosenkavalier by Strauss in Frankfurt conducted by Sebastian Weigle, at the Semperoper conducted by Christian Thielemann, and at La Scala. She made her debut at the Royal Opera House in 2015 as Pamina in Mozart's Die Zauberflöte. She performed the solo soprano part in Mahler's Second Symphony in a performance of the Rheingau Musik Festival 2017 at Eberbach Abbey, conducted by Christoph Eschenbach, with Gerhild Romberger, the SWR Vokalensemble, and SWR Symphonieorchester. Awards 2007: Neue Stimmen (6th prize) 2008: Special Prize for Oratorio/Lied at the International singing competition "Tenor Viñas" of the Liceu in Barcelona 2008: Award of the Hamel Foundation at the Schleswig-Holstein Musik Festival 2009: Opernwelt Young Artist of the Year 2009 2010: Winner of the ECHO Klassik 2010 in the category "Young Artist", "singing" 2016: Winner of the ECHO Klassik 2016 in the category "Solistische Einspielung" (solo recording) 2018: Brahms-Preis Literature Ursula Ehrensberger: Das Porträt – Christiane Karg. In: Das Opernglas 2010, No. 3, , pp 26–30. References External links Christiane Karg / Biografie Gasteig Christiane Karg (Soprano) Bach Cantatas Website Category:1980 births Category:Living people Category:People from Ansbach (district) Category:German operatic sopranos Category:Mozarteum University Salzburg alumni Category:21st-century German opera singers Category:21st-century women singers

By Kevin Kelly (KK) Church music Halifax violinist Chris Church ventures beyond the notes, and his North American tour, to return home for a one-night Jazz Fest show. Chris Church is sweating out a three-month tour around the continent. It started on May 4 in Scottsdale, Arizona, including stops in such far-flung places as Rouyn-Noranda, Quebec, and Pomona, New Jersey, winding up in Boulder, Colorado, on July 29. Halifax isn’t among the 25 stops listed on the tour schedule at chrischurch.ca, but all the same he will be here on July 20 for a show at Ginger’s, as part of the Atlantic Jazz Festival. You see a North America tour schedule and think it might be all in place, but, says Church on the phone from Toronto, “It’s fucking hard to get it together to come down and play.” In the course of the few days last week when I speak with Church on the phone he is variously in Toronto, on a bus from Montreal to Ottawa, actually in Ottawa and at Pearson Airport in Toronto, flying to Philadelphia to play in Atlantic City. He’s touring with different musicians; with Jesse Cook at the Montreal Jazz Festival and in PEI, just before he hits Halifax. Also joining him here are flamenco guitarist Nicolas Hernandez and the Egyptian-Canadian vocalist Maryem Hassan Tollar, who works often with composer Christos Hatzis, plus Tom Roach and Tony Tucker. “It’s all insanely busy,” says Church. “On Canada Day I played on Parliament Hill and took a charter to Toronto that evening and to Montreal the next day to rehearse for a DVD and then did six shows in three days at the Montreal Jazz Festival, with Jesse Cook and guests like Ron Sexsmith, Sophie Millman and Melissa McClelland.” Halifax is his hometown. Church’s violin career began here, in grade four or five at Sir Charles Tupper school. He joined the Halifax Schools string program, where kids take violin, viola, cello or bass, mostly as part of their school day. Pat Wyman was his first teacher. “What I remember,” she says, “is his wire-rim glasses. He was small and really keen and he had those glasses. He showed a real ability and it was recommended that he take private lessons.” Church did, from Symphony Nova Scotia violinist Anne Rapson. He was part of a crew that came through the program in those years, including Paula Caballero, John Scott, Molly Read, Laura Conrad, Gillian Smith, Shen Tan (who is principal second violinist in Oregon) and Debbie Miles (who now plays in the PEI Summer Festival and as an extra for SNS). Church has three brothers. “Everybody plays something,” he says, “and my parents are very supportive.” Church went to Dalhousie for a degree in violin performance. In the summers he studied at the Cleveland Institute of Music and the Meadowmount School for Strings in upstate New York. He played with SNS off and on for eight or nine years, and did some orchestral arranging. “I just whored around,” laughs Church, “for SNS and Big Picture and Lennie Gallant.” He moved to Toronto, continuing to write arrangements for SNS and other Canadian orchestras. In Toronto, Church found sounds and influences to add to his foundation in classical music. He became more and more interested in improvisation. “I was drawn towards other musical styles,” he says. “I got into jazz, pop, rock and more avant-garde improvisation with a range of talented musicians. I liked performing classical music but I was more passionate about music when I improvised in these other groups.” The composer in him didn’t surface until late. “I am a latent bloomer,” he laughs. “The composer was always there in me but I never put time into him until my late 20s when I found more peace.” Church will turn 34 on October 16. “It was kind of frustrating,” he says. “For a long time the stuff I tried to convey through the violin was roadblocked by my classical training, but as I lived more life through music and learned to express myself on violin, it got better.” In the winter of 2003 he decided to start working on a record that would reflect the last few years of his life. That CD, here, came out in 2004. Church describes it as being songs of himself. “The songs on here are stories, moments, thoughts and various ideas that were going on in my life at that time. I coloured these stories with the sounds that surrounded me as I toured the world and collaborated with other artists.” here contains styles from bossa nova to Irish lament, sounds from the Middle East to more familiar sounds from back home in Halifax and the co-writing input of Ian Janes and Gordie Sampson. Church continues to grow as a musician. “I’m only now beginning to feel that I sing on the violin,” he says. “To tap the surface. Being on stage is such a joy, I try not to think about anything and to let it come naturally.” He plays with the improvisation group SuddenlyLISTEN with the likes of Dani Oore and Norman Adams. “Classical music is ingrained in my head but now I can leave it there and really go for it.” Church is passionate and forceful as he talks, and the same on stage. To see and hear Chris Church play the violin is to see and hear the work of someone with a terrific amount of what classical musicians call intent, the going beyond the page, going beyond the notes in performance and giving up more than the head and the muscles to the making of music and its delivery. Intent, at a basic level of string instrument work, is to put the bow to the strings intending to make a particular sound and then making it, like calling the shot in a game of pool and then sinking the ball. This technique, getting out the intended notes (or a reasonable facsimile) will take a musician far enough; often a display of high notes or fast notes or loud notes gives an audience something it feels is out of the ordinary, but it’s a deeper and higher calling to give something extraordinary—to intend particular inflection and phrasing—to do this, musicians must venture further within themselves, further off the map that is the page of musical notation. Church does this. He calls on all of himself when he performs. He scouts deep into himself—heart, soul, gut and entrails—to mine a glittering nugget of emotion. And then he brings that chunk to the surface and delivers it into the audience. He moves with the music, standing, swaying, flying. “I fucking sweat a lot,” he says. “On the DVD shoot probably 17 people were playing. I was the only one with matted, wet, sweaty hair. I’m on a four towel system.” Church thinks for a moment. “That’s what I need,” he laughs. “I need a towel check! Like I need both a sound check and a towel check.”

355 F.Supp. 314 (1972) Modesto Montañez ALAMO, Plaintiff, v. Elliot RICHARDSON, Secretary of Health, Education and Welfare, Defendant. Civ. No. 921-70. United States District Court, D. Puerto Rico. December 22, 1972. *315 James A. Toro, San Juan, P. R., for plaintiff. Julio Morales-Sanchez, U. S. Atty., for defendant. TOLEDO, District Judge. ORDER This is an action brought by plaintiff pursuant to Section 205(g) of the Social Security Act (hereinafter called the Act), Title 42, United States Code, Section 405(g), seeking a judicial review of a final decision of the Secretary of Health, Education and Welfare (hereinafter named the Secretary), holding that he is not entitled to a period of disability and to disability insurance benefits, under Sections 216(i) and 223 of the Act, Title 42, United States Code, Sections 416(i) and 423. Plaintiff, a 55 year old man with first grade education, alleged before the Secretary that he became unable to work in 1962, due to impaired discs and arthritis.[1] The Secretary determined plaintiff met the special earnings requirements at the date he alleged his impairments originated,[2] but found he was not disabled within the meaning of the Act. All of the administrative remedies were exhausted when the decision of the hearing examiner became the final decision of the Secretary, once the Appeals Council, on September 4, 1970, approved said decision. The statutory scheme of judicial review being limited in nature, this Court is bound to ascertain only whether the record contains substantial evidence to support the Secretary's findings. Santiago v. Secretary of Health, Education and Welfare (D.C.P.R.1971), 336 F.Supp. 1071; Rosario v. Secretary of Health, Education and Welfare (D.C.P. R.1971), 324 F.Supp. 1321. The plaintiff asserts he has been unable to work since 1962, with the exception of 2 or 3 days in 1965 (when he worked only four hours); that the record before the Court is less than impartial, occasioned principally by the absence of a counsel to represent him before the Secretary; and that the said absence of counsel representation has prevented him from meeting his evidentiary burden and properly presenting his case before the Secretary. Plaintiff also asserts that the hearing examiner committed abuse of discretion and, hence, reversible error, for he failed to act as an impartial arbitrator, seeing that all relevant facts were fully developed during the course of the hearing. Lastly, plaintiff contends that the hearing examiner's decision was not based on substantial evidence, for exclusion was made of some documentary evidence plaintiff intended to introduce at the hearing; for *316 the hearing examiner made misleading unqualified findings of facts regarding his education and ability to read, count, give change and solve simple arithmetic problems without any foundation on the record; for the hearing examiner has concluded he had no significant mental impairment on or before March 31, 1965, based on a document which merely specifies the date he began to receive treatment (see note 1); and for the hearing examiner found he was able to do his prior job on or before March 31, 1965, when the record show the contrary. Plaintiff submits that the hearing examiner's failure to explore the facts adequately, as well as his making findings of fact which are clearly unsubstantiated by any evidence taking the record as a whole, constitutes reversible error, thereby entitling him to judgment in his favor. In the alternative, plaintiff requests the Court to remand the cause to the Secretary for a rehearing, where he would present evidence that will establish his physical and mental condition during the critical period in which he last met the earnings requirements (see note 2); at which rehearing, he will be represented by his appointed counsel, for this cause. It is defendant's basic position that the only medical evidence relating to the period during which plaintiff was insured for disability purposes is a report from the Puerto Rico Industrial Commission dated April 17, 1963, and that the remaining medical evidence relates to examinations made after March 31, 1965, when plaintiff's insured status expired. Defendant adds that the pertinent medical evidence, when considered with the non-medical evidence, fails to satisfy the disability requirements of the law; that is, that said evidence fails to establish that plaintiff is unable to engage in any substantial gainful activity by reason of a medically determinable physical or mental impairment, which can be expected to result in death or which has lasted or can be expected to last for a continuous period of not less than 12 months; and which can be demonstrated by evidence supported by objective data obtained by medically acceptable clinical and laboratory techniques. With respect to the medical evidence subsequent to the date plaintiff last met the insured status requirements, defendant contends that said evidence does not entitle plaintiff to disability benefits, for at most, it can reflect that plaintiff's impairment reached disabling severity after the expiration of his insured status, and that for said reason, is not relevant, even though the impairment itself may have existed before plaintiff's insured status. In relation to plaintiff's lack of legal representation at the hearing before the hearing examiner, it alleges that it is no ground for remand; that plaintiff was notified of his right to be represented by counsel, thus afforded due process by the defendant, but apparently, chose to proceed with the hearing without counsel; and that plaintiff has failed to show that he was prejudiced or that the proceedings were unfair. Finally, defendant asserts there was no abuse of discretion in the hearing examiner's development of the evidence. Although we are aware that the burden is upon the plaintiff to prove he is entitled to the benefits of the Act, Reyes Robles v. Finch (1 Cir. 1969), 409 F.2d 84; De Jesús Faría v. Finch (D. C.P.R.1971), 336 F.Supp. 1069; Torres v. Secretary of Health, Education and Welfare (D.C.P.R.1971), 333 F.Supp. 676, and not upon the Secretary to make an initial showing of nondisability to perform a job, De Jesús Faría v. Finch, supra; de la Cruz v. Secretary of Health, Education and Welfare (D.C.P. R.1971), 331 F.Supp. 522, we are of the opinion that said rules, even though being very convenient for the expediency of process on the administrative level, should be applied with caution in cases like the one before us. Román v. Secretary of Health, Education and Welfare (D.C.P.R.1972), 355 F.Supp. 646 (Memorandum Opinion and Order of November 15, 1972). We have before us a plaintiff with only first grade education, *317 who can hardly understand the administrative process before the defendant, who was the subject of a hearing which, although held in the Spanish language, was in our opinion, too technical for him to fully understand, especially being him at the time of the hearing under treatment for a mental condition and not represented or assisted by counsel. In Leon v. Secretary of Health, Education and Welfare (D.C.P.R.1971), 337 F.Supp. 905, we dealt with a situation similar to the present one. Likewise, in Román v. Secretary of Health, Education and Welfare, supra. In this respect, it suffices to say that when a claimant appears without counsel at the administrative hearing and the presence of a mental impairment is obvious, be it because it is observable by the hearing examiner or because medical evidence is presented in that respect, it is the duty of the hearing examiner to adequately explore all aspects of the claim before him. Otherwise, legitimate claims, such as mental impairments, could unjustly go unattended for failure to properly act upon them. We are of the opinion that in this case the hearing examiner failed to adequately explore all aspects of the claim. It appears to us that although the physical impairments alleged by the plaintiff are his main complaint, the record shows that his mental condition may be a significant factor in determining the severity of his disability. Román v. Secretary of Health, Education and Welfare, supra. The evidence in record before us, in relation to the mental condition of plaintiff, is insufficient for us to conclude that the record contains substantial evidence to support the findings of the Secretary (see Note 1). Likewise, it is our opinion that in view of the factors we previously mentioned, this plaintiff may have been prevented from adequately presenting his claim before the Secretary. Moreover, plaintiff has satisfied us that with representation of counsel at the rehearing, new medical evidence and a better case will be presented. Toledo v. Secretary of Health, Education and Welfare (1 Cir. 1971), 435 F.2d 1297. Furthermore, by failing to adequately inquire about plaintiff's mental condition, defendant has acted unfairly. Toledo v. Secretary of Health, Education and Welfare, supra; Torres v. Secretary of Health, Education and Welfare (D.C.P.R.1971), 337 F.Supp. 1329. In view of the foregoing, we are of the opinion that a rehearing to further inquire into plaintiff's physical and mental impairments is proper and necessary, at which time the plaintiff is to be assisted by counsel. Wherefore, this Court finds there is good cause to remand under Section 205(g) of the Act, Title 42, United States Code, Section 405(g), and the Court, accordingly, hereby Orders, adjudges and decrees, that the present action be remanded to the Secretary of Health, Education and Welfare, with specific instructions that the complete psychiatric record of the plaintiff be made available and made part of the record; that a consultative psychiatric examination be made if necessary to clearly establish whether the mental impairment constitutes a disability as defined by the Act at the time plaintiff last met the insured status requirements; that any necessary and appropriate examinations with regard to his physical impairments be made; and that the defendant carry out any other further proceeding they may deem proper consistent with this memorandum opinion. The Court also deems proper to recommend that plaintiff be assisted at the rehearing by his appointed counsel on this cause. It is so ordered. NOTES [1] The record before the Court reveals plaintiff has been receiving psychiatric treatment at the Caguas Mental Health Clinic since September 9, 1969, but no mention is made as to the onset date of the mental impairment, the diagnosis and prognosis, etc. [2] Plaintiff met the earnings requirement through the quarter ending March 31, 1965. Therefore, on the basis of his applications of November 28, 1967, plaintiff must establish that he was under a disability which commenced prior to March 31, 1965, when he last met the special insured status requirements.

Q: Join two of the same tables to another table and output the info of the (same) table in the same row Sorry for the bad/long title but I don't know how else to put it. What I want to do is join to 'A' tables and join it to the 'B' table where both 'A' have a foreign key in common and display info from both 'A' tables in the same row while preventing duplicates such as the example in the pic: I know the query is just doing it's job, but is there a way to prevent 'duplicates' by comparing between the rows before output? Here's what I tried, I know it may be bad performance-wise and there may be better ways but this is for a mini-project with a small DB, where performance shouldn't really matter: SELECT w.emp_id AS emp1_id, w2.emp_id AS emp2_id, e.fname || ' ' || e.lname AS emp1_name, e1.fname || ' ' || e1.lname AS emp2_name, e.jobtitle AS emp1_jobtitle, e1.jobtitle AS emp2_jobtitle, e2.fname || ' ' || e2.lname AS cs_name FROM work_on w LEFT JOIN work_on w2 on w.emp_id != w2.emp_id and w.ticket_id = w2.ticket_id LEFT JOIN employee e on w.emp_id = e.emp_id LEFT JOIN employee e1 on w2.emp_id = e1.emp_id LEFT JOIN ticket t on t.ticket_id = w.ticket_id LEFT JOIN customer_problem p on p.problem_id = t.problem_id LEFT JOIN employee e2 on e2.emp_id = p.emp_id WHERE e2.emp_id = 20 and p.submit_date >= '2018-04-08' and p.submit_date <= '2018-04-11' and e1.emp_id != e.emp_id ORDER BY w.emp_id; My tables: Employee: | Work_On: | Ticket: | Problem ----------+------------+--------------+------------ emp_id work_id ticket_id problem_id fname emp_id problem_id emp_id lname ticket_id In this case I'm trying to combine two Employee on Work_On where they have the Ticket in common and another Employee which connects to the ticket via the Problem table. A: Here is one option using least/greatest: SELECT DISTINCT LEAST(w.emp_id, w2.emp_id) AS emp1_id, GREATEST(w.emp_id, w2.emp_id) AS emp2_id, LEAST(e.fname || ' ' || e.lname, e1.fname || ' ' || e1.lname) AS emp1_name, GREATEST(e.fname || ' ' || e.lname, e1.fname || ' ' || e1.lname) AS emp2_name, LEAST(e.jobtitle, e1.jobtitle) AS emp1_jobtitle, GREATEST(e.jobtitle, e1.jobtitle) AS emp2_jobtitle, e2.fname || ' ' || e2.lname AS cs_name FROM work_on w LEFT JOIN work_on w2 ON w.emp_id != w2.emp_id AND w.ticket_id = w2.ticket_id LEFT JOIN employee e ON w.emp_id = e.emp_id LEFT JOIN employee e1 ON w2.emp_id = e1.emp_id LEFT JOIN ticket t ON t.ticket_id = w.ticket_id LEFT JOIN customer_problem p ON p.problem_id = t.problem_id LEFT JOIN employee e2 ON e2.emp_id = p.emp_id WHERE e2.emp_id = 20 AND p.submit_date >= '2018-04-08' AND p.submit_date <= '2018-04-11' AND e1.emp_id != e.emp_id ORDER BY w.emp_id; To see why the least/greatest trick works, consider the following two records/columns: emp1_id | emp2_id 2 | 15 15 | 2 It should be clear that while these records are distinct now, if we instead choose the least id followed by the greatest id, they appear identical: LEAST(emp_id1, emp_id2) | GREATEST(emp_id1, emp_id2) 2 | 15 2 | 15 Then, using SELECT DISTINCT removes one of the two duplicate rows.

Sunday, October 25, 2009 Good Morning!! Welcome to our 10th challenge at Stampin' Sisters in Christ. We are so blessed that so many of you participate each week, we just want to thank you for your support. Our sponsor this week is the Rubber Stamp Shack , owned by Barbara and John DiPrizio. They are graciously donating the Scriptures stamp sheet #16. Thank you for that donation.Our host for this week is Laurie . She is one of our newest DT sisters and we are blessed to have her on board. Laurie chose the verse from I Chronicles 16:34“Give thanks to the Lord, for he is good; his love endures forever.”The challenge for you this week is to make a card that expresses your thankfulness for someone or something they have done or to make a card that encourages someone to be thankful. This is a wonderful challenge that I know many people will enjoy doing. Each one of us has someone in our lives that we want to express our thankfulness to. I know I do:). So here is my project I came up with... I made this watering can from the SCS weekly inkling #181. I found this tutorial here. It's just the perfect gift to give someone that I'm thankful for. I know they'll enjoy this. I added pearls at the bottom and on each point of the flourish. I punched out some flowers and added some 'dum dum' suckers to complete the watering can. Now it's your turn to create an amazing card/project. Can't wait to see what you do. Please visit the SSCC blog and see what the other DT sisters have created. Kathy, this is AWESOME! I was just looking at the watering can tutorial and gallery on SCS last week but yours is the cutest one I've seen so far. Wouldn't these make precious Mother's Day gifts, too? Love it! I've got to add this to my list of techniques to try. Have a great day! THIS Totally ROCKS Kathy! LOVE all your details on it!! What a fantastic idea for a gift and I just love the fun colors and everything. YUM-O! =) OOOO, yes, this would be perfect for a Mother's Day gift!! I'm it! I called it first!! LOL - just kidding. Actually, I just might if you don't make it over here to give it in person, cause... I'm pretty sure this would not mail well, LOL But it was a great thought! Love ya sis! ♥ This is an amazing project, I could hardly wait for it to appear on the blog once I saw it in the gallery, so everyone could enjoy it! The pearls and detail and color on it, wonderful! It is bursting with joy and will make someone very happy as a thank you remembrance.

Q: Using anonymous functions in Haskell CIS194 Week 2 Exercise 2 I am trying to learn haskell through the "CIS 194: Introduction to Haskell 2013" course and have run into a snag with exercise 2 in the second week. I am trying to define my build function using an auxiliary recursive function reduce :: (a -> b -> b) -> [a] -> b -> b reduce _ [] x = x reduce f (x:xs) y = reduce f xs (f x y) The goal of this function is to recursively apply a function on the list [a] to the b. This is because I want to recursively insert a list of LogMessage's into a MessageTree build :: [LogMessage] -> MessageTree build [] = Node Leaf (Unknown "") Leaf build messages = reduce (\x y -> (insert x y)) tail(messages) (Node Leaf head(messages) Leaf) but when I try and compile this, I am receiving a long list of compilation errors: LogAnalysis.hs:42:22: Couldn't match expected type `t0 -> MessageTree' with actual type `MessageTree' In the return type of a call of `insert' Probable cause: `insert' is applied to too many arguments In the expression: (insert x y) In the first argument of `reduce', namely `(\ x y -> (insert x y))' LogAnalysis.hs:42:31: Couldn't match expected type `MessageTree' with actual type `t0 -> MessageTree' In the second argument of `insert', namely `y' In the expression: (insert x y) In the first argument of `reduce', namely `(\ x y -> (insert x y))' LogAnalysis.hs:42:35: Couldn't match expected type `[LogMessage]' with actual type `[a0] -> [a0]' In the second argument of `reduce', namely `tail' In the expression: reduce (\ x y -> (insert x y)) tail (messages) (Node Leaf head (messages) Leaf) In an equation for `build': build messages = reduce (\ x y -> (insert x y)) tail (messages) (Node Leaf head (messages) Leaf) LogAnalysis.hs:42:40: Couldn't match expected type `t0 -> MessageTree' with actual type `[LogMessage]' In the third argument of `reduce', namely `(messages)' In the expression: reduce (\ x y -> (insert x y)) tail (messages) (Node Leaf head (messages) Leaf) In an equation for `build': build messages = reduce (\ x y -> (insert x y)) tail (messages) (Node Leaf head (messages) Leaf) LogAnalysis.hs:42:51: Couldn't match expected type `MessageTree -> t0' with actual type `MessageTree' The function `Node' is applied to four arguments, but its type `MessageTree -> LogMessage -> MessageTree -> MessageTree' has only three In the fourth argument of `reduce', namely `(Node Leaf head (messages) Leaf)' In the expression: reduce (\ x y -> (insert x y)) tail (messages) (Node Leaf head (messages) Leaf) LogAnalysis.hs:42:61: Couldn't match expected type `LogMessage' with actual type `[a1] -> a1' In the second argument of `Node', namely `head' In the fourth argument of `reduce', namely `(Node Leaf head (messages) Leaf)' In the expression: reduce (\ x y -> (insert x y)) tail (messages) (Node Leaf head (messages) Leaf) LogAnalysis.hs:42:66: Couldn't match expected type `MessageTree' with actual type `[LogMessage]' In the third argument of `Node', namely `(messages)' In the fourth argument of `reduce', namely `(Node Leaf head (messages) Leaf)' In the expression: reduce (\ x y -> (insert x y)) tail (messages) (Node Leaf head (messages) Leaf) Failed, modules loaded: Log. For reference, here is my insert function, which compiles correctly insert :: LogMessage -> MessageTree -> MessageTree insert (Unknown _) tree = tree insert _ tree@(Node _ (Unknown _) _) = tree insert message Leaf = Node Leaf message Leaf insert message@(LogMessage _ timeStampMessage _) (Node left m@(LogMessage _ timeStampTree _) right) | timeStampMessage < timeStampTree = Node (insert message left) m right | timeStampMessage > timeStampTree = Node left m (insert message right) | timeStampMessage == timeStampTree = Node left message right A: Parentheses do not indicate function calls; they simply affect operator precedence. Function application is represented by juxtaposition (which can be thought of as the operator with highest precedence). This line build messages = reduce (\x y -> (insert x y)) tail(messages) (Node Leaf head(messages) Leaf) should first be shortened to build messages = reduce insert tail(messages) (Node Leaf head(messages) Leaf) which makes it easier to see the correct way to parenthesize: build messages = reduce insert (tail messages) (Node Leaf (head messages) Leaf) This is, however, better written using pattern matching than using head and tail: build (x:xs) = reduce insert xs (Node Leaf x Leaf)

Credit card payments evolve beyond the mobile wallet Mobile wallets can make paying by credit or debit card seamless: Tap your phone at checkout and you're on your way. But mobile wallets are just the beginning. Payment networks and manufacturers are building payment functions into more devices — expanding your options as well as freeing up your hands. You could find yourself buying gas from the dashboard of your car, groceries from your refrigerator door or dinner by flashing a smile. And you won't even need your phone with you to make purchases on the go. MORE DEVICES ADD PAYMENT CAPABILITY Payment options already available or on the horizon include: — Wearables. Connected "smart" accessories such as watches, bands and rings travel lighter than a phone. To use, the wearer holds a wrist or hand up to a contactless payment terminal. Visa tested these devices at the 2016 Rio Olympics to demonstrate possibilities, says Mark Jamison, global head of innovation and design at Visa. The market will determine, he says, if fashion designers want to "embed payments into a ring or any other device." One company privately testing similar tech is Token, whose smart ring — which performs a variety of functions, from opening doors to paying for purchases — has a waiting list. In 2017, payment capabilities branched out from Apple and Android smartwatches to some Fitbit and Garmin fitness devices, meaning more people could leave their phone behind while working out. By the end of this year, Visa expects merchants to have tap-to-pay capability at 50 percent of U.S. locations where face-to-face transactions take place. — Virtual assistants. When voice payments are enabled on virtual assistants like Amazon Echo and Google Home, you can multitask and take care of "errands" in the moment with verbal commands. CONNECTED DEVICES WILL BE THE NORM Consider the number of mobile applications with saved payment information on your mobile device. In the future, you could free up some data and save a little battery life by using other connected devices: — Cars. Visa and Mastercard are working with manufacturers to embed options in car models. Manufacturers are also testing ways to pay for gas, groceries, takeout, metered parking and other things from screens on vehicle dashboards. "It's still early, but we are focused on bringing that to life this year, to have the ability for you, as the driver, to not just order from one type of merchant," says Stephane Wyper, senior vice president of new commerce partnerships and commercialization at Mastercard. — Appliances. Appliances will get smarter in the future. A glimpse of what's possible: Samsung's Family Hub refrigerator, which lets you order groceries from the Groceries by Mastercard app; Whirlpool's Smart Dishwasher, which, when synced with an Amazon account, can estimate when you're low on detergent and order it automatically. — Your body. Going totally device-free could also become an option. Biometric payments make it possible to pay by voice, face, iris scan or fingerprint. It's not a big stretch from the biometric authentication currently used by some phones or applications. "The technology itself has been around for a while, but consumers were skeptical of it," Jamison says. They've since become accustomed to authenticating using a fingerprint via phone, and their preference has shifted from user ID and password to biometrics, he says. In January 2018, CaliBurger restaurants launched a pay-by-face pilot program in Pasadena, California. Customers pay for their order by smiling for the camera and entering the three-digit number from the back of a credit card. MORE OPTIONS, FEWER PASSWORDS Current devices, apps and websites generally each require a separate profile with payment details. In the future, you could keep payment information in one place and sync it to all devices. In 2017, Mastercard launched Consumer Control to offer consumers a central view of their cards across different channels. With access through their issuer, cardholders can add their cards to their favorite shopping sites and devices from one location. The forecast for the future includes more convenient payment options. Visa estimates that 50 billion smart devices will be connected to the internet by 2020. Merchants won't begin accepting payments from everything overnight; for certain devices it may take a few years. For now, you can start exploring the options outside of your mobile wallet. This article was provided to The Associated Press by the personal finance website NerdWallet. Melissa Lambarena is a writer at NerdWallet. Email: mlambarena@nerdwallet.com. Twitter: @LissaLambarena. RELATED LINKS: Apple Pay, Android Pay and Samsung Pay: What to know https://nerd.me/mobile-payments-roundup The gold standard: Visa introduces the future of digital transactions to the world

Dear Beloved Reader, we're going to be real with you. We're asking you to join our membership program so we can become fully financially sustainable (and you get some cool perks too!) With dropping ad rates across the media industry, we're at continuous risk of shutting down. And we don't want you to face Trump and his kind without the unique resources we provide. If everyone reading this only gave $10, we could raise enough money for the entire year in just one day. That's right, with the price of a single lunch out, you can save us. We're an independent feminist media site, led entirely by people of color, and that pays everyone who writes for us. If Everyday Feminism has been useful to you, please take one minute to keep us publishing the articles you've come to rely on us for. Thank you! Click here to join! everyday feminism What ‘Passing’ Says About Our Expectations of Women – And Why This Trans Woman’s Getting It Wrong on Purpose Originally published on Ravishly and republished here with their permission. “Do I pass?” I get this question every day. From strangers, lovers, and others. Every trans woman holds a pair of broken pistols, one marked “Pride” and the other “Fear.” You never know which one will go off in your hand when you draw them. Some days you twirl around your living room in a flowing floral skirt, espousing the emphatic, maniacal laughter of someone who finally showed them all. And others—you’re in front of a mirror or a computer monitor, inviting interrogation of your face, to document any offending angle of your jaw or amateur application of your makeup. Your queendom for a pointier wing in your eyes. We want to live our truth, but that requires “living,” which may necessitate mimicry. Some women get their hair cut like Tina Fey to look attractive — others to avoid harassment when outside of their homes. I reckon the right-wing gumshoes out there hyped on “exposing” trans women for our wicked deceptions have never felt the hands of a man whose cornered you in a gas station at midnight, combing your neck for stubble with calloused, greasy hands. And they should never have to. No one should ever be punished for not “looking like a woman right”—when a man physically assaults a trans woman, it is (in part) to make her pay for the inflexibility of his sexual identity, that desires but will not allow an escape from a gender binary. Ask any trans woman who is not me and who dates men and you will find yourself inundated with a deluge of doubletalk in realtime, men who insult trans women to their face but check for a ring and a sign there’s “still a chance for him” behind her back. A man has never asked me out, but he has followed me in his car for five blocks to call me a faggot. There is not time enough to explain to him that I actually like being called that in a consensual scenario and therefore the word has no real impact on me before he drives away, high-fiving himself all the way into that hellish normalcy. I do not pass, as such. I tower, even for a “tall girl” (a little insider scoop on trans lady lingo for you). I am tall, muscular, and have a deeper voice than many cis men. I don’t like how I look in jeans, so I don’t wear them. I don’t shave or wear makeup everyday, but I keep the dress and stockings and combat boots, even when I hike along the trails of Santa Cruz. I am a terrific TERF nightmare in the flesh. And off goes the good gun. I am beautiful. I do not “pass” as a cis woman but I am still a woman, even when I don’t do it “right.” But my work puts me in the (grateful) company of straight, cisgender women, who, for a number of reasons — including a lifetime of media educating them to judge other women by their appearances and limit personal contact to terse and superficial compliments on one another’s clothing — use fashion as a primary means of starting conversation with each other. “I am having the stocking envy at you right now.” I am trying to facilitate a friendly moment with you with the tools I have and they aren’t much and I wish it wasn’t this way. Dysphoria (which is the distress or discomfort that occurs when the gender someone is assigned does not align with their actual gender) and a desire to emulate — that dynamic duo of our demons — would keep me hunkered and heartbroken when around other women, if I let it. I can never look like them. I could buy the right clothes, I could excise all extracurricular activities in my life to make way for makeup tutorials, but I will never look like them. I am a woman in a world meticulously manicured to only allow certain women to occupy womanhood. The rest of us — trans women, fat women,women of color, disabled women — are told to camouflage ourselves if we want jobs or loving relationships and then eviscerated when our camouflage is not “convincing,” chided for not being content to just “be ourselves.” I will never do woman right, so I’m getting it wrong on purpose. I’m not covering up my garters anymore. The gloves would come off if I ever wore gloves; you told me women like me didn’t belong in the nail salon so I learned to do them myself and they always look too nice to cover in scratchy wool bodybags. You can jeer and sneer at the little black bands bracing my legs against the rolling tide of cisheteronormativity — but I’m not falling for it. If I hike my dress down to cover them up, you would dock me for wearing a dress in this weather, for “being too fancy” for a day job look, which is totally what some fetishist crossdresser would do, probably, if you had ever actually met one in real life. I don’t mind looking like an apathetic crossdresser. A crossdresser is not a bad thing to be. It’s not who I am, but no amount of patiently explaining to you the nuances of identity and presentation will pry free your arm, slung over your face as if in mid-faint. This is Lilith myth on repeat: You want women to “dress like women” but you are scandalized at the sight of scaffolding. You impose on women to wear bras, but then chide them when their straps show underneath their tops, just as you raise a hackle at me when I don’t shave, when I don’t cover my pimples with makeup, when I over- or under-dress. If I continue to try and mimic “how proper women dress”, I will shudder and shrink myself into utter fatigue. I can’t look like you — but I’ll always look like me. It seems such an obvious conclusion for an adult to make, but look at what I’ve had to work with: “proper, real, natural” men and women, scouring our Facebooks and livejournals to find the most unflattering, most “un-passing” pictures in our galleries, to mock us for not living up to the standards that have eroded their self worth so minutely. As if your haircut and suit and even the swaying of your shoulders while you walk weren’t decided for you by generations and generations of gender architecture. Do I pass? Do I care? Not today. Come back tomorrow to see which pistol signals the day’s races to proceed.

Amphianthus Amphianthus is a genus of sea anemones. It is the only genus in the monotypic family Amphianthidae. Species The following species are recognized: Amphianthus armatus Carlgren, 1928 Amphianthus bathybium Hertwig, 1882 Amphianthus brunneus (Pax, 1909) Amphianthus californicus Carlgren, 1936 Amphianthus capensis Carlgren, 1928 Amphianthus caribaeus (Verrill, 1899) Amphianthus dohrnii (Koch, 1878) Amphianthus ingolfi Carlgren, 1942 Amphianthus islandicus Carlgren, 1942 Amphianthus lacteus (Mc Murrich, 1893) Amphianthus laevis Carlgren, 1938 Amphianthus margaritaceus (Danielssen, 1890) Amphianthus michaelsarsi Carlgren, 1934 Amphianthus minutus (Hertwig, 1882) Amphianthus mirabilis (Verrill, 1879) Amphianthus mopseae (Danielssen, 1890) Amphianthus natalensis Carlgren, 1938 Amphianthus nitidus (Verrill, 1899) Amphianthus norvegicus Carlgren, 1942 Amphianthus radiatus Carlgren, 1928 Amphianthus rosaceus Wassilieff, 1908 Amphianthus sanctaehelenae Carlgren, 1941 Amphianthus valdiviae Carlgren, 1928 Amphianthus verruculatus Carlgren, 1942 References Category:Hormathiidae Category:Hexacorallia genera

Speakers Topics Blog Posts What We Learned from the NY Times’ Saturday Profile on Kimberley Motley Last week The New York Times ran an excellent Saturday Profile on Kimberley Motley: TED Speaker, star of the award-winning documentary Motley’s Law, and the only Western litigator working in Afghanistan. While serving as a fantastic introduction to Motley’s inspiring work (and captivating personality), it also sheds light on her commitments, genuinely daring cases, sexism in and outside the courtroom, and the current state of law and stability in Afghanistan. On Being a Woman in a Male-Dominated Environment In a country where many women can’t leave the home without a burqa, Motley isn’t exactly inconspicuous. Knowing what we do about Afghanistan’s patriarchal structure—and recent history under the Taliban—one might think Motley would encounter countless obstacles merely by virtue of being an outspoken woman. But that’s not quite the full story, as it turns out. “To tell the truth, I get more sexist crap from foreign men than I do from Afghans,” she tells the Times. And any abuse she does receive from male colleagues demanding deference doesn’t slow her down. “I try not to complain, because that’s what people want you to do—they want to see weakness,” she says. Motley’s earned the respect of her Afghan peers, she reveals, by showing it herself. “I’m not coming in and saying, ‘This is the way we do things in America.’ I’m coming in and I’m using the Afghan laws and the Holy Quran. People appreciate real people, and I try to be as real as I can possibly be.” On Helping a Teenaged Girl Escape from Imprisonment While working in law isn’t always exciting, Motley has seen her fair share of dangerous, daring cases. As the Times reports, she recently helped a teen girl, lured to Afghanistan under false pretences and imprisoned by her family, escape the country and return to her home in Vienna. The daring rescue involved teams of armed local men, multiple attempts, and surreptitious communications—not your average work for a litigator, certainly. On Law and Order in Afghanistan Despite President Ashraf Ghani’s promise that Afghanistan has “overcome the past,” the country is definitely in a precarious position. According to the Times article, numerous NGOs have given up, and billions in aid have been wasted. “This place should look like Dubai, considering all the money that’s been poured into it,” Motley reveals. Now, she’s uncertain of how long she’ll be able to keep up the good fight. “And things are getting worse, not better,” she says. As coalition forces withdraw, the threat of a returning Taliban (some 60,000 fighters strong) grows ever more potent. As stated in a report from the Council on Foreign Relations, “The Taliban has outlasted the world’s most potent military forces and its two main factions now challenge the governments of Afghanistan and Pakistan.” In this environment, it’s incredible that Motley continues on. But despite the dangers she faces, she’s still at least part-way optimistic. “The fact that as an American lawyer I’m allowed to practice here, that’s progress”—and perhaps a sign of things to come, as the number of licensed lawyers in the country has rocketed into the thousands since she arrived. “We can all be contributors to a global human rights economy,” says Motley. “We can create a culture of transparency and accountability to the laws, and make governments more accountable to us as we are to them.” To book Kimberley Motley as the keynote speaker for your next event, contact The Lavin Agency speakers bureau.

Local car crash survivors Pat Henman (centre) and her daughter Maia Vezina (right) were manning a table for Mothers Against Drunk Driving at the Chahko Mika Mall on Monday. They were joined by Henman's husband Larry Vezina Mother-daughter duo spread MADD message Pat Henman and Maia Vezina manned a booth at Chahko Mika Mall on Monday to remind people of the dangers of drunk driving. Nelson shopper Pam Hall had an unexpectedly emotional encounter at the Chahko Mika Mall on Monday afternoon when she met local car crash survivors Pat Henman and Maia Vezina. The mother-daughter pair were running a booth to raise awareness for Mothers Against Drunk Driving. “Oh, wait a minute. You are the two?” Hall asked, after receiving a ribbon, a look of realization dawning on her — she’d heard about their fateful 2013 head-on collision. “I met you at the park one day with your puppy. Oh, wow. I’m so glad you’re well.” And then Hall burst into tears, embracing both women as Henman’s husband Larry Vezina looked on. When asked by the Star how it feels to meet someone who has survived a drunk driving crash, Hall was still tearful. “I’ve been thinking about these two for months and to see them here in one piece and beautiful as they are, it’s a blessing they’ve survived and here they are doing this. They’re living to tell the story.” Hall plans to place the ribbon on her Christmas tree. Henman was thrilled with that outcome. “We’re just trying to create awareness of the Red Ribbon program and Campaign 9-1-1. We know each other, we recognized each others’ faces, but then she remembered our story and that is the kind of connection we’re here for. Her awareness leapt 100-fold.” Vezina noted drinking and driving spikes during the holidays, which is part of the reason they decided to host this table during the Christmas season. The pair are both struggling with ongoing health concerns, and Vezina is waiting for surgery, but they’ve started getting back into their lives. “It’s still a struggle,” Vezina said. “This is a permanent disability. As a student they’ve acknowledged that now, so at least they understand and work around my needs, but it’s still a struggle walking around school.”

Q: Spark - Which instance type is preferred for AWS EMR cluster? I am running some machine learning algorithms on EMR Spark cluster. I am curious about which kind of instance to use so I can get the optimal cost/performance gain? For the same level of prices, I can choose among: vCPU ECU Memory(GiB) m3.xlarge 4 13 15 c4.xlarge 4 16 7.5 r3.xlarge 4 13 30.5 Which kind of instance should be used in EMR Spark cluster? A: Generally speaking, it depends on your use case, needs, etc... But I can suggest a minimum configuration considering the information that you have shared. You seem to be trying to train an ALS factorization or SVD on matrices between 2 ~ 4 GBs of data. So actually that's not too much of data. You'll be needing at least 1 master and 2 nodes to setup and configure a small distributed cluster. The master won't be doing any computing whatsoever so it won't need much resources but of course I would be dealing task scheduling, etc. You can add slaves (instances) according to your needs. 1 x master : m3.xlarge m5.xlarge - vCPU : 4 , RAM : 16 GB with EBS storage. 2 x slaves : c3.4xlarge c5.xlarge - vCPU : 16, RAM : 32 GB with EBS storage. EDIT : As mentioned in the comments, 5th generation instances are now available for each of the instance types mentioned in this thread: R5, M5, and C5. In general, latest-generation instance types are cheaper and more performant than their older counterparts. C3, C4, and C5 are compute optimized instances featuring high performance processors and with a lowest price/compute performance in EC2 compared to R3, R4 or R5 although it's recommended use cases are distributed memory caches and in-memory analytics. But C5 will do the job for you for a lower price. Performance Optimizations : Amazon EMR charges on hourly increments. This means once you run a cluster, you are paying for the entire hour. That's important to remember because if you are paying for a full hour of Amazon EMR cluster, improving your data processing time by matter of minutes may not be worth your time and effort. Don't forget that adding more nodes to increase performance is cheaper than spending time optimizing your cluster. Reference : Amazon EMR Best Practices - Parviz Deyhim. EDIT : You might also consider enabling Ganglia to monitor your cluster resources: CPU, RAM, Network I/O. This would help you also tuning your EMR cluster. Practically, you don't have any configuration to do. Just follow the documentation to add it to your EMR cluster on creation.

Q: How to rotate image clockwise? According to this the gstreamer pipeline gst-launch videotestsrc ! videoflip method=clockwise ! ffmpegcolorspace ! ximagesink rotates a video stream clockwise. I have tested this video pipeline in the past on an embedded-Linux Leopardboard successfully but I need to rotate individual images so I modified the pipeline like this: gst-launch filesrc location=test.jpeg ! videoflip method=clockwise ! ffmpegcolorspace ! filesink location=testClockwise.jpeg My modified pipeline causes the following errors: Setting pipeline to PAUSED ... Pipeline is PREROLLING ... ERROR: from element /GstPipeline:pipeline0/GstVideoFlip:videoflip0: not negotiated Additional debug info: ../../../../src/libs/gst/base/gstbasetransform.c(2253): gst_base_transform_handle_buffer (): /GstPipeline:pipeline0/GstVideoFlip:videoflip0: not negotiated ERROR: pipeline doesn't want to preroll. Why is GstVideoFlip:videoflip0 not negotiated? Why doesn't the pipeline want to preroll? How do I fix these errors? Edit: So. I add jpegdec and jpegenc to my pipeline like this: gst-launch filesrc location=test.jpeg ! jpegdec ! videoflip method=clockwise ! ffmpegcolorspace ! jpegenc ! filesink location=testClockwise.jpeg but now get this error: WARNING: erroneous pipeline: no element "jpegdec" But why, since jpegdec and jpegenc are both in gst-plugins-good Elements? A: Your pipeline reads a jpeg. Not a YUV/RGB. So you need to decode your jpeg file, flip it the way you are doing, reencode it to jpeg and then write it to a file.

A program of industrial consultation by a community mental health center. A staff member of a community mental health center provided consultation services to untrained industrial counselors at a large private corporation. Characteristics of "newer' workers, ethnically different from supervisors, are described. Attitude change is the key, but difficult to bring about. The consultant had full cooperation of the counselors, but never gained full support from management. The article describes procedures for training counselors and successful and unsuccessful efforts to reach foremen.

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Shameless Editorial Director Sheila, and Arts Editor Ronak are working alongside two super rad teens Kaya and Sahlla (the mastermind behind the Fight Like A Girl zine) to craft together a zine activity book on DIY interactive guides to revolution. We have one hour to make it and would love for you to come by and tell us your thoughts, opinions, and what revolution means to you. Tweet us at @shamelessmag use the hashtags #DIYrevNB #12hrzinemachine to let us know what revolution means to you and what you’d like to see in the zine! Also, make sure to check out the 12 Hr. Zine Machine website throughout the night to download the finished zines being made throughout the night! The only thing 12 Hr. Zine Machine participants can make ahead of time is the zine cover. This is ours! More info on the event: Drop by the Gladstone Hotel and behold the vibrant and bustling “12 HR Zine Machine” as ten featured Toronto artist groups are challenged to create a handmade zine art book every hour, all night long! Join the zine machine by writing your suggestions on a large “Idea Board” in the workroom or tweeting them to @maddycollective. Nuit Blanche audience members are also invited to sit with the zine artists and contribute their own drawings and writings. Free limited edition copies of each hour’s unique zine will be given out to Nuit Blanche guests. The books will also be uploaded online and downloadable throughout the night on our project website at: www.12hrzinemachine.com.

/* $NetBSD: if_ieee1394.h,v 1.6 2005/12/10 23:21:38 elad Exp $ */ /* * Copyright (c) 2000 The NetBSD Foundation, Inc. * All rights reserved. * * This code is derived from software contributed to The NetBSD Foundation * by Atsushi Onoe. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * 3. All advertising materials mentioning features or use of this software * must display the following acknowledgement: * This product includes software developed by the NetBSD * Foundation, Inc. and its contributors. * 4. Neither the name of The NetBSD Foundation nor the names of its * contributors may be used to endorse or promote products derived * from this software without specific prior written permission. * * THIS SOFTWARE IS PROVIDED BY THE NETBSD FOUNDATION, INC. AND CONTRIBUTORS * ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED * TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE FOUNDATION OR CONTRIBUTORS * BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE * POSSIBILITY OF SUCH DAMAGE. */ #ifndef _NET_IF_IEEE1394_H_ #define _NET_IF_IEEE1394_H_ /* hardware address information for arp / nd */ struct ieee1394_hwaddr { u_int8_t iha_uid[8]; /* node unique ID */ u_int8_t iha_maxrec; /* max_rec in the config ROM */ u_int8_t iha_speed; /* min of link/PHY speed */ u_int8_t iha_offset[6]; /* unicast FIFO address */ }; /* * BPF wants to see one of these. */ struct ieee1394_bpfhdr { uint8_t ibh_dhost[8]; uint8_t ibh_shost[8]; uint16_t ibh_type; }; #ifdef _KERNEL /* pseudo header */ struct ieee1394_header { u_int8_t ih_uid[8]; /* dst/src uid */ u_int8_t ih_maxrec; /* dst maxrec for tx */ u_int8_t ih_speed; /* speed */ u_int8_t ih_offset[6]; /* dst offset */ }; /* unfragment encapsulation header */ struct ieee1394_unfraghdr { u_int16_t iuh_ft; /* fragment type == 0 */ u_int16_t iuh_etype; /* ether_type */ }; /* fragmented encapsulation header */ struct ieee1394_fraghdr { u_int16_t ifh_ft_size; /* fragment type, data size-1 */ u_int16_t ifh_etype_off; /* etype for first fragment */ /* offset for subseq frag */ u_int16_t ifh_dgl; /* datagram label */ u_int16_t ifh_reserved; }; #define IEEE1394_FT_SUBSEQ 0x8000 #define IEEE1394_FT_MORE 0x4000 #define IEEE1394MTU 1500 #define IEEE1394_GASP_LEN 8 /* GASP header for Stream */ #define IEEE1394_ADDR_LEN 8 #define IEEE1394_CRC_LEN 4 struct ieee1394_reass_pkt { LIST_ENTRY(ieee1394_reass_pkt) rp_next; struct mbuf *rp_m; u_int16_t rp_size; u_int16_t rp_etype; u_int16_t rp_off; u_int16_t rp_dgl; u_int16_t rp_len; u_int16_t rp_ttl; }; struct ieee1394_reassq { LIST_ENTRY(ieee1394_reassq) rq_node; LIST_HEAD(, ieee1394_reass_pkt) rq_pkt; u_int32_t fr_id; }; struct ieee1394com { struct ifnet fc_if; struct ieee1394_hwaddr ic_hwaddr; u_int16_t ic_dgl; LIST_HEAD(, ieee1394_reassq) ic_reassq; }; const char *ieee1394_sprintf(const u_int8_t *); void ieee1394_input(struct ifnet *, struct mbuf *, u_int16_t); void ieee1394_ifattach(struct ifnet *, const struct ieee1394_hwaddr *); void ieee1394_ifdetach(struct ifnet *); int ieee1394_ioctl(struct ifnet *, u_long, caddr_t); struct mbuf * ieee1394_fragment(struct ifnet *, struct mbuf *, int, u_int16_t); void ieee1394_drain(struct ifnet *); void ieee1394_watchdog(struct ifnet *); #endif /* _KERNEL */ #endif /* !_NET_IF_IEEE1394_H_ */

Economy of North America The economy of North America comprises more than 579 million people (8% of the world population) in its 23 sovereign states and 15 dependent territories. It is marked by a sharp division between the predominantly English speaking countries of Canada and the United States, which are among the wealthiest and most developed nations in the world, and countries of Central America and the Caribbean in the former Latin America that are less developed. Mexico and Caribbean nations of the Commonwealth of Nations are between the economic extremes of the development of North America. Mexico lies in between these two extremes as a newly industrialized country (NIC), and is a part of the North American Free Trade Agreement (NAFTA) and a member of the Organisation for Economic Co-operation and Development (OECD), being one of the only two Latin American members of this organisation (together with Chile). The United States is by far the largest economy in North America and the largest national economy in the world. The US, Canada and Mexico have significant and multifaceted economic systems. In 2011, the US has an estimated per capita gross domestic product (PPP) of $47,200, and is the most technologically developed economy in North America. The United States' services sector comprises 76.7% of the country's GDP (estimated in 2010), industry comprises 22.2% and agriculture comprises 1.2%. Canada's economic trends are similar to that of the United States, with significant growth in the sectors of services, mining and manufacturing. Canada's GDP (PPP) was estimated at $39,400 in 2010. Canada's services sector comprises 78% of the country's GDP (estimated in 2010), industry comprises 20% and agriculture comprises 2%. Mexico has a GDP (PPP) of $15,312, and per capital income is estimated at approximately one-third of the United States'. The country has both modern and outdated industrial and agricultural facilities and operations, and is modernizing in sectors such as energy production, telecommunications and airports. Economic development Great Depression The Great Depression began in North America in October 1929. The start is often dated to the stock market collapse of Black Tuesday although this was not the cause of the Great Depression. Canada and the United States experienced especially large declines, with the gross domestic product falling 37% from 1929 to 1933 in the United States, and 43% in Canada over the same period. The economy reached its lowest point in 1933, however recovery was slow. The outbreak of World War II in 1939 created demand for war materials that brought about the end of the depression. The Great Depression spurred increased government intervention in the economy in North America. The United States introduced unemployment insurance, a minimum wage and standardised working hours under the New Deal. Canada introduced similar measures. Mexico nationalised some key industries during the Great Depression, with the railroads nationalised by 1937 and the oil industry nationalised in 1938. World War II The large scale enlistment of men into armed forces during World War II women entered the workforce en masse, filling many jobs in manufacturing and technical areas that had previously been closed to women. This led to the "We can do it!" campaign. The economic output in North America increased substantially, with unemployment practically eliminated in the United States. Rationing severely reduced the availability of consumer goods, with the increase in industrial production coming from the demand for war materials. During the peak of World War II activity, nearly 40 per cent of US GDP was devoted to war production. Cold War US-Canada Free Trade Agreement and NAFTA - a new era of economic integration The Canada-United States Free Trade Agreement of 1989 and the subsequent expansion to the North American Free Trade Agreement (NAFTA) triggered a dramatic increase in trade between these three countries, with Mexican trade with the United States and Canada increasing threefold. Over 85% of Canadian exports in 2006 went to the United States. Regional variation With various climate zones, agricultural products vary from country to country. Job sectors are also different, with industrialized countries having more service workers, and developing countries relling on agriculture. Trade blocs Asia-Pacific Economic Cooperation The Asia-Pacific Economic Cooperation (APEC) is a group of Pacific Rim countries which meet with the purpose of improving economic and political ties. APEC's stated goals are aimed at free and open trade and investments by cutting tariffs between zero and five percent in the Asia-Pacific area for industrialised economies by 2010 and for developing economies by 2020. The organisation has members from four continents, those from North America are Canada, Mexico and the United States. Caribbean Community The Caribbean Community (CARICOM) was created "To provide dynamic leadership and service, in partnership with Community institutions and Groups, toward the attainment of a viable, internationally competitive and sustainable Community, with improved quality of life for all". Its secretariat is based in Georgetown, Guyana, South America. - On January 1, 2006 six members: (Barbados, Belize, Guyana, Jamaica, Suriname and Trinidad and Tobago) unofficially ushered in the Caribbean (CARICOM) Single Market and Economy (CSME). - At the official signing of the protocol on January 30, 2006 in Jamaica, A further six members: (Antigua and Barbuda, Dominica, Grenada, Saint Kitts and Nevis, Saint Lucia and Saint Vincent and the Grenadines) announced their intention to join by the second quarter of 2006. Montserrat, a British Oversees territory is awaiting approval by the United Kingdom. Haiti and the Bahamas have no immediate plans to join. Central American Free Trade Agreement The Central American Free Trade Agreement (CAFTA) is an agreement between the United States and the Central American countries of Costa Rica, Guatemala, El Salvador, Honduras, and Nicaragua. The treaty is aimed at promoting free trade between its members. Canada and Mexico are negotiating membership. North American Free Trade Agreement The North American Free Trade Agreement (NAFTA) is an agreement between Canada, Mexico and the United States to eliminate tariffs on goods traded between themselves. Although currently only a trade agreement, with no supranational bodies or laws as in the European Union, there have been various proposals to move towards a customs union or a North American currency union. It is unknown whether this may eventually develop into a North American Union similar to that of Europe. Currency Below is a list of the currencies of North America, with exchange rates between each currency and both the euro and US dollars as of 12 April 2008. This list may vary as it is not current. Table correct as of 12 April 2008 Economic sectors Agriculture Agriculture is very important in Central American and Caribbean nations. In western Canada, in the provinces of Saskatchewan, Alberta, British Columbia and Manitoba, wheat and other various main agricultural products are grown. The U.S. also has many states with significant agriculture production, mainly in the central continental U.S. Mexico produces many tropical fruits and vegetables as well as edible animals. Manufacturing North America has developed and its manufacturing sector has grown. In the beginning the European nations were the large manufacturing powers. At the start of the 1950s, the United States was a top manufacturing power, with Canada and Mexico also making significant progress. Service In Canada, the US and the Caribbean, service-based employment is a significant percentage of overall employment. Many people work in stores and other retail locations. In Canada more than 70% work in the services sector, with a similar percentage in the United States. Investment and banking The United States leads North America in investment and banking. Canada, Mexico and most recently, February 2011, El Salvador is growing in this sector. And smaller economic powers such as Guatemala, Honduras, Costa Rica, and Panama are also growing slowly in this sector. Tourism Tourism is extremely important for the Caribbean economies, as they contain many beaches and have warm climates. Skiing in Canada and the US is also important. Tourism of national parks and natural landmarks, such as Mount Rushmore and the Grand Canyon in the United States, and Niagara Falls and Moraine lake in Canada, contribute to the economy in these regions. See also North America North American Industry Classification System Statistics: List of North American countries by GDP (PPP) List of North American countries by GDP (nominal) List of North American countries by GDP per capita References North America

Q: Regex - Issues with using Boundary to excluding words In my authentification web site, I'm using regex to control a blacklist password. (example of blacklisted password : 12345678, 123456789, baseball, football) I would like to add new regex rule (using boundary), which will exclude words (black listed password). I have read some similar questions on StackOverflow and tried to declare it with something like this: ^(?!\b12345678\b|\b123456789\b|\bbaseball\b|\bfootball\b|\bsuperman\b).*$ this regex doesn't match the words above, it's correct. For exemple "Baseball" with a letter, number or special character (before or after the "baseball") must match. But "baseball!" doesn't match contrary to "!baseball". Can you give me some advices how to do it? A: But "baseball!" doesn't match contrary to "!baseball"… baseball! doesn't match because your pattern doesn't allow baseball at the beginning (^ followed by a negative lookahead for baseball). !baseball in contrast matches because ! is placed at the beginning, and the negative lookahead is done only there, not aft. One could think of putting the .* at different places, but that will lead to nothing. Just include the anchors ^ $ in the lookahead: (?!^(12345678|123456789|baseball|football|superman)$)^.*$ (in fact, we could even drop the initial ^).

2017 Fed Cup Europe/Africa Zone Group I – Pool B Pool B of the 2017 Fed Cup Europe/Africa Zone Group I was one of four pools in the Europe/Africa zone of the 2017 Fed Cup. Three teams competed in a round robin competition, with the top team and the bottom team proceeding to their respective sections of the play-offs: the top team played for advancement to the World Group II Play-offs, while the bottom team faced potential relegation to Group II. Standings Standings are determined by: 1) Number of wins; 2) Number of matches; 3) In two-team ties, head-to-head records; 4) In three-team ties, (a) percentage of sets won (head-to-head records if two teams remain tied), then (b) percentage of games won (head-to-head records if two teams remain tied), then (c) Fed Cup rankings Round-robin Croatia vs. Bosnia and Herzegovina Hungary vs. Bosnia and Herzegovina Croatia vs. Hungary References External links Fed Cup website Category:2017 Fed Cup Europe/Africa Zone

Pages Friday, February 12, 2010 McQueen exits the world of fashion Who would ever forget these fluffy, hard-to-tame, eccentric ensembles? The first time I saw his work was in an old magazine from my mother. Apparently, the pages of that fashion magazine were now all over my wall. Autumn/Winter 2006Fall 2009 Who wouldn't recognize who's the mastermind behind this shoes? My friend Joanarc once posted a photo of this in her tumblr and I couldn't help but wonder how on earth could a woman walk with these shoes (which you probably spotted in Lady Gaga's videos): Hair Spring 2010Spring/Summer 2008The 40-year old designer was found dead in his apartment in London, Thursday morning (February 11). Learned about the news here: NYTIMES. Sadly, the suicide issue is all over the internet. The website MailOnline even mentioned suicide in its headline. No comments: Désaveu Unless noted, I do not claim the right of all the images posted. The copyright belongs to their respective owners and I will try my best to credit the masterpieces. If you do not want to see your photos, or any work of art here, kindly send me an email and your request will be approved immediately.

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[Energy metabolism of growing rats in relation to the environmental temperature]. 8 experiments were carried out with 9 albino rats each (Wistar line, bred at the institute) in the live weight range between 70 and 200 g and at environmental temperatures (ET) of 34, 32, 30, 28, 26, 24, 22 and 20 degrees C. In the course of each individual experiment the rats were alternatively fed for maintenance and weight gain (semi ad libitum) with feed mixtures containing 10, 25 and 40% crude protein (3 animals/variant). Energy metabolism was measured according to the method of indirect calorimetry over a total of 780 metabolism periods. In the temperature range studied there was no compensation between thermoregulatory heat and heat from other processes of the metabolism. The partial utilization of metabolizable energy for energy retention in the body was independent of ET and ranged between 73 and 80% for the 7 experiments with ET between 32 and 20 degrees C. Energy utilization depended on the protein content of the feed and decreased from 81 to 79 or 73 resp. when the protein content increased from 10 to 25% or to 40% resp. Energy requirement for protein retention varied between 1.61 and 2.09 kJ metabolizable energy/kJ and was independent of ET. Energy maintenance requirement (measured at 28, 30 and 32 degrees C) increased with the growing protein content from 415 to 439 and 447 kJ/kg LW0.75.d resp. (regression analysis) and from 411 to 420 and 432 kJ/kg LW0.75.d (measuring at maintenance level). The relative weight gain with the increased protein content of the feed largely corresponds to the expected values according to the efficiency of ATP synthesis in the oxidative degradation of nutrients. The relationship between heat production and ET is parabolic. In the live weight range studied the average thermoneutral temperature (TNT) was 32 degrees C. It decreased during the course of development from 34 to 30 degrees C. TNT decreased with the growing protein content of the feed. Thermoregulatory heat production depended on both environmental temperature and the stage of development. Its average value in the development range studied decreased with an increase of the environmental temperature by 2 K each, starting from 20 degrees C and rising to 32 degrees C, in the following linear sequence: 23.3, 21.0, 16.8, 12.5, 8.3, 4.0 and 0.3 kJ/kg LW0.75.d.K.

Sesskia’s Diary, part 7 17 Senessay I’m feeling overwhelmed, so I’m just going to start at the beginning and hope writing it all down calms me. I’m fairly certain about the date, but that’s the only thing I’m sure of anymore. The new bedroom was still a cell, if a nicer one. People brought me meals, and the lights dimmed by themselves after a time—I think the lights in my first cell didn’t work properly—so I slept when it was dark and paced the room and practiced pouvrin when it was light. I gained enough control over the mind-moving pouvra that I could lift the bed, the dresser, and the wardrobe all at once. Only an inch or two, and only for a few seconds, but it was exciting. But that’s not what has me overwhelmed. I just went back and re-read the first page of this book, just to be certain I haven’t forgotten my own language. Though if I’m writing in it now—see how flustered it’s made me? But I’m getting ahead of myself again. I didn’t see Terrael yesterday or today, and I was a little surprised at how disappointed I was. I mean, I couldn’t understand him, but at least he was nice and didn’t treat me like a problem. I poked my head out of the door a few times and there was a single guard, so either they were feeling more sure of me or they’ve given up on trying to contain me and that was just a token. I smiled and waved at the guard (a man) and he just watched me impassively until I became bored and went back inside. I decided I was going to make another escape attempt tonight when the lights went dark again. Except before that happened, Terrael appeared. He no longer looked confident. He looked like a boy about to do something that would get him into trouble. He came into my room, shut the door, and made a pinching gesture in front of his lips that I guessed meant “be quiet.” As if anything I might say would be meaningful, no matter how loudly I said it. Then he opened the door and gestured for me to precede him. In the hall, he said something to the guard, who nodded. He looked bored. I couldn’t blame him. I followed Terrael down the corridor and into the cavern again. It was quieter, less busy, like a marketplace where almost everyone has closed up shop for the day. Terrael was walking very casually now, greeting the people we passed, stopping to exchange a few words with a pretty young woman whose hair was fastened with a jeweled clasp, polished jasper with cabochon garnets, reasonably valuable if only for the craftsmanship, and the first sign of individuality I’d seen among these drones. Eventually we made it around the perimeter of the cavern to a door, metal like all the ones in the corridor, but wider, and Terrael took out a large key and unlocked it, then shooed me inside with the first hint of nervousness he’d displayed so far. The room beyond was much larger than the corridor rooms, though of course nothing near as big as the cavern, and was brightly lit. And it was filled with castoffs—I didn’t recognize a single thing there, but I’ve been stealing from great estates long enough to recognize a room where unwanted things are stored. Almost all of the things were made primarily of metal, and they were all intricately decorated with engravings that reminded me of the maybe-letters on the glass light baskets. I went to pick up a sphere of overlapping bronze strips like an enclosed basket, and Terrael yanked my hand away, shaking his head vigorously in a way that told me, first, that ‘no’ was in fact a universal gesture, and second, that he absolutely did not want me to touch anything. Naturally, this made me want to touch everything I could get my hands on, but there was fear on Terrael’s face that made me put my hands in my pockets. I was planning to go back there for some real exploration, but after what’s happened, I’m not sure I’ll be able to. Terrael went to the back of the room, carefully not touching anything himself, and soon disappeared behind this tall slab of greenish copper that looked like a horse trough stood on end. I waited, jamming my hands firmly into my trouser pockets just in case they decided to do a little exploring on their own, and eventually he came back holding some kind of helmet. No, it was more of a cap made of black iron, and for a wonder it wasn’t covered with scribbles; there was just a blank band all the way around the rim that was smoother and shinier than the rest of the cap. Terrael held it out to me, and I took it. It felt like cold metal, and nothing happened to me when I touched it, so I turned it upside down to look into it. The inside of the cap had these hair-fine traceries all over it, as if someone had done lacework on it in molten iron. I ran my finger over the lines, and it still only felt cold. Terrael nudged me, and made a gesture like he was putting something on his head. I looked at the cap again. Suddenly it seemed a little sinister, all this secrecy, Terrael acting tense and telling me not to touch anything, and then handing this thing over as if it were nothing. When I didn’t respond right away, Terrael made an exasperated sound, took the cap from me, and put it on his own head. Nothing happened. He took it off and offered it to me with a “see, it’s harmless” look. So I put it on. It was far too big for my head, and canted a little over my left ear. I must have looked so stupid—I certainly felt stupid, standing there in that room surrounded by mysterious cast-off things, with Terrael beaming at me as if, once again, I’d performed a trick and deserved a reward. Then he looked around, made that exasperated noise again, and began clearing a spot on a nearby counter until he had a bare space about five feet across. He pointed at it, but it wasn’t until he sat on the counter himself that I figured out that’s what he wanted me to do. It wasn’t a very tall counter, but I’m not a very tall woman, and my feet dangled a little. Terrael started muttering to himself. It was the kind of muttering you do when you’re going over a complicated project in your head, like planning to break into one of the royal manors, so I didn’t feel obliged to pay any attention to him. He reached inside his robe and pulled out a pot with a stoppered lid and a small brush, its skinny bristles no longer than my pinky nail. The pot turned out to be full of silvery ink or paint. Terrael came to stand close in front of me and began painting on the brim of the cap. Every few minutes he would rotate the cap on my head to paint a new section. I really wished I could ask him questions—hah! That’s funny now. Anyway, I stayed patient because I was curious about what he was doing. I don’t know if it’s good or not that I didn’t just run away. Finally, he stepped back, and his eyes focused on mine again. He looked very serious, like saying goodbye forever serious, and I got nervous and was about to take the cap off when he reached out with the brush and made a final mark on the cap. I thought my head had exploded. It hurt worse than anything I’d ever imagined possible, and I wanted to rip the cap off my head and throw it at Terrael’s face, but my entire body was paralyzed. I found later that I’d fallen off the counter, but at the time I couldn’t feel anything but the pain that radiated from my forehead through my entire body. Phantom smells of ash and rainwater filled my nostrils, and I tasted salt, and I couldn’t see or hear anything at all, not even the screaming I’m sure I was doing. And then I could hear too much, all these voices shouting in hundreds of languages, none of which I understood. Somewhere in there I blacked out, I think, because the sound went from being hundreds of voices to just one, high-pitched like a woman’s, chanting. I still couldn’t understand it, but then I realized I could move—that’s when I found I was on the floor. I had the cap off my head and flung across the room before I discovered I wasn’t in pain anymore, and I could see. What I saw, from my perspective on the floor, were two pairs of sandaled feet attached to two pairs of black trousers. Terrael was arguing with Smug Git, and this is the overwhelming part—I listened to their conversation for nearly a minute before I realized I understood what they were saying. It staggered me to the point that I can’t remember now what their exact words were, just that Smug Git was furious with Terrael about what he’d done with the cap, and Terrael, surprisingly, was standing up to him and saying something like “it was worth the risk.” I got to my feet, and they both stopped arguing. Smug Git said, “We will have to watch her to see if any permanent damage was done.” The way he said it, like I was some kind of animal, made me angry, so I said—I can’t remember exactly, that’s how angry I was—“Oh, yes, let’s hope she didn’t sustain any permanent damage, that would be so inconvenient for you” and that’s as far as my anger took me before I realized I was speaking their language, and that startled me so much I shrieked and clapped my hands over my mouth. Terrael’s mouth fell open. Smug Git raised one eyebrow again—really, that makes him look even more arrogant and annoying than he naturally does. “It worked,” he said. He made it sound like the whole thing was his idea. It sounded like Terrael felt the same way, and he said, “Just as I said, Sai Aleynten,” and I could practically hear him thinking I told you so, though he was careful not to sound rude. Smug Git nodded once, and said, “Take her back to her room, Master Peressten, and I will interrogate her in the morning.” I didn’t like the sound of “interrogate,” and I said, “You brought me here, maybe I should be interrogating you.” It wasn’t much, but I couldn’t stand there and not defy him. It’s his face. He just turned that cold, indifferent gaze on me, then said “In the morning, Master Peressten,” and walked away. So I lost my temper and summoned the fire in a circle around him. Terrael cried out and took a step toward the git, who just turned smoothly on his heel, made a few gestures like writing on the air—and I flew back into the counter I’d been sitting on. It knocked the air out of me, and I lost control of the fire and it went out, but obviously what really stunned me was seeing him work that pouvra. Never mind that I couldn’t do anything nearly so powerful; what was the gesturing for? Pouvrin are meant to come from inside you, something you encompass with your mind and then turn outward. If I gestured all the time when I did magic, I’d be captured instantly. So— All right. I’m still overwhelmed. I was overwhelmed enough then that I didn’t strike back at Smug Git or whatever it was Terrael called him. Sai Aleynten. He walked away without another word, and Terrael helped me stand, babbling something about how I shouldn’t attack people and Smug Git could have done far worse because he’s some word I didn’t understand. Whatever Terrael’s cap did to me, there are apparently words it didn’t bother translating, and there’s probably some logic to it, but I can’t see it at the moment. He brought me back to my room, and now I’m hurrying to write this before the lights go out. There’s just too much. Here’s what I know. That cap did something to me that lets me speak their language. These people have magic. Powerful magic, if Smug Git is representative. They don’t work magic the way I do. They want to learn something from me, hence the promised interrogation. I ought to escape. I have no reason to believe that just because I haven’t been hurt before, their interrogation won’t involve…maybe not torture, but physical duress at least. But—this is the first place I’ve ever been where magic not only isn’t feared, but is openly practiced. Even if the way they use pouvrin is not at all like mine. I can’t leave until I’ve at least learned why that is. And I’m increasingly curious about why I’m here at all. I think Smug Git’s interrogation may give me more information than I give him. At least, that’s my plan.

Cutting Vinyl Records In Mono And Stereo Wondering how records are actually made? As vinyl comes back into fashion, it's worth taking a look at the process with one of the biggest names in the biz. One of the most sought after vinyl-cutting systems in the world is the nearly indestructible VMS-70 and VMS-80 cutting systems built by Neumann. The VMS-82 was the last of these produced. I’m thankful to say that we get to use our VMS-82 lathe every day to cut lacquers for clients around the globe. The actual cutting happens at the cutter head. In this case, the BMW of cutter heads, the SX-74. Though it was initially built in 1974, this design was never dramatically improved. It was capable of cutting with sufficient level and flat frequency response to please nearly everyone. The underside of a SX-74 cutter head. A closer look. The two round “cans” on either side are the voice coils. You can also see the cutting stylus: a faceted sapphire glued to a pin that mounts in the tube that connects to each voice coil. Also in the foreground are two fine wires. These carry a small voltage that heats the stylus to an optimal temperature so that it slices smoothly through the lacquer instead of dragging and causing extra noise from a jagged cut. The drive coils of the stereo cutter head are mounted at right angles. When there is audio in the left channel, the left coil goes in and out just like a speaker does. And when there is audio in the right channel, the right coil goes in and out. One voice coil in the cutter head is wired out of phase on purpose so that, when a mono signal is cut, the left coil is moving in as the right coil is moving out. Thus, a mono signal cuts a lateral groove. The Neumann VMS-80 with SX-74 cutter head remains a gold standard in vinyl-cutting systems. Why is it done this way? We have to go back to mono to find out. Early records—initially 78s and then LPs—were mono. Systems that cut mono records had only one drive coil and it moved the cutting stylus back and forth creating a lateral, constant-depth groove. There was little concern about the depth of the cut as long as it was deep enough to hold the playback stylus in the groove. Then along came stereo. Researchers needed to find a way to carve two channels of audio into a record but make the new technology compatible with mono records and players. Unfortunately, today’s technology designers don’t put quite so much effort into forward- and backward-compatibility. But that’s a soapbox speech for another time. So what they came up with was to record the mono component of the stereo audio laterally—like on a mono record. Then, by adding a second coil and wiring it out of phase with the first coil, they created depth modulation which records the stereo or side signal. Going deeper. Researchers needed to find a way to carve two channels of audio into a record but make the new technology compatible with mono records and players. Stereo is made up of a left signal and a right signal. OK, that’s simple. But stereo can also be described as the "mono component" (everything that is exactly the same in both speakers) and the "difference component" (everything that is different). This is commonly called Middle and Side, or M-S for short. A stereo signal can be converted into an M-S signal and back again with nearly no change at all. FM radio is transmitted in M-S. The middle signal is a strong full-wave signal. And it is this signal that you hear when you are far away from the radio tower. That signal is mono. As you get closer to the radio tower, your radio can tune in the sub carrier signal, which carries the difference (side channel). When you receive a strong enough signal, the FM station now plays back in full stereo because it has both the middle and the side signals. It can be hard to believe because we commonly think in "left and right" rather than "middle and side". But it’s true. It’s a matter of physics and alternating current electronics. The groove shows us the difference signal by it’s depth. So a mastering engineer says “lateral” and means the mono (or, middle) signal. And when the engineer says “vertical” he or she is referring to the difference(or, side) signals. Once you have a hold of that concept, we can start to talk about why some records seem to make the vocals spitty and sibilant. And why some recordings have to be modified with equalization to minimize out-of-phase bass. But there is one more thing to understand before we can control our quality. It was a standard developed in the 1950s called the RIAA Curve. Next week I'll talk about what the RIAA curve is, why it was standardized, and what steps we have to take to make records sound really good.

Influence of tapeworm infection on the production of aggregation pheromone and defensive compounds in Tribolium castaneum. Recent studies suggest that parasites affect host development, reproduction, and behavior through alterations of host hormones and pheromones, or other hormone-triggered biochemical events. We previously reported that Hymenolepis diminuta infection affects surface-seeking and cannibalism behaviors, and reduces male sperm precedence of Tribolium castaneum beetles. This study examined the quantitative effects of H. diminuta on the production of aggregation pheromone and 3 defensive compounds in male T. castaneum beetles, using 2 wild-caught, geographically distinct T. castaneum strains. For the c-Madison strain, infected beetles exhibited a 2- to 22-fold increase in defensive compounds; conversely, no changes were observed in strain c-Africa. Parasite infection did not significantly influence aggregation pheromone secretion in either strain. Because defensive compounds function as repellents or deterrents to other insects, parasite-induced increases in the secretion of defensive compounds may be a physiologic clue for the behavioral changes in infected T. castaneum beetles. Significant among-strain variation in defensive compound production seen in infected beetles suggests that caution is needed before generalizing about changes in volatile production and in host behavior induced by a parasite.

The minimal underlying data necessary to replicate this study are included in the manuscript and its Supporting Information files. Qualifying researchers may request confidential data such as interview transcripts by contacting Anna Barker at <anna.barker@monash.edu> Introduction {#sec001} ============ Despite advances in clinical practice and research, falls remain the most common adverse event in hospitals. More than 240,000 in-hospital falls occur each year in England and Wales with falls being the most commonly reported safety incident in National Health Service hospitals \[[@pone.0171932.ref001], [@pone.0171932.ref002]\]. Falls prevention programs for hospitalised older people are multifaceted, reflective of the complex causal pathway for falls. With increased complexity comes increased risk of implementation failure. Implementation of falls prevention programs can be influenced by several factors including environmental and contextual issues; staff knowledge, beliefs and attitudes; organisational culture and climate; staff workloads; and access to appropriate equipment and resources \[[@pone.0171932.ref003]\]. An understanding of these factors can inform the development of an implementation plan that addresses the barriers and enablers to the implementation of the intervention. There is limited information about the barriers and enablers to the implementation of falls prevention in acute hospitals. Two survey based studies implemented across five acute care hospitals in Singapore showed that nurses perceived the greatest barriers to implementation of fall prevention practices to be: staff and patient education; lack of motivation in staff; availability of support staff; and access to facilities and equipment \[[@pone.0171932.ref004]--[@pone.0171932.ref005]\]. These barriers were also reported in a recent Cochrane review of 11 RCTs \[[@pone.0171932.ref006]\]. Other barriers reported in the review included: leadership support at the organisational and unit level; engagement of front-line staff in program design; pilot-testing to identify potential barriers to implementation; provision of data about falls; and changes in nihilistic staff attitudes about falls prevention were associated with successful implementation of inpatient falls prevention programs in hospitals \[[@pone.0171932.ref006]\]. Tailoring the implementation of falls prevention programs to the local context optimises implementation. The 6-PACK falls prevention program is nurse-led ([Box 1](#pone.0171932.box001){ref-type="boxed-text"}) and was developed as part of continuous quality improvement activities at an Australian acute hospital. An evaluation reported that fall-related injuries appeared to reduce following the implementation of the program \[[@pone.0171932.ref007]\]. This led to a multi-centre randomised controlled trial (RCT) to further establish the efficacy of the 6-PACK program \[[@pone.0171932.ref008], [@pone.0171932.ref009]\]. Whilst 6-PACK intervention components are required to remain fixed in an RCT, the implementation of the program was tailored to the local context of the intervention wards to ensure implementation was optimised. Box 1. The 6-PACK program {#sec002} ------------------------- The 9 item fall-risk tool \[[@pone.0171932.ref010]\] is updated for each patient each shift by their treating nurse. Patients identified as high falls risk receive: 1. A 'falls alert' sign positioned above their bed, and one or more of the following interventions: 2. Supervision of patients in the bathroom 3. Ensuring patients' walking aids are within reach 4. A toileting regime 5. A low-low bed 6. A bed/chair alarm The COM-B model was developed by condensing concepts from 19 frameworks of behaviour change identified in a systematic review by Michie and colleagues \[[@pone.0171932.ref011]\]. The COM-B model demonstrates human behaviour (B) as the interaction between physical and psychological capabilities (C) that utilise social and environmental opportunities (O) via motivators (M) that are reflective ('thinking' with the head) or automatic ('thinking' with the heart). It has been widely adopted in implementation and health services research \[[@pone.0171932.ref012], [@pone.0171932.ref013]\]. The aim of this study was to use the COM-B model to identify the perceived barriers to, and enablers of, implementation of the 6-PACK program from the perspectives of nurses and senior staff to inform the implementation plan. Specifically we sought to identify physical and psychological factors (capability); environmental and social contexts (opportunity); and reflective and autonomic processes (motivation) that are perceived to be barriers or enablers of the successful implementation of the 6-PACK program. In addition, we sought to gain insights into what strategies could be applied to optimise successful implementation of the 6-PACK program in the RCT. Materials and methods {#sec003} ===================== Design {#sec004} ------ A multi-centre mixed methods study. This study was part of the 6-PACK project that incorporated a three-year research plan: 1) Studies of current falls prevention practice and moderators (pre-implementation) \[[@pone.0171932.ref014]\]; 2) A cluster RCT testing 6-PACK effectiveness ([S1 Appendix](#pone.0171932.s001){ref-type="supplementary-material"}), including economic \[[@pone.0171932.ref015]\] and program evaluations (implementation); and 3) An assessment of sustainability of practice change and outcomes (maintenance). The study reported here forms part of the pre-implementation stage. Participants and setting {#sec005} ------------------------ Detailed information about participants, recruitment and data collection are reported elsewhere ([S2 Appendix](#pone.0171932.s002){ref-type="supplementary-material"}). In brief, this study involved staff from 16 medical and 8 surgical wards participating in the 6-PACK RCT. Nurses were invited to complete the survey and participate in focus groups. Key informant interviews were conducted with senior staff (Nurse Unit Managers (NUMs), senior physicians, Directors of Nursing (DONs) and clinical services, falls prevention leaders and senior personnel involved in quality, safety and risk management). Nurse survey {#sec006} ------------ The 42 item survey was developed with items related to beliefs about falls; current falls prevention practice; 6-PACK program components; best practice guidelines and key recommendations; and reporting practices were included. Participants indicated their level of agreement using a five point Likert scale ranging from strongly disagree to strongly agree. Seven items related to the COM-B domains: one to capability, three to opportunity and three to motivation ([Table 1](#pone.0171932.t001){ref-type="table"}). 10.1371/journal.pone.0171932.t001 ###### Mapping of survey, focus group and interview questions to the COM-B domains \[[@pone.0171932.ref011]\]. {#pone.0171932.t001g} Survey Focus group Interview Questions/Statements ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------- ----------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ **Capability: The individual's psychological and physical capacity to engage in the activity concerned.** ✓ ✓ What strategies would you recommend we use when implementing the 6-PACK program? Why? ✓ ✓ What learning can we take from other program implementation experiences on your ward? What were some of the barriers? What would you do differently next time? What worked well? ✓ You can't stop older people from falling. ✓ ✓ Do you believe falls can be prevented? What interventions do you feel are most important? **Opportunity**: **The factors that lie outside the individual that make the behaviour possible or prompt it.** ✓ ✓ Who are the critical people that need to be involved in falls prevention activities at your hospital? ✓ ✓ What strategies/factors would you consider to be essential to sustaining programs like the 6-PACK? Please explain. ✓ What falls prevention activities are currently occurring/or planned for the hospital? Do you perceive these activities to be complementary or inhibitory to the 6-PACK implementation on the intervention wards? Please explain. ✓ Who should we involve in the processes of implementing the 6-PACK this hospital? What do you see their role will be? How do you rate the relative importance of these individuals or group in terms of making the implementation successful? ✓ Who do you anticipate may be obstructive/resistive to the implementation of 6-PACK? Why? (Knowledge, beliefs and skills? Attitudes and opinions? Conflicting demands?) What strategies do you recommend to better engage these people? (Incentives and motivators?) What strategies do you recommend to inform/approach/involve key staff in the change process? ✓ What system level barriers do you feel may exist to implementing the 6-PACK program? E.g. Equipment and staffing resources, communication, leadership and teamwork, environmental constraints (e.g. budgets, redevelopments, restructuring) ✓ Leadership and supervision for falls prevention practice. ✓ An active falls prevention leader is essential for falls prevention programs to be successful on my ward. ✓ This feedback \[about how I use falls prevention interventions\] helps me use falls prevention interventions more effectively. **Motivation: Reflective and automatic mechanisms that activate or inhibit behaviour. Includes habitual processes, emotional responding, as well as analytical decision-making.** ✓ ✓ What effect do you feel audit, feedback and reminders will have on the effectiveness of the 6-PACK program implementation? Can you provide examples of when these have been effectively used previously? ✓ There are more important things I should do than falls prevention interventions for my high falls risk patients. ✓ Incident reporting provides us with a way of measuring how we are going with patient falls. ✓ It is not my responsibility to stop patients from falling. Focus groups and key informant interviews {#sec007} ----------------------------------------- Discussion guides for the focus groups and key informant interviews based on the COM-B framework were developed to elicit ward nurse and senior staff views on barriers and enablers to implementing the 6-PACK program ([Table 1](#pone.0171932.t001){ref-type="table"}). Focus groups and key informant interviews at each hospital were scheduled and conducted. Senior staff nominated by the DON at each hospital received a letter of invitation to participate in an interview from the research team. The perspectives of senior staff were sought to understand hospital practices, policies and the organisational context influencing falls prevention interventions. Data analysis {#sec008} ------------- Descriptive statistics were calculated for survey responses using Stata MP v13 statistical software. Analysis of interview and focus group data was continuous with deductive coding being applied for the three COM-B domains and emerging themes explored and tested for applicability and consistency. Three researchers independently coded and recoded transcripts using Nvivo (QSR International 2012), continually working back and forth between data sources in a process of open, axial and thematic coding \[[@pone.0171932.ref016], [@pone.0171932.ref017]\]. Discrepancies were resolved by discussion and consultation with the investigator team as required. Quantitative and qualitative data were analysed separately with a process of triangulation applied at the interpretation stage of the analysis whereby findings from each component were considered to determine whether findings were convergent, complementary or contradictory \[[@pone.0171932.ref018]\]. Ethics {#sec009} ------ This study was approved by Monash University Human Research Ethics Committee--CF11/0229--2011000072 and each of the relevant hospital ethics committees. Participants were given verbal information about the study and asked to sign consent forms if they were interested in participating. Results {#sec010} ======= Study participants {#sec011} ------------------ Overall, 702 surveys were distributed with 420 (60%) returned. The majority of respondents were registered nurses (74%); staff working on medical wards (75%); and staff with at least one year of experience at the hospital (74%). Twelve focus groups involving 96 nurses and 24 interviews with senior staff (SS) were conducted. Six DONs, seven NUMs, one Clinical Risk Coordinator, one Quality and Safety Manager, one clinical program nurse manager, and eight nursing educators participated in the interviews. Each of the COM-B domains and arising sub-themes are described below ([Table 2](#pone.0171932.t002){ref-type="table"}) in the context of barriers and enablers to the implementation of the 6-PACK program. Implementation strategies suggested by the participants have also been described and summarised in [Table 3](#pone.0171932.t003){ref-type="table"}. 10.1371/journal.pone.0171932.t002 ###### Mapping of barrier and enabler themes to COM-B domains. {#pone.0171932.t002g} ------------------------------------------------------------------------------------------------------- COM-B domain Theme -------------- ---------------------------------------------- ----------------------------------------- Capability Barrier • Management of complex patients (N)\ • Belief that falls are inevitable (N)\ • Ward layout (N) Enabler • Training and education (N and SS)\     ■ Face-to-face education\     ■ Case study based teaching Opportunity Barrier • Lack of resources (N) Enabler • Use of falls data (SS)\     • Feedback on progress (N)\     • Competition (SS)\ • Leadership (SS) Motivation Barrier • Lack of ownership (SS)\ • Complacency (SS, N) Enabler • Goal to reduce falls (SS)\ • Engaging staff in falls prevention\     • Emotional impact of patient falls (N)\     • Improved patient outcomes (SS)\ •Audit, reminders and feedback (N and SS) ------------------------------------------------------------------------------------------------------- N = nurses, SS = senior staff 10.1371/journal.pone.0171932.t003 ###### Strategies to optimise successful implementation of the 6-PACK program {#pone.0171932.t003g} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- COM-B domain Rationale Implementation strategy -------------- -------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------- Capability • Improve knowledge and skills\ • Regular practical face-to-face education and training for nurses (ward walk arounds, small interactive group sessions) • Support attitudinal change\ • Model new behaviours Opportunity • Provide and discuss data\ • Provision of falls data\ • Inform about progress • Leadership and champions (ward champions, Nurse Unit Managers)\ • Provision of equipment\ • Newsletters and posters communicating progress, achievements and stories Motivation • Reinforce key strategies for falls prevention\ • Compliance audits\ • Troubleshoot and provide support\ • Reminders and feedback\ • Demonstrate commitment to project • Reward and recognise change in practice and leadership -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Capabilities {#sec012} ------------ ### Management of complex patients {#sec013} Implementing falls prevention interventions was viewed as difficult, particularly when treating patients with complex health issues. A nurse's ability to manage multiple risks including pressure areas, medications, nutrition and falls was described as a "*daunting balancing act"*. > When you're looking at those more elderly, confused, aggressive patients, it's weighing up between the falls risk versus the medication management to keep them settled...it is a balancing act. (SS3, Hospital (H) 3) ### Belief that falls are inevitable {#sec014} Many nurses reported that they were unable to prevent falls, despite feeling they had knowledge in falls prevention. They identified a number of patient characteristics that they perceived were associated with high falls risk and not amenable to falls prevention interventions. > We've got patients on the ward who are in the high visibility area, on low-low beds, have the pressure sensor, \[yet\] they are still falling... I don't think falls can be prevented. (Nurse, H3) > > We've got dementia patients... You can do as much as you can, and \[falls are\] still just going to happen...I don't think falls can be prevented. (Nurse, H3) Only 46% of nurses responding to the survey disagreed with the statement 'You can't stop older people from falling', while 23% were undecided. This suggests discord in beliefs regarding the inevitability of falls. Senior staff were less likely than nurses to accept the inevitability of falls based on patient characteristics, and emphasised the need to minimise the impact of falls. > \[Falls\] should be preventable. We shouldn't have them. I think it's about changing that perception and that belief, and that awareness, \[to be\] that actually any fall is wrong, it shouldn't have happened. (SS2, H6) ### Ward layout {#sec015} The layout of the ward was often perceived by nurses as a hindrance to surveillance. Single rooms made it difficult for nurses to physically move efficiently from one patient to another. > Sometimes we have four \[high risk\] patients in three different rooms, it's a disaster...how do you get to look at everyone at the same time? (Nurse, H5) > > Say if you're stuck in a room or a bathroom with someone and someone else buzzed...you mightn't see that \[patient\] for 20 minutes because you're in doing a massive dressing or \[something else\]. (Nurse, H1) ### Training and education {#sec016} Improving knowledge and skills through training and education sessions were identified as enablers to falls prevention practice. Survey data indicated only 32% of nurses felt they received useful training from falls prevention leaders. Senior staff valued e-learning methods as they believed that information could be conveyed efficiently. > I put a module of falls strategies on e-Learning so nurses can access information on falls. (SS1, H5) Nurses raised issues such as lack of access to computers and *"no time to get to the computers*" as a barriers to e-learning education. Nurses specified that although e-learning was convenient, practical and hands-on training on the ward with case studies was preferable to increase their capabilities in falls prevention. > We all prefer face-to-face learning rather than e-learning...I think you learn more with real-life situations. (Nurse, H5) Senior staff valued ongoing feedback and case review as an effective means of enhancing falls prevention knowledge. > I educate the staff every month about falls we've had...I explain strategies that could have been improved. I go and speak to staff who have been involved in a fall and find out why a strategy wasn't put in place, what were the obstructions to that, and the circumstances around it. (SS3, H3) In addition to discussions on the delivery mode for education, staff raised suggestions for education content. Nurses identified the specific need for education on the treatment of delirium and management of patients with cognitive impairment. Senior staff highlighted the need for training on how to connect fall-risk tool scores to appropriate interventions. Opportunity {#sec017} ----------- ### Access to resources {#sec018} A key barrier identified in the implementation of the 6-PACK program was access to resources. One of the interventions of the 6-PACK program was to put high risk patients on a low-low bed, however there was not a sufficient number of beds available on the wards for nurses to use. This was further complicated by a lack of tracking systems of where the beds are within a hospital. > We have about 12 low-low beds, which is not sufficient. There is no system of tracking where the Low-Low bed is. The poor nurse has to ring Environmental Services or six different wards to see if they have a Low-Low bed. (SS2, H2) > > If I've identified someone as a high-falls risk, I've got to put an intervention in place, \[but\] we don't have the resources \[equipment\] to do that. (Nurse, H3) ### Use of data to drive practice change {#sec019} Senior staff highlighted the need to ensure that nursing staff understood the extent of the problem of falls on the wards. This involved presenting data on the trends and benchmarking of ward falls across wards. > Here's our data, this is what we're looking like and your patient safety boards...I think that's really valuable because it puts your performance up there to be seen as a trend; they can be benchmarking against themselves. (SS3, H1) The majority of nurses surveyed (75%) agreed with the statement *'incident reporting provides us with a way of measuring how we are going with patient falls'*. Providing this ***feedback on progress*** in falls prevention to nurses was seen as an opportunity to encourage and promote practice change. Participants were asked if using data to promote competition between wards would encourage falls prevention action. While senior staff believed *"a bit of competition between wards"* was a good idea, nurses were less positive as they felt ward experiences would vary due to different patient characteristics. > Oh, I don't think it would make any difference. We've all got different patients. (Nurse, H1) ### Leadership {#sec020} Leadership, including the establishment of champions for falls prevention was identified as a key enabler for practice change. Leaders were identified by staff as playing a critical role in providing guidance and support to those less experienced, and to develop and promote standardised practices in terms of implementing falls prevention interventions. Nurses were either neutral (35%) or agreed (42%) that there was strong leadership support for falls on their ward and that their supervisors have assisted them when issues of falls have been raised (64%). Senior staff reported that the NUM has a critical role in falls prevention. > The NUMs are important players in \[falls prevention\]...to educate staff and support them about the right techniques. (SS3, H3) NUMs were also seen as vital in ensuring the sustainability of the program. > \[NUMs\] are going to be the drivers, not just from the beginning but in six months' time when it's implemented. (SS1, H3) Champions were identified as a practice change strategy for other projects including infection control, pain management and wound care. They were able to provide a link between committees, senior management and the ward staff and provide education and support while on the wards. > The falls champion on that ward will play a very active role in delivering the education and doing the assessments ...because that links back to the Falls Committee. (SS3, H4) Senior staff emphasised that the key to a successful champion is finding staff who have *"the passion for falls and wants to make a difference to patient care"* and willing to push the agenda of falls prevention on the wards. One staff member described champions as '*resource people*'. Motivation {#sec021} ---------- ### Lack of ownership {#sec022} A perceived barrier to the implementation of the 6-PACK program was a lack of ownership for falls prevention in some hospitals. > Who drives falls? Nobody owns falls. (SS3, H2) The majority of nurses (80%) believed that they were responsible for falls prevention. Senior staff agreed that nurses were primarily responsible but recognised the value of multidisciplinary input into falls prevention. > It's everyone's responsibility to work together to reduce falls. But I suppose primarily it comes back to nurses as they're there with the patient 24/7. (SS1, H1) ### Complacency {#sec023} Reflecting on previous and current falls prevention practice, staff recognised that one barrier to practice change was complacency. Complacency was often discussed in relation to the completion of fall-risk tools. Prior experience of staff suggested that complacency in completing these tools could be an issue with nurses stating "*we all just go tick*, *tick*, *tick*, *tick"*. > Staff just tick the same boxes that were done yesterday without really assessing...There's that difficulty of just that complacency of ticking the same boxes...that doesn't give you the best outcome. (SS1, H3) To address issues of complacency, ***audits*, *reminders and feedback*** were suggested by staff. > Better to be reminded to do this, and reminded all the time. (Nurse, H6) > > The other thing that we have a gap in is that we don't do regular auditing...It's about the audits and the feedback that's given. (SS1, H1) ### Falls prevention goals and commitment {#sec024} An enabler to falls prevention was a commitment to falls prevention by senior staff demonstrated through provision of resources (equipment and staff) as well as clearly articulated goals. Participants believed this provided motivation and was also a source of pride and achievement when progress was being made. > So it's pride in falls, reduction in falls. Commitment by staff. And it's commitment by management...if they're going to have the need for low-low beds or whatever you need, \[they will get it\]. Implementation care is paramount. (SS2, H3) ### Engaging staff in falls prevention {#sec025} As highlighted by one senior staff participant, staff engagement is important and can be facilitated through '*engaging hearts and minds'---*both the emotional and logical aspects of falls prevention. Nurses described feeling '*guilty'*, '*stressed'* and '*distressed'* when a patient under their care experienced a fall. They also described the '*worry'* experienced if a patient suffered a fall-related injury. The ***emotional impact of a patient fall*** was seen as something that could be a motivating factor. A senior staff member at one hospital highlighted that nurses responded to interventions that ***emphasised the benefit to the patient***. This also had implications for sustaining the project long term. > If you always promote it as best for the patient and patient focused you'll get staff on-board, and continuing to help drive the program. You've got to be able to sell it to them...first of all say this is going to be so much better for your patient outcomes. (SS1, H1) Discussion {#sec026} ========== This study identified a number of implementation targets, particularly in the areas of motivation and opportunity. These included education and training to address skills, knowledge and beliefs of nurses and developing systems to encourage falls prevention practice such as audits, reminders and feedback, provision of equipment and facilitating a culture of falls prevention through leadership and champions. Previous studies have also reported the above enablers \[[@pone.0171932.ref004]--[@pone.0171932.ref006]\]. Unlike prior research, this study details differences between nurses and senior staff beliefs regarding falls prevention. Learnings from this study were used to develop an implementation plan for the RCT \[[@pone.0171932.ref008]\]. The belief in the inevitability of falls is consistent with findings from other studies \[[@pone.0171932.ref006], [@pone.0171932.ref019]\]. Although survey results suggest nurses thought falls could be prevented, nurses in focus groups identified patient groups where they believed falls could not be prevented. There was disagreement between nurse and senior staff perspectives as to whether in-hospital falls could be prevented. Incongruity between nurses' and senior staff perceptions of the inevitability of falls has implications for the success of a falls prevention program. If nurses do not believe falls can be prevented, it may be difficult to implement interventions that aim to prevent falls. Senior staff recommended that education and training was required to increase nurse confidence and knowledge in how to prevent falls and to utilise the resources provided effectively. Education was identified as a strategy to improve capabilities. However, implementation did raise some practical challenges. While both senior staff and nurses valued face to face case studies, senior staff favoured e-learning due to convenience and efficiency. Carefully designed e-learning packages can be effective in disseminating best practice education and have the potential to reach less accessible night and casual staff \[[@pone.0171932.ref020]\]. However, if a model of e-learning was adopted it would be important to ensure nurses have adequate access to computers and that these packages address aspects of falls prevention that are of greatest concern to nurses. A motivator identified by senior staff was to harness the emotional impact of falls, for example through 'story telling' of falls incidents at handover. Case studies with patient stories and experiences of falls may also prove powerful in highlighting the need to address in-hospital falls in education sessions. Communicating clearly the patient benefits of the 6-PACK program was also seen to be a strategy to enhance engagement by staff. A challenge to motivation is complacency in falls prevention practice. The acute setting is a crowded landscape of patient safety initiatives that can compete for the attention and time of nurses. Previous research has described the phenomenon of 'missed care' or 'unfinished care' where nurses can find it difficult to achieve all their tasks in caring for a patient. This can lead to adverse patient events such as falls \[[@pone.0171932.ref021]\]. To promote continuing engagement in strategies and to assist in care prioritisation, senior staff and nurses highlighted the importance of regular audits, reminders and feedback. Audits, reminders and feedback are generally an effective approach in guiding the implementation of an intervention \[[@pone.0171932.ref022]\]. Providing data to demonstrate the extent of the problem of falls on wards and to benchmark progress was another strategy identified by participants. Incident reporting has also been identified as a useful approach to change the attitudes, perceptions and practice of staff and promote engagement in patient safety initiatives \[[@pone.0171932.ref023]\]. The majority of falls prevention programs are focused on nurses and nursing interventions with falls often considered a nursing sensitive patient outcome \[[@pone.0171932.ref024]\]. However, a barrier to motivation identified is the lack of ownership for falls prevention. Senior staff stated that falls prevention should involve a multidisciplinary team approach and is everyone's responsibility. Conversely, a recent study in Australian hospitals reported that doctors perceived time limitations as a major barrier to their involvement in falls prevention and acknowledged that medical priorities were more important for them \[[@pone.0171932.ref025]\]. While 6-PACK is a nurse led program, it does not discourage involvement of other clinicians. Indeed, nurses are a critical link between the patient and other care team members and often are responsible for communicating on progress and changes in patient status. This importance of the nurse role in multi-disciplinary management of falls should be communicated to staff in training. The opportunity domain examined factors outside of the individual which enable or prompt falls prevention action. The key themes related to opportunity included lack of availability of resources, provision of falls data and leadership for falls prevention. The lack of availability of falls prevention equipment, such as low-low beds, has previously been described \[[@pone.0171932.ref019], [@pone.0171932.ref024]\]. Leadership is both an opportunity and motivation strategy and was recognised as important by both nurses and senior staff. NUMs and champions were identified as key individuals in the implementation and sustainability of falls prevention interventions. The need for leadership and champions has been reported as an important implementation strategy in the literature \[[@pone.0171932.ref006], [@pone.0171932.ref026], [@pone.0171932.ref027]\]. Limitations and future research {#sec027} ------------------------------- The 6-PACK program is a nurse delivered intervention and therefore the focus of this research was to seek the perspective of nursing staff. The perspectives of other health professionals (doctors, allied health professionals such as physiotherapists, occupational therapists) involved in direct patient care were not captured in this study. Further research to explore whether the barriers and enablers identified by nurses and senior staff are also identified by other hospital staff is required. The wards that participated in this study volunteered to take part in the 6-PACK RCT which may have introduced selection bias. This may have impacted on the results with participants being more likely to recognise the importance of falls prevention practice. Conclusions {#sec028} =========== This study identified barriers and enablers to the implementation of the 6-PACK program corresponding to the constructs of capability, opportunity and motivation. Barriers identified included beliefs that falls could not be prevented; limited knowledge on falls prevention in patients with complex care needs (e.g. cognitive impairment); lack of resources; and lack of ownership in falls prevention efforts. Enablers included education and training, particularly face to face case study based approaches; improved leadership; using data to drive practice change; and use of reminders, audits and feedback. Successful falls prevention program implementation in acute hospital wards are likely to require a multifaceted, planned approach that includes: regular practical face-to-face education and training for nurses to modify skills and established beliefs; provision of equipment; audit, reminders and feedback; leadership and champions; and the provision of falls data. Supporting information {#sec029} ====================== ###### 6-PACK programme to decrease fall injuries in acute hospitals: cluster randomised controlled trial (published article). (PDF) ###### Click here for additional data file. ###### Development of an implementation plan for the 6-PACK falls prevention programme as part of a randomised controlled trial: protocol for a series of preimplementation studies (published article). (PDF) ###### Click here for additional data file. We acknowledge Jeannette Kamar and the Injury Prevention Unit, The Northern Hospital, Northern Health, Melbourne, Australia who developed the 6-PACK Program. The study could not have been completed without the collaboration and support from the participating hospitals, site clinical leaders and nursing staff. [^1]: **Competing Interests:**The authors have declared that no competing interests exist. [^2]: **Conceptualization:** AB CB FL SB KH PL MB.**Data curation:** DA AB RM JT MB.**Formal analysis:** DA AB JT.**Funding acquisition:** AB CB FL SB KH PL MB.**Investigation:** AB RM CB FL SB KH PL MB.**Methodology:** DA AB RM CB FL SB KH PL MB.**Project administration:** RM AB DA JT.**Resources:** DA AB RM JT.**Software:** DA AB JT MB.**Supervision:** AB RM MB.**Validation:** DA AB RM MB.**Visualization:** DA AB RM.**Writing -- original draft:** DA JT AB RM.**Writing -- review & editing:** DA JT AB RM CB FL MM EB SB KH PL MB. [^3]: ‡ These authors also contributed equally to this work.

I am disappointed. Very.When the final tally came in, turns out you guys sent a total of 567 emails during the course of the fourth ODI at Ahmedabad.That is less than the number of balls bowled in......

Q: Script works on laptop, but not on PI I have a streaming server / motion detection script that works on my laptop, but when I move it to my PI it gives the following error when i try and access the stream: my laptop ip address - - [23/Jun/2015 13:04:34] "GET / HTTP/1.1" 200 - Exception in thread Thread-2: Traceback (most recent call last): File "/usr/lib/python2.7/threading.py", line 552, in __bootstrap_inner self.run() File "/usr/lib/python2.7/threading.py", line 505, in run self.__target(*self.__args, **self.__kwargs) File "/usr/src/scripts/areadetect_movie__33.py", line 300, in server server = HTTPServer(('',port),CamHandler) File "/usr/lib/python2.7/SocketServer.py", line 419, in __init__ self.server_bind() File "/usr/lib/python2.7/BaseHTTPServer.py", line 108, in server_bind SocketServer.TCPServer.server_bind(self) File "/usr/lib/python2.7/SocketServer.py", line 430, in server_bind self.socket.bind(self.server_address) File "/usr/lib/python2.7/socket.py", line 224, in meth return getattr(self._sock,name)(*args) error: [Errno 98] Address already in use a) The motion side of the script works, but the server doesn't. b) I can serve a simple webpage using the same IP address and port, using a different script, so I don't think something else is using that port. c) Both PI and laptop are on the same local network. 1) Are there any other tests I can do, that could point to what is wrong? 2) What do these errors suggest? Edit: ps aux | grep streaming_server_variation_16.py gives: root 2508 79.7 11.9 111392 45560 ? Sl 13:19 1:53 /usr/bin/python ./streaming_server_variation_16.py root 2560 0.0 0.8 5380 3200 pts/0 S+ 13:21 0:00 grep streaming_server_variation_16.py I'm starting two threads like this: if runtime_frame_count == 0: Thread(target = main).start() time.sleep(10) Thread(target = server).start() note 'runtime_frame_count' increments, so this should only run once. The "main" thread gathers the frames from the camera and performs the motion detection, and the "server" tries to serve the frames gathered. Could this be why I'm having the same script trying to access the port, but in different threads? edit 2 to explain exactly what i'm doing: a) I'm accessing the headless PI, via two open SSH putty connections, from my laptop which has a local IP address of 192.168.1.4 and the PI has a local IP address of 192.168.1.61. b) I run the script using one SSH connection and watch the script increment the runtime_frame_count which prints out on each new frame gathered; i'll call this SSH window run-ssh-window, then I use the other SSH connection to send the commands, i'll call this command-ssh-window. c) When thread 2 starts (server) the script outputs from the run-ssh-window server started on: 127.0.1.1 because of the following code inside that thread: print ("server started on: ") print(socket.gethostbyname(socket.gethostname())) d) Then I try and access the stream from my laptop's browser by placing the text http://192.168.1.61:1991/ in the browser, when I get the same errors as listed above from the running incrementing script (in the run-ssh-window). e) following the error, the script continues to increment (in the run-ssh-window), so in the command-ssh-window i sudo netstat -atpen which gives: Active Internet connections (servers and established) Proto Recv-Q Send-Q Local Address Foreign Address State User Inode PID/Program name tcp 0 0 0.0.0.0:8080 0.0.0.0:* LISTEN 0 6188 1962/uv4l tcp 0 0 0.0.0.0:22 0.0.0.0:* LISTEN 0 6509 2306/sshd tcp 0 0 127.0.0.1:6010 0.0.0.0:* LISTEN 0 35991 11960/0 tcp 0 0 127.0.0.1:6011 0.0.0.0:* LISTEN 0 36885 12135/1 tcp 0 352 192.168.1.61:22 192.168.1.4:50175 ESTABLISHED 0 36843 12135/1 tcp 0 0 192.168.1.61:22 192.168.1.4:49853 ESTABLISHED 0 36031 11984/sshd: root@no tcp 1 0 192.168.1.61:49540 editexternaladdress:443 CLOSE_WAIT 108 7075 2284/mount.davfs tcp 0 0 192.168.1.61:22 192.168.1.4:49812 ESTABLISHED 0 35928 11960/0 tcp 0 0 192.168.1.61:22 192.168.1.4:50355 ESTABLISHED 0 37350 12214/sshd: root@no where I cannot see two processes trying to access the port 1991 on 192.168.1.61. Am I doing anything incorrectly? I'm still confused what could be happening with the "address already in use" error listed above. Another edit resulting from comments I start the socket using the command: server = HTTPServer(('',port),CamHandler) which calls the CamHandler class to deal with constructing the stream. the CamHandler class looks like this: class CamHandler(BaseHTTPRequestHandler): def do_GET(self): print (self.path) if self.path.endswith('.mjpg'): self.send_response(200) self.send_header('Content-type','multipart/x-mixed-replace; boundary=--jpgboundary') self.end_headers() while(True): global imgRGB #imgRGB is gathered from the camera in the main thread, which starts 10 seconds prior to the server thread (this one) r, buf = cv2.imencode(".jpg",imgRGB) # imencode Encodes an image into a memory buffer. # the following 7 lines output the contents of buf to the webpage self.wfile.write("--jpgboundary\r\n") self.send_header('Content-type','image/jpeg') self.send_header('Content-length',str(len(buf))) self.end_headers() self.wfile.write(bytearray(buf)) self.wfile.write('\r\n') time.sleep(0.01) k = cv2.waitKey(20) if k == 27: break cv2.destroyAllWindows() capture.release() if self.path.endswith('.html') or self.path=="/":# jumps into this block of code at start of request to set up the html page but doesn't come back self.send_response(200) self.send_header('Content-type','text/html') self.end_headers() self.wfile.write('<html><head></head><body>') self.wfile.write('<img src="http://'+socket.gethostbyname(socket.gethostname())+':'+ str(port) +'/cam.mjpg"/>') self.wfile.write('</body></html>') return I too thought there was an issue with the script outputting 127.0.1.1 to I tried:self.wfile.write('<img src="http://'+'192.168.1.61'+':'+ str(port) +'/cam.mjpg"/>') but still I got the same error. I'm using windows 7 too. Yet another edit with a simplified version of the whole script. #==================================================================== # a function to get the image from the camera (this is the only place the camera is accessed) #==================================================================== def get_cv_frame(): global img while(1): rc,img = capture.read() if not rc: continue return (img) def cv_frame(frame): #all the cv2 movement magic happens and it returns a modified image (numpy array) return background # the server function def server(): global runtime_frame_count, runtime_frame while(1): #======================================= server started here #======================================= server = HTTPServer(('',port),CamHandler) #======================================= #======================================= print ("server started on: ") print(socket.gethostbyname(socket.gethostname())) print("server started") q1 = cv2.waitKey(20) if (q1 == ord('q')) or (runtime_frame_count >= runtime_frame): capture.release() server.socket.close() print("server stopped") cv2.destroyAllWindows() cv2.waitKey(10) time.sleep(0.1) cv2.waitKey(10) cv2.waitKey(10) break server.handle_request() return class CamHandler(BaseHTTPRequestHandler): def do_GET(self): print (self.path) if self.path.endswith('.mjpg'): self.send_response(200) self.send_header('Content-type','multipart/x-mixed-replace; boundary=--jpgboundary') self.end_headers() while(True): global imgRGB r, buf = cv2.imencode(".jpg",imgRGB) # imencode Encodes an image into a memory buffer. self.wfile.write("--jpgboundary\r\n") self.send_header('Content-type','image/jpeg') self.send_header('Content-length',str(len(buf))) self.end_headers() self.wfile.write(bytearray(buf)) self.wfile.write('\r\n') time.sleep(0.01) k = cv2.waitKey(20) if k == 27: break cv2.destroyAllWindows() capture.release() return if self.path.endswith('.html') or self.path=="/":# jumps into this block of code at start of request to set up the html page but doesn't come back self.send_response(200) self.send_header('Content-type','text/html') self.end_headers() self.wfile.write('<html><head></head><body>') self.wfile.write('<img src="http://'+socket.gethostbyname(socket.gethostname())+':'+ str(port) +'/cam.mjpg"/>') self.wfile.write('</body></html>') return # the main function def main(): global capture, average, img, imgRGB, runtime_frame_count, runtime_frame capture = cv2.VideoCapture(0) capture.set(3,REDUCED_SIZE[0]) capture.set(4,REDUCED_SIZE[1]) while(1): img = get_cv_frame() # gets a frame from the camera imgRGB = cv_frame(img) #all the cv2 movement magic happens and it returns a modified image (numpy array) time.sleep(0.1) print (runtime_frame_count) runtime_frame_count = runtime_frame_count+1 q2 = cv2.waitKey(20) if (q2 == ord('q')) or (runtime_frame_count >= runtime_frame): capture.release() server.socket.close() print("main stopped") cv2.destroyAllWindows() cv2.waitKey(10) time.sleep(0.1) cv2.waitKey(10) cv2.waitKey(10) break return # this is where the threads are started. if runtime_frame_count == 0: Thread(target = main).start()# starts the main() function as a thread time.sleep(1) Thread(target = server).start()# starts the server() function as a new thread a time delay after the first thread was started A: I'm no Python expert but I'm surprised that in the server function you create the HTTP socket within the infinite loop. So basically (if I'm not mistaken) you create after each request handling a new server binding. Therefore, I'm not surprised that you get address already in use. Solution: Try to move the server = HTTPServer(('',port),CamHandler) before while(1): # the server function def server(): global runtime_frame_count, runtime_frame server = HTTPServer(('',port),CamHandler) print ("server started on: ") print(socket.gethostbyname(socket.gethostname())) print("server started") while(1): q1 = cv2.waitKey(20) if (q1 == ord('q')) or (runtime_frame_count >= runtime_frame): capture.release() server.socket.close() print("server stopped") cv2.destroyAllWindows() cv2.waitKey(10) time.sleep(0.1) cv2.waitKey(10) cv2.waitKey(10) break server.handle_request() return Previous answer Which port are you using? Is there another application using the same port? You can check the list of used port with this command: sudo netstat -atpen And if you know the port number, you can just grep it (e.g. port 8080): sudo netstat -atpen | grep 8080 Note: sudo will allow you to see the process which is using a particular port. If you don't use sudo, then some processes might not be displayed (but the port usage will be) PS: I've updated my answer to remove UDP as obviously you are trying to bind a TCP socket.

This invention relates to the field of methods for catalytically oxidizing carbon monoxide to carbon dioxide, and more particularly, to methods for selectively catalytically oxidizing carbon monoxide present in a hydrocarbon stream using a specially prepared supported cobalt oxide catalyst. Numerous synthetic routes to such products as acetic acid, acrylic acid, vinyl acetate and benzyl acetate, rely upon the catalytic oxidation of a hydrocarbon, for example, ethylene in the case of acetic acid and vinyl acetate, propylene in the case of acrylic acid, and toluene in the case of benzyl acetate. In addition to the principal reaction product, small quantities of carbon monoxide are also produced in these reactions. Following the recovery of the principal product, the reaction effluent is recycled so that any unreacted starting material can be fully utilized. It is known that carbon monoxide deactivates the catalysts, for example, palladium metal, which are commonly used for these syntheses. Accordingly, it is highly desirable that the carbon monoxide be converted to the dioxide prior to the recycling of the reaction effluent without, however, causing any significant oxidation of the unreacted hydrocarbon present in the effluent. It is well known that cobalt oxide is a useful catalyst for the oxidation of carbon monoxide to carbon dioxide. Ismailov et al., Azerb. Khim. Zh., 1969, (4), 75-80 (Russian), describes the oxidation of carbon monoxide and gaseous hydrocarbon present in a simulated internal combustion engine exhaust using a supported cobalt oxide catalyst. The maximum level of carbon monoxide conversion was 50% and was achieved at 800.degree. C. Belgium Pat. No. 814,130 describes a cobalt aluminate catalyst useful in gas masks and as a component of cigarette filters. U.S. Pat. No. 3,839,545 describes the accelerated combustion of both carbon monoxide and hydrocarbons in exhaust gases using a catalyst containing a mixture of copper and manganese oxides and the oxides of other metals such as cobalt. The oxidation of hydrocarbons and CO is described in the article entitled "The Oxidation of Hydrocarbons and CO over Metal Oxides III. Co.sub.3 O.sub.4 " by Yung-Fan Yu Yao, Journal of Catalysis, 33, 108-122 (1974). In none of the foregoing is there a selective oxidation of carbon monoxide to the virtual exclusion of the hydrocarbon component which may be present. The catalytic oxidation of propylene is described in the article entitled "Regularities in Catalytic Properties of Metal Oxides in Propylene Oxidation" by U. Morooka and A. Ozaki, Journal of Catalysis, 5, 116-124 (1966). A series of experiments is discussed therein and shows that a cobalt oxide catalyst, prepared by calcining silicon carbide pellets impregnated with cobalt nitrate, readily oxidized propylene.

Responses of tooth eruption and alveolar bone subject to somatic growth retardation in the rat. The morphogenesis and regression of the osteodental fissure formed by alveolar bone in the maxillary and mandibular regions has been investigated in relation to eruption of the dentition during and following a period of somatic retardation stemming from nutritional suppression. Fissural formation occurred above the first and second molars of the maxilla and mandible, morphologically within normal limits but retarded by two days. During eruption a sequence of cuspal perforations of the alveolar bone took place behind the edge of the alveolar crest which later disintegrated as the bulk of the crown moved upwards. The appearance of the specific cusps of the experimental animals was two days behind that of the controls. Eruption through the oral mucosa was fairly rapid and the deficit noted in the bone emergence phase was reduced to one day. As in control animals, fissural formation did not occur over the third molars which were totally encapsulated by bone. Eruption of third molars in experimental animals was similar to control observations- no difference in timing and the removal of the encapsulating bone being achieved by the rapid enlarging of a wedge-shaped area at the mesio-occlusal aspect. Observation of eruption through alveolar bone is regarded as a more accurate assessment of changes in the early phases.

Kurt Müller (footballer) Kurt "Kudi" Müller (born 9 May 1948) is a Swiss former footballer who was capped on 38 occasions by the Switzerland national football team, scoring eight goals. He started his career with FC Emmenbrücke and then played for FC Luzern and Grasshopper Club Zürich. In January 1973, he moved to Germany to play club football for Hertha BSC. Müller went on to make 77 league appearances for the Berlin side over the next two and a half seasons, and in 1974–75 he helped Hertha finish as runners-up in the Bundesliga. After leaving Hertha, he returned to Switzerland to play for Servette FC for the 1975–76 season, and he played for them in the final of the 1975 Coppa delle Alpi tournament, in which they defeated FC Basel 3–0. Müller finished his career with the Bern team BSC Young Boys. Müller made his international debut for Switzerland on 15 December 1970, in a UEFA Euro 1972 qualifying match against Greece. His final international cap came almost seven years later in a 1978 FIFA World Cup qualification match against Norway on 30 October 1977. Müller coached SC Kriens in 1986–87. He later became manager at FC Luzern, and he was in charge when they reached the final of the Swiss Cup in 1996–97, although they lost on penalties to FC Sion. References Category:1948 births Category:Living people Category:Swiss footballers Category:Association football forwards Category:Switzerland international footballers Category:Swiss Super League players Category:Bundesliga players Category:FC Luzern players Category:Grasshopper Club Zürich players Category:Hertha BSC players Category:Servette FC players Category:BSC Young Boys players Category:Swiss expatriate footballers Category:Expatriate footballers in Germany Category:Swiss expatriate sportspeople in Germany Category:Swiss football managers Category:FC Luzern managers

1. Technical Field The present invention relates to a positioning device, a mobile terminal, a positioning method, and a positioning program and particularly to those suitably used for a GPS (Global Positioning System). 2. Related Art At present, as a ground position determination system, a GPS which is called NAVSTAR and operated by the U.S. government is known as disclosed in Parkinson, Bradford W Gilbert, Stephen W.; “NAVSTAR: Global Positioning System—Ten Years Later”; Proceedings of the IEEE; Vol. 71; No. 10. This GPS is a wireless navigation system which is designed for providing highly accurate three-dimensional information to a recipient on or near the ground, the GPS using a satellite as a transmitter. Also, the former Union of Soviet Socialist Republics government operates a GPS known as GLONASS. Further, a system known as GALILEO is under development by the European Community. Recently, a GPS receiving device is used as a car navigation system. Also, in JP-A-6-118156, for example, a method of downsizing the GPS receiving device to obtain a direction and a distance of a person's movement while the person is walking so that the person can carry the device is disclosed. Further, JP-A-2001-133535, for example, discloses a method of reducing positioning time by using location information of a base station acquired by a PHS communication as an initial location for the positioning using the GPS. JP-A-2001-235337, for example, discloses a method of reducing positioning time by using a field intensity map of a base station in a mobile communication network as an initial location for the positioning using the GPS. However, in the method of reducing GPS positioning time by using the location information of the base station on the mobile communication network as the initial location, it is necessary to receive the location information by communicating with the base station or to store the location information of the base station in a GPS receiver. The method of communicating with the base station for the purpose of acquiring the location information of the base station has the problem of communication cost, thereby undesirably increasing a burden on a user. Also, the method of storing the location information of the base station in the GPS receiver has the problem that, in the case where multiple base stations exist at an interval of a several kilometers, a large memory capacity is required for storing the base stations. Further, with the method of using the location information of the base station on the mobile communication network as the initial location, even in the case where the previous positioning information can be used as the initial location for the current positioning, it is necessary to confirm the location information of the base terminal by communicating again with the base terminal and to use the newly obtained location information of the base terminal as the initial location for the current positioning, thereby raising a problem of wasteful communication. Accordingly, an object of the present invention is to provide a positioning device, a mobile terminal, a positioning method, and a positioning program capable of reducing the time required for positioning by a GPS without allowing specific parties to occupy a communication channel.

THIRD DIVISION March 28, 2007 No. 1-06-2216 CINCINNATI INSURANCE COMPANY, ) Appeal from the as Subrogee of Harbour Contractors, Inc., ) Circuit Court of BAKER CONCRETE CONSTRUCTION, and ) Cook County. NICHOLAS NOWICKI, ) ) Plaintiffs, ) ) v. ) ) Nos. 95 CH 10953, GATEWAY CONSTRUCTION COMPANY, INC.,) 95 CH 6802, ) 97 CH 1126 Defendant and Counterplaintiff-Appellant ) ) ) (Lexington Insurance Company, ) The Honorable ) David R. Donnersberger, Defendant and Counterdefendant-Appellee). ) Judge Presiding. PRESIDING JUSTICE THEIS delivered the opinion of the court: This appeal arises from an order of the circuit court granting summary judgment in favor of defendant Lexington Insurance Company (Lexington) against plaintiff Cincinnati Insurance Company (Cincinnati)1 and defendant, counterplaintiff Gateway Construction Company, Inc. (Gateway). On appeal, the dispute concerns whether certain parties were covered as additional insureds under the terms of Gateway’s excess liability policy issued by Lexington. Gateway contends that the trial court erred in erroneously interpreting the underlying policy endorsement language, arguing that an oral promise to name someone as an additional insured, memorialized 1 Cincinnati is not a party to this appeal. 1-06-2216 in writing after the injury for which coverage is sought, is sufficient to create additional insured status under the policy. For the following reasons, we disagree and affirm the judgment of the circuit court. BACKGROUND Harbour Contractors, Inc. (Harbour), a general contractor, entered into an agreement with Willowbrook Center Associates to construct a Mark Shale Warehouse facility designed by architect Nicholas Nowicki. Harbour also entered into an agreement with Baker Concrete Construction (Baker), a subcontractor, under which Baker was responsible for the concrete work on the project. Baker, in turn, entered into an informal, unwritten agreement with subcontractor Gateway to install certain concrete reinforcements for the project, which agreement was later memorialized in writing and executed by Baker on June 4, 1990. On January 10, 1990, prior to the execution of the written agreement, Thomas Scully, a metal worker employed by Gateway, was injured on the jobsite. On the date of the injury, Baker was insured under a comprehensive general liability (CGL) policy issued by Liberty Mutual Insurance Company (Liberty) and under an excess liability policy issued by Cincinnati. Harbour and Nowicki were covered as additional insureds under Baker’s policies. Gateway was insured under a CGL policy issued by National Union Insurance Company (National) and under an excess liability policy issued by Lexington. It is alleged, for purposes of summary judgment, that Gateway’s representative had orally agreed to name Baker, Harbour and Nowicki as additional insureds under its policy with National. The Baker/Gateway written agreement was drafted on January 26, 1990, after the injury. The original 2 1-06-2216 agreement did not contain any additional insured requirements. Subsequently, an addendum to the agreement, dated February 7, 1990, included an additional insured provision and that contract was executed by Baker five months after the accident on June 4, 1990. A certificate of insurance naming Baker, Harbour and Nowicki as additional insureds was issued two months after the accident, on March 15, 1990. In 1991, Scully brought suit against Harbour, Baker and Nowicki to recover for his injuries. Harbour, Baker and Nowicki tendered the defense of the suit to National. National denied its duty to defend, maintaining that Harbour, Baker and Nowicki were not additional insureds under its policy with Gateway because a written agreement to add them had not been executed at the time of Scully’s accident. As a result, the defense was taken over by Liberty, Baker’s primary insurer. In 1996, Scully’s suit settled for $2.5 million. Liberty paid its policy limit, and Cincinnati, Baker’s excess insurer, paid the remainder of the settlement. Liberty then assigned its right to seek recovery of its portion of the settlement to Cincinnati. Meanwhile, in 1995, Harbour, Baker and Nowicki filed suit against National seeking a declaration that National had a duty to defend and indemnify them in Scully’s underlying suit. They argued that they were additional insureds under National’s policy with Gateway by virtue of Gateway’s oral promise to make them additional insureds. Cincinnati, as the subrogee of Harbour, Baker and Nowicki, was substituted as the plaintiff in that suit. Cincinnati and National filed cross-motions for summary judgment. After the circuit court denied those motions, Cincinnati and National subsequently settled their dispute for the limits of the National policy. 3 1-06-2216 Cincinnati then filed suit against Lexington and Gateway, alleging that Harbour, Baker and Nowicki were additional insureds under Gateway’s policies with both National and Lexington, and, therefore, were entitled to excess coverage from Lexington. Alternatively, Cincinnati alleged that Gateway breached its oral contract with Baker when it failed to obtain insurance for Baker, Harbour and Nowicki. Gateway filed a counterclaim against Lexington alleging that it had breached its contract with Gateway in denying excess coverage to Harbour, Baker and Nowicki. Thereafter, Gateway filed a motion for summary judgment on Cincinnati’s claims against it, and also filed a motion for summary judgment with respect to its claims against Lexington. Lexington then filed a motion for summary judgment on Cincinnati’s and Gateway’s claims against it. The court found that the threshold issue was whether the alleged oral agreement between Gateway and Baker would be sufficient to provide additional insured coverage under the National and Lexington policies. The court held that the terms of National’s policy were unambiguous and that pursuant to the language of the policy, a mere oral promise was insufficient to grant coverage. Accordingly, the circuit court denied Gateway’s motions and granted summary judgment in favor of Lexington. Gateway filed its timely appeal. ANALYSIS A motion for summary judgment is properly granted when the pleadings, depositions, admissions, and affidavits on file, taken in the light most favorable to the nonmoving party, establish that no genuine issue of material fact exists and the moving party is entitled to judgment as a matter of law. 735 ILCS 5/2-1005(c) (West 2004). The standard of review of an order 4 1-06-2216 granting summary judgment is de novo. Chatham Foot Specialists, P.C. v. Health Care Service Corp., 216 Ill. 2d 366, 376, 837 N.E.2d 48, 54-55 (2005). The threshold issue raised by Gateway is whether the alleged oral agreement between Baker and Gateway to procure additional insured coverage is sufficient to provide coverage under the language of National’s policy. In interpreting the language of the policy, we must consider that an insurance policy is a contract and, thus, the rules governing the construction of other types of contracts also apply to insurance policies. Nicor, Inc. v. Associated Electric & Gas Insurance Services Ltd., 223 Ill. 2d 407, 416, 860 N.E.2d 280, 285 (2006). The primary objective is to ascertain and give effect to the intent of the parties as expressed in the agreement. Nicor, Inc., 223 Ill. 2d at 416, 860 N.E.2d at 286. In reaching that objective, the court construes the policy as a whole, taking into account the type of insurance, the nature of the risks, and the overall purpose of the contract. Nicor, Inc., 223 Ill. 2d at 416, 860 N.E.2d at 286. Where the provisions of the policy are clear and unambiguous, they will generally be applied as written, unless it contravenes public policy. Nicor, Inc., 223 Ill. 2d at 417, 860 N.E.2d at 286. Nevertheless, a term is not ambiguous merely because it is not defined or “because the parties can suggest creative possibilities for its meaning.” Nicor, Inc., 223 Ill. 2d at 417, 860 N.E.2d at 286. Rather, it is only ambiguous “if the term is susceptible to more than one reasonable interpretation.” Nicor, Inc., 223 Ill. 2d at 417, 860 N.E.2d at 286. Whether a contract is ambiguous is a question of law, subject to de novo review. Nicor, Inc., 223 Ill. 2d at 416, 860 N.E.2d at 285. National’s policy endorsement provides, in pertinent part as follows: 5 1-06-2216 “[T]he following are Additional Insureds under this policy: All corporations, partnership[s] and or/[sic] affiliated individuals promised to be added as additional insured[s] under a written contract with the Named Insured.” Gateway initially contends that the clear and unambiguous language of the policy merely requires an oral promise to name someone as an additional insured in a written contract at a later date. Alternatively, Gateway argues that the language could be construed to mean that the insured must promise, in writing, to add the additional insured, and as a result of that ambiguity, it must be construed in favor of coverage for Gateway. The only reasonable interpretation of National’s endorsement is that a promise in writing is required to grant an additional insured coverage under the policy. To hold otherwise would effectively nullify the import of the words “under a written contract” in the endorsement. A policy must not be interpreted in a manner that renders provisions of the policy meaningless. Atwood v. St. Paul Fire & Marine Insurance Co., 363 Ill. App. 3d 861, 864, 845 N.E.2d 68, 71 (2006). Indeed, Gateway concedes that a written agreement is ultimately necessary under the language of the endorsement, but contemplates that the written agreement could be made “at a later time.” Gateway’s interpretation would render the need for a written agreement meaningless because it would allow the insured to reduce an oral agreement to writing after the loss has occurred, effectively making coverage retroactive. That construct is inconsistent with the provisions of the policy that indicate that coverage is triggered at the time of the “bodily injury.” Alternatively, Gateway’s interpretation is unreasonable if it maintains that coverage is 6 1-06-2216 triggered at the time of the oral promise. Under that scenario, there may ultimately be no written agreement to procure additional insured coverage, again rendering the words “under a written contract” superfluous. If that were the case, the endorsement would merely have been drafted to provide coverage for any entity the insured promised to add as an additional insured. Nevertheless, it was not drafted in that limited way. Thus, giving reasonable meaning to each term, and construing the policy as a whole, it is evident that under the terms of National’s endorsement, the only reasonable interpretation of the language is that it requires a promise under a written agreement by the insured, in effect at the time of the claimed loss. Here, there was no promise under a written agreement at the time of the accident, and no other documentation confirming additional insured coverage at the time of the accident. Even the original draft agreement between Baker and Gateway, dated after the accident, did not provide for additional insured coverage, and the subsequent addendum adding that requirement was not executed until five months after the Gateway employee was injured. A certificate of insurance was not issued until March 1990, two months after the accident. Under these circumstances, there is no coverage. See, e.g., West American Insurance Co. v. J.R. Construction Co., 334 Ill. App. 3d 75, 80-81, 777 N.E.2d 610, 615 (2002) (although there was no written agreement as required by the endorsement, there was other evidence establishing a contractual commitment including a certificate of insurance, a letter from the insurer, and several internal memoranda from the insurer confirming additional insured status); United States Fire Insurance Co. v. Hartford Insurance Co., 312 Ill. App. 3d 153, 156-57, 726 N.E.2d 126, 129 (2000) (where insuring agreement requires a written contract, an oral contract alone is 7 1-06-2216 insufficient). Additionally, although not alleged here, Gateway’s interpretation to allow the insured to reduce an oral agreement to writing after the loss has occurred could lead to collusion by the interested parties to create coverage by manufacturing an oral promise after the injury occurs. Thus, Gateway’s interpretation of National’s endorsement could lead to a violation of public policy and would be contrary to the intent of the insuring agreement. For all of the foregoing reasons, a mere oral promise to add Harbour, Baker and Nowicki to the National policy as additional insureds was insufficient under the terms of the policy to bind National to provide them with coverage for the accident. As a result, there is also no coverage under Lexington’s excess policy as it only covers those additional insureds covered under National’s policy. We therefore need not address Lexington’s additional contentions. Accordingly, we affirm the judgment of the circuit court. Affirmed. GREIMAN and CUNNINGHAM, JJ., concur. 8  REPORTER OF DECISIONS - ILLINOIS APPELLATE COURT _________________________________________________________________ CINCINNATI INSURANCE COMPANY, as Subrogee of HARBOUR CONTRACTORS, INC., BAKER CONTRACTORS, INC., and NICHOLAS NOWICKI, Plaintiffs, v. GATEWAY CONSTRUCTION COMPANY, INC., Defendant - Counter Plaintiff - Appellant, (Lexington Insurance Company, Defendant - Counter Defendant - Appellee.) ________________________________________________________________ No. 1-06-2216 Appellate Court of Illinois First District, Third Division Filed: March 28, 2007 _________________________________________________________________ PRESIDING JUSTICE THEIS delivered the opinion of the court. Greiman and Cunningham, JJ., concur. _________________________________________________________________ Appeal from the Circuit Court of Cook County Honorable David R. Donnersberger, Judge Presiding _________________________________________________________________ For DEFENDANTS - William J. McKenna, Jr. APPELLANTS Michael S. Shapiro Foley & Lardner LLP 321 N. Clark St. Chicago, IL 60610 For PLAINTIFF - James P. McCarthy APPELLEE Patricia M. Kelly Gunty & McCarthy 150 S. Wacker Dr. Chicago, IL 60606 

/* * Copyright 2000-2018 Vaadin Ltd. * * Licensed under the Apache License, Version 2.0 (the "License"); you may not * use this file except in compliance with the License. You may obtain a copy of * the License at * * http://www.apache.org/licenses/LICENSE-2.0 * * Unless required by applicable law or agreed to in writing, software * distributed under the License is distributed on an "AS IS" BASIS, WITHOUT * WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the * License for the specific language governing permissions and limitations under * the License. */ package com.vaadin.ui.declarative; @SuppressWarnings("serial") /** * An exception that is used when reading or writing a design fails. * * @since 7.4 * @author Vaadin Ltd */ public class DesignException extends RuntimeException { public DesignException() { super(); } public DesignException(String message) { super(message); } public DesignException(String message, Throwable e) { super(message, e); } }

Latest perk on Google buses: security guards - vellum http://in.reuters.com/article/2014/01/16/us-techbuses-security-idINBREA0F1O320140116 ====== yetanotherphd Good for Google for doing this. I like this a lot better than making concessions to the thugs that threaten Google employees or Google's property.

Abellen language Abellen, Abenlen, Aburlin, or Ayta Abellen, is a Sambalic language. It has about 3,500 speakers and is spoken in a few Aeta communities in Tarlac province, Philippines Ayta Abellen itself is part of the Sambalic language family in the Philippines and is closely related to not only the 5 other Ayta dialects, but also the Botolan dialect of Sambal. History Early history The Ayta Abellen people are historically a semi-nomadic people. Also known as Negritos, they are said to be descendants of the earliest inhabitants of the Philippines, dating back to the late Pleistocene Era. The Ayta Abellen are distinguishable by their curly black hair, and darker skin tone as compared to other Filipinos. Since their language is similar to Austronesian languages, there is a theory of an Austronesian migration that occurred. In this theory, there were two different migrations, one from the southern coast of Sundaland eastward and from Wallacea to Mindanao, causing there to be a separation of Ayta people and the Mamanwa for about 20,000 to 30,000 years. Prior to the Austronesian migration, there was not much similarity between the original languages of the Negritos. Modern History and Revitalization After the eruption of Mt. Pinatubo in the 1990s, some of the Ayta Abellen have relocated from the mountains and have intermarried and mixed in with the local Ilocano people. As a result, there are Ilocano loan words in the language. Much of the population also speaks Ilocano as a second language along with Tagalog as well. They Ayta people rely on natural resources, however, due to shrinking forests, it has become harder to sustain that life style. This problem along with diseases, and its remoteness from modern health care centers are correlated with the higher death rate as compared to birth rate in the Ayta Abellen people Phonology Additionally, s, r, c (for [k]), j,among other phonemes are used in loan words and names. In Sambal and Ayta languages, the glottal stop tends to replace a word final non-obstruent when proceeded by a stressed high central vowel. Grammar Ayta Abellen shares the same Verb-Subject-Object sentence structure as other languages in the Philippines. It shares similar phonology with other Ayta dialects as well as Botolan Sambal. Not only does it share an identical pronoun system with other Sambalic languages, but between other Ayta languages, it is around 70% similar. This language is a CV(consonant and vowel) and CVC language, although sometimes, it is ambiguously a VC and V language. In this language, vowel deletion as well as consonant deletion are evident when words are combined. In this language, placement of stress can be unpredictable. Poly-syllabic words have primary stress whereas words with more than three syllables contain a secondary stress. However, suffixation also causes a shift in stress placement. Writing System Ayta Abellen is written using Latin text. Ilocano is a second language to much of the Abellen and the lingua franca of where many of the Abellen people reside, while Tagalog is the national language of the Philippines, transcribers are trying to document the language in text that is similar to both Ilocano and Tagalog. Much of the hymnals used in that area are written in Botolan Sambal, and thus they are also trying to have Ayta Abellen orthography conform to it as well. Locations Abellen Ayta speakers in the following locations of Maamot, San Jose, Tarlac Province. Station Juliana, Mayantoc, Tarlac Province. Capas, Tarlac Province. Sitio Loob-Bunga, barangay Poon Bato, Botolan, Zambales. See also Languages of the Philippines Aeta People Ilocano People Ilocano Language Tagalog Philippines References Category:Endangered Austronesian languages Category:Sambalic languages Category:Aeta languages Category:Languages of Tarlac

The primary symptoms of hypoparathyroidism include muscle cramping, convulsions, intellectual disabilities, cataracts and abnormal heart rhythm. Symptoms are due to low serum calcium (hypocalcemia). Replacement of parathyroid hormone (PTH) has been explored to remedy the calcium deficiency, but maintaining an optimal calcium level has proven problematic because hypercalcemia can occur as a result of excess PTH. Multiple efforts are under way targeting either full-length PTH (PTH 1-84) or the active amino-terminal domain (PTH 1-34), but these molecules have undesirable pharmacokinetic properties for chronic daily management of calcium levels in patients with hypoparathyroidism. Eli Lilly scientists have identified a PTH receptor modulator (PTH-RM) that can normalize serum calcium. At fairly low doses, the PTH-RM was shown to normalize calcium levels in parathyroidectomized rats. The investigators are collaborating with TRND to develop this PTH-RM toward a Phase II proof-of-concept study for hypoparathyroidism by leveraging the existing data package. TRND scientists, in collaboration with researchers from Eli Lilly & Company and the Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD), have further developed the animal model of hypoparathyroidism to generate robust efficacy data. The team will execute the full preclinical development plan. TRND will support the preparation and filing of the Investigational New Drug application with the Food and Drug Administration, with NICHD providing support for subsequent clinical trials in patients.

James Kolstad Alumni, Class of 2008 Spent many years in Chicago climbing the ladders of various fields, also did some writing and performance, before throwing it all overboard, moving to New York and getting my (first) master's. Now it appears that I am addicted to sleepless nights and fragile economic circumstances. I don't know specifically where ITP will take me, but I have become increasingly interested in the intersections among media, architecture, and public space. Selecting this education source as 'Public' will make this the main information displayed on the ITP webpage. Existing public educational sources will be available to the ITP Community. Selecting this professional source as 'Public' will make this source the main information displayed on the ITP webpage. Existing public professional sources will be available to the ITP Community.

Auditory cortical plasticity induced by intracortical microstimulation under pharmacological blockage of inhibitory synapses. Electrical stimulation that can reorganize our neural system has a potential for promising neurorehabilitation. We previously demonstrated that temporally controlled intracortical microstimulation (ICMS) could induce the spike time-dependant plasticity and modify tuning properties of cortical neurons as desired. A 'pairing' ICMS following tone-induced excitatory post-synaptic potentials (EPSPs) produced potentiation in response to the paired tones, while an 'anti-pairing' ICMS preceding the tone-induced EPSPs resulted in depression. However, the conventional ICMS affected both excitatory and inhibitory synapses, and thereby could not quantify net excitatory synaptic effects. In the present work, we evaluated the ICMS effects under a pharmacological blockage of inhibitory inputs. The pharmacological blockage enhanced the ICMS effects, suggesting that inhibitory inputs determine a plastic degree of the neural system. Alternatively, the conventional ICMS had an inadequate timing to control excitatory synaptic inputs, because inhibitory synapse determined the latency of total neural inputs.

A radio frequency (RF) connector is an electrical connector designed to work at radio frequencies in the multi-megahertz range. Typically, RF connectors are used in a variety of applications such as wireless telecommunications applications, including WiFi, PCS, radio, computer networks, test instruments, and antenna devices. In some instances, a number of individual connectors are ganged together into a single, larger connector housing for electrically and physically connecting two or more printed circuit boards. One example of an RF connector interface is the sub-miniature push-on (SMP) interface. SMP is commonly used in miniaturized high frequency coaxial modules and is offered in both push-on and snap-on mating styles and is often used for PC board-to-board interconnects. For these applications, the conventional SMP interface utilizes a male connector on each of the PC boards and a female-to-female adapter mounted in between to complete the connection. One problem with conventional RF connectors is that such connectors typically do not have the flexibility to customize the degree of axial or radial float between connectors. Another problem associated with conventional RF connectors is that the density of individual connectors is limited by the shape and design of the adapter. As RF connector applications have begun to require a greater number of individual connections between components, RF connectors using conventional designs have necessarily increased in size to accommodate this. Larger connectors require more physical space in order to provide the necessary contacts, which make the connectors less applicable to high density systems requiring smaller connectors and more expensive to produce. Accordingly, there is a need for an electrical connector, such an RF connector, with improved axial and radial float while also having a smaller profile.

#region Copyright (C) 2007-2018 Team MediaPortal /* Copyright (C) 2007-2018 Team MediaPortal http://www.team-mediaportal.com This file is part of MediaPortal 2 MediaPortal 2 is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. MediaPortal 2 is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with MediaPortal 2. If not, see <http://www.gnu.org/licenses/>. */ #endregion using System.Collections.Generic; using System.Runtime.Serialization; namespace MediaPortal.Extensions.OnlineLibraries.Libraries.MusicBrainzV2.Data { //{ // "created": "2016-04-27T11:11:27.118Z", // "count": 1, // "offset": 0, // "artists": [ // { // "id": "8538e728-ca0b-4321-b7e5-cff6565dd4c0", // "type": "Group", // "score": "100", // "name": "Depeche Mode", // "sort-name": "Depeche Mode", // "country": "GB", // "area": { // "id": "8a754a16-0027-3a29-b6d7-2b40ea0481ed", // "name": "United Kingdom", // "sort-name": "United Kingdom" // }, // "begin-area": { // "id": "9b4cb463-9777-46c3-8190-e1cb3da2749f", // "name": "Basildon", // "sort-name": "Basildon" // }, // "life-span": { // "begin": "1980", // "ended": null // }, // "aliases": [ // { // "sort-name": "Depech Mode", // "name": "Depech Mode", // "locale": null, // "type": null, // "primary": null, // "begin-date": null, // "end-date": null // }, // { // "sort-name": "DM", // "name": "DM", // "locale": null, // "type": "Search hint", // "primary": null, // "begin-date": null, // "end-date": null // } // ], // "tags": [ // { // "count": 1, // "name": "electronica" // }, // { // "count": 1, // "name": "post punk" // }, // { // "count": 1, // "name": "alternative dance" // }, // { // "count": 6, // "name": "electronic" // }, // { // "count": 1, // "name": "dark wave" // }, // { // "count": 0, // "name": "britannique" // }, // { // "count": 4, // "name": "british" // }, // { // "count": 1, // "name": "english" // }, // { // "count": 2, // "name": "uk" // }, // { // "count": 0, // "name": "rock and indie" // }, // { // "count": 1, // "name": "electronic rock" // }, // { // "count": 1, // "name": "remix" // }, // { // "count": 0, // "name": "synth pop" // }, // { // "count": 2, // "name": "alternative rock" // }, // { // "count": 0, // "name": "barrel" // }, // { // "count": 6, // "name": "synthpop" // }, // { // "count": 4, // "name": "new wave" // }, // { // "count": 1, // "name": "new romantic" // }, // { // "count": 1, // "name": "downtempo" // }, // { // "count": 0, // "name": "producteur" // }, // { // "count": 0, // "name": "producer" // }, // { // "count": 1, // "name": "synth-pop" // } // ] // } // ] //} [DataContract] public class TrackArtistResult { [DataMember(Name = "artists")] public List<TrackArtist> Results { get; set; } } }

Diabetes update: long-term treatment of adults. Current evidence supports a less interventional, less aggressive, and more patient-oriented approach to the care of patients with diabetes than is commonly followed. When treating an adult patient with type 2 diabetes, the physician must focus on the following (in order of importance): smoking cessation and other lifestyle interventions, blood pressure control, metformin use, lipid control, and glycemic control. Patients also should receive influenza and pneumococcal vaccinations. Management goals should be individualized, but general target values are blood pressure of 140/80 mm Hg, low-density lipoprotein less than 100 mg/dL (or 70 mg/dL in a patient with diabetes and coronary artery disease, according to consensus opinion), and A1c less than 8%. Hypertension control is important; a thiazide or angiotensin-converting enzyme inhibitor might be the best first-line treatment. Metformin is the foundation of treatment for most patients with type 2 diabetes; in patients who are overweight, use of metformin delays premature mortality regardless of achieved glucose levels. Statin drugs are superior to other drugs for cholesterol reduction. The use of combination or high-intensity drugs does not appear to confer additional benefit.

Er Sajad To Be RSP(S) Candidate For Srinagar Seat; Sanjay Dhar From Anantanag Srinagar March. 14: Rashtriya Samajwadi Party(S) is going to field State president of party Er Sajad Mushtaq Reshi from Srinagar and Sanjay Dhar from Anantanag parliamentary constituency for the by-polls for the two Lok Sabha seats. As per Report that the decsions has been taken in a meeting which was Chaired by Party President Shri S U chaturvedi. Loading... It is pertinently Mention here that Srinagar is going for by-polls on April 9 whereas on April 12, Anantnag will go for poll. Anantnag seat had fallen vacant after Mehbooba Mufti took over as the Chief Minister of the State whereas Srinagar seat fell vacant after Tariq Karra, resigned from LoK Sabha and PDP in September last year in protest against what he called “brutalities” against last year’s summer unrest protesters.

Q: Create table for each id existent in another table I was wondering if there is a way to create a MySQL table for each ID that exists in another table. I think that would be fairly easy doing it with PHP, but I'm not sure if that can be done with MySQL. So for instance I have a table called users which has an X amount of columns. One column is the IDs column. So I would like to iterate through that column and grab the IDs and create for each of those IDs a new table which will have the name of "user_specific_id_ " + ID. So for the ID 1 the name of the newly created table would be user_specific_id_1. Could the above be done just with MySQL, or is it necessary to use PHP ? And if I need to use PHP what would be the approach ? A: I'm not familiar with a pure MySQL way. Using PHP you'll select all id's from your table, and then in a foreach loop issue a CREATE TABLE user_specific_id + $id query That being said, creating a separate table for each user doesn't sound like the correct way of handling a DB.

3*k**5 - 3*k**5. 5*k**5 + 6*k**2 Expand 2*s**5 + s**5 - 5*s**5 + (0*s**4 + s**4 + 2*s**4)*(-8 - s + 8) - 3*s**4 + 3*s**4 + s**5. -4*s**5 Expand (-2*a**3 + 2*a**3 + a**3)*(-a**2 - 2 + 2) - 2*a**5 - 2*a**5 + a**5 + 0*a**3 + a**5 + 0*a**3. -3*a**5 Expand (2*c**2 + 0*c + 0*c)*(-25 + 25 + 25*c**3). 50*c**5 Expand (79*l**4 + 18*l**3 - 52*l**4 - 31*l**4)*(3*l - l + 0*l). -8*l**5 + 36*l**4 Expand (-5 - 9 - 2 - 1 + 2 + 1 + (4 - 4 - 2)*(4 - 4 + 1))*(3*t**3 - 5*t**3 + 3*t**3). -16*t**3 Expand (u**3 + 2*u**2 - 2*u**2)*(-19 - 4 + 1). -22*u**3 Expand (-3*h - 2*h - 2*h**3 + 6*h)*(-13 + 6 + 6). 2*h**3 - h Expand (-x + 2*x - 2*x)*(87*x**3 + 5*x**4 - x**2 - 87*x**3). -5*x**5 + x**3 Expand -m**3 - 2*m - 2*m**3 + 4*m**3 + (3*m - 7*m + 2*m)*(-5*m + 5*m - 6*m**2). 13*m**3 - 2*m Expand (-2*k + 0 - 2 + 0*k)*(7*k - 7*k - 56*k**2)*(-k**2 + 2*k**2 + k**2). 224*k**5 + 224*k**4 Expand (7*x - x + 3*x + (1 - 1 - 2)*(2*x + 2*x - 3*x))*(-3 - 4 + 5). -14*x Expand 0*v + 0*v + v + (-2 + 1 + 3 - v)*(-1 - 2 + 2). 2*v - 2 Expand (-2*a - 3*a + 3*a)*(-2*a**4 + 7*a**4 - 3*a**4) + 73*a**3 - 4*a**4 + a**5 - 73*a**3. -3*a**5 - 4*a**4 Expand 5 - 5 - 2*u + (3 - 3 - 2)*(3 - 2*u - 3) + u + 3*u - 2*u. 4*u Expand (-b + 0*b + 3*b)*(-31*b**2 - 2*b**2 - 8*b**2)*(-2*b**2 + 2*b**2 - b**2). 82*b**5 Expand (2*h**2 - 2 + 2)*(h**2 - h**2 + h**2) + h**4 - 71*h - 25 + 71*h. 3*h**4 - 25 Expand (-w + 4 - 4*w + 6*w)*(-2*w + 1 - 1). -2*w**2 - 8*w Expand (4 + 0 - 2)*(-15 + 14*o + 15). 28*o Expand (7 - 6 + 7)*(-3 + l + 3). 8*l Expand (-a - 1 + 1)*(-12 + 12 + 2*a)*(a**3 + 2 - 2) + 0*a**4 - 2*a**5 + 0*a**4. -4*a**5 Expand (2 - 36*p + 19*p + 11*p)*(-3*p - p**2 + 3*p). 6*p**3 - 2*p**2 Expand b + 0*b - 4*b + (-3 - 3 + 4)*(-2 - 2*b + 2) - 2 + 2*b + 2 + 2*b + b - 4*b - 3*b - 4*b + 6*b - 5*b - b + 3*b. -2*b Expand (142*o - 177*o + 80*o)*(4 - 6 + 3). 45*o Expand (3*g - 20*g - 2*g)*((3 + 0 - 2)*(-g - 5*g + 4*g) + g + 4*g - 4*g). 19*g**2 Expand (1 - 1 - h**2)*(-22 + h + 22) + 0 + 0 - 2*h**3 + (0*h + 0*h + 2*h**2)*(-5*h + 5*h - 2*h). -7*h**3 Expand (-59 + 59 - 18*w**2)*(3*w + 5 - 5). -54*w**3 Expand (0 - 4 + 3)*(5*g - 3*g + 0*g)*(3 - 2 + 1). -4*g Expand (-2*c**3 + 2*c**3 + 2*c**4)*(5 - 1 - 2) - c**4 + 0*c**3 + 0*c**3. 3*c**4 Expand (3*d + 0 + 0 - 8*d + 4*d - 4*d + (-2 + 2 - d)*(-4 + 0 + 5))*(d - 2 + 2). -6*d**2 Expand (-14*q**2 + 3*q**2 - q**2)*(q**2 + 3*q**2 - 3*q**2) - 2*q**2 - 2*q**4 + 2*q**2 + (2*q**4 - q**4 + q**4)*(1 - 5 + 2). -18*q**4 Expand 0*v + v + v + (-3 + 3 + 2)*(-7*v - 8*v + 5*v) + (0 + 0 + 2)*(2*v - 5*v + v). -22*v Expand (-u**4 + 4*u**4 + 4*u**4 + 2*u**4 + 2*u**4 - 5*u**4 + (0 + 0 - 2*u)*(u - u**3 - u) - 3*u**4 + 2*u**4 + 3*u**4)*(3 - 3 - 2*u)*(-2 - 1 + 4). -20*u**5 Expand 4*q**3 - 4*q**3 + 2*q**3 + q**3 + 2*q**3 + 12*q**3 + (q**3 - 2 + 2)*(-2 + 1 + 3). 19*q**3 Expand (0*u**3 + 0*u**3 + 5*u**5)*(-1 + 3 + 0)*(1 - 4 + 2)*(0 - 3 + 1). 20*u**5 Expand 5*x - 5*x + 2*x**4 + 6 + (-x**3 + x**4 + x**3)*(-5 - 1 + 5) - 4*x**4 + x**4 + 2*x**4 - 2*x**4 + 3*x**4 - 3*x**4. -2*x**4 + 6 Expand (0 + 0 - 2)*(2*u - 4 + 4)*(-208 + 3*u + 208). -12*u**2 Expand (b + 4 - 4)*(-5*b + 8*b + b + (-4*b + b + 2*b)*(-2 - 4 + 3))*(-2 + 2 + 1)*(-2 - 2 + 1). -21*b**2 Expand (-3*z + 4*z + 4*z)*(-3*z + 4*z - 3*z) + (z - 3*z + 3*z)*(0*z - 3*z + 5*z). -8*z**2 Expand (-2*h**4 + 2*h**4 - 2*h**4)*(h - h - 8*h). 16*h**5 Expand -p**5 - 2*p**3 + 2*p**3 + (p**3 - p**3 + 3*p**3)*(0*p**2 + 3*p**2 - 4*p + 6*p - 3). 8*p**5 + 6*p**4 - 9*p**3 Expand -q**2 + q**2 - 2*q**3 + (8*q**2 - 10*q**2 + q**2)*(2*q + 0*q + 0*q). -4*q**3 Expand (-1 + 1 + 3)*(j**3 + 3*j**3 - 6*j**3)*(-j**2 + 2*j**2 - 3*j**2)*(0 + 1 - 3). -24*j**5 Expand ((4 + 11 + 5)*(2 + 5 - 5) + 1 + 5 - 5)*(-w + w - 2*w**2). -82*w**2 Expand (0*n + n - 4*n)*(-4*n + 0*n + 5*n) + n**2 + 0*n + 0*n. -2*n**2 Expand (-1 + 3 + 2)*(-7*h - 5*h**2 + 7*h). -20*h**2 Expand -2*d**4 - 1 + 1 - 3*d**4 + d**4 + 4*d**4 + (4*d - d - 2*d)*(d - d**3 - d). -d**4 Expand (1 + 0 - 3)*(6 - 2 - 2)*(-4*h - 10*h + 4*h). 40*h Expand (-2*k**3 + k**2 - k**2)*(-3 - 2 + 1)*(-1 + 0 + 0)*(-45 + 16 + 2). 216*k**3 Expand (-4*r**2 + 4 - 4)*(3 - 2*r - 3) - 5*r**3 + 0*r**3 + 6*r**3. 9*r**3 Expand (1 - 1 - a**2)*(-2*a + 0 + 0) + (0*a**2 + 0*a**2 + a**2)*(-2*a - a + 2*a) + 2*a**3 + 2*a**3 - 2*a**3. 3*a**3 Expand a**3 + 2*a**3 - a**3 + (4*a - 5*a + 2*a)*(-5 + 5 + 4*a**2) - a**3 + 2*a**3 - 3*a**3. 4*a**3 Expand (-16 + 16 + g**5)*(-1 + 2 + 3). 4*g**5 Expand (11*x - x - 6*x)*(-3 + 3 + x)*(-2 - 2 + 3). -4*x**2 Expand 4*w - 2*w - 4*w + (0 + 2 - 1)*(4*w + w - 4*w) + (0 + 1 - 3)*(3*w - 2*w + 0*w). -3*w Expand (-10 + 5 - 5)*(2*g**2 + 4*g**2 - 4*g**2). -20*g**2 Expand g**5 + 0*g**4 + 0*g**4 + g**4 + g**5 - g**4 - g**5 + 2 - 2 + (0 + 0 + g**3)*(-3 + 3 - g**2) + 0*g - 2*g**5 + 0*g + 0*g**4 - 2*g**4 - 6*g**5 + 0*g**4. -8*g**5 - 2*g**4 Expand (-3*n + 0*n + 2*n + (0 - 1 + 3)*(n + 1 - 1))*(4*n**4 + 7*n**4 - 3*n**4). 8*n**5 Expand (-1 + 1 + 2*n)*(-4*n**2 + n**2 + 0*n**2) + 13*n - 8*n**3 - 13*n. -14*n**3 Expand (4*o - 4 + 4)*(2*o + 2*o - 2*o) + 4*o**2 + 4*o**2 - 2*o**2. 14*o**2 Expand (y**3 + 2*y**3 - 4*y**3)*(-1 - y**2 + 1) - 2*y**5 + y**5 + 0*y**5 + (1 - 2 - 1)*(-1 - 2*y**5 + 1) + 4*y**5 - 2*y**5 - y**5. 5*y**5 Expand (-3*t + t + 5*t)*(1 + 1 - 1) + 3 - 3 - 2*t + 0*t - t - t + (3*t + t - 2*t)*(-1 + 6 - 3). 3*t Expand -3*t**5 + t**5 + t**5 + 0*t**5 - 5*t**5 + 2*t**5 + (-2*t**2 + t**3 + 2*t**2)*(t**2 + t**2 - 4*t**2) + 3*t**2 - t**5 - 3*t**2. -7*t**5 Expand (-b**2 + 3 - 3)*(-24*b**2 + 11*b + 25*b**2 + 0*b). -b**4 - 11*b**3 Expand ((-7*y**2 + 4*y**2 + 4*y**2)*(2 + 0 - 4) - 2*y**2 + 2*y**2 - 2*y**2)*(y**2 - 4*y**2 - 3*y**2). 24*y**4 Expand (2*r - 1 + 1)*(-3*r**2 + 6*r**2 + 5*r**2) + (r + r - 3*r)*(0 - 2*r**2 + 0). 18*r**3 Expand (-w**2 + 0*w**2 + 2*w**2)*(590*w**2 + 3*w - 2*w - 592*w**2 + 32). -2*w**4 + w**3 + 32*w**2 Expand (6 - 4 - 3 - r)*(0 - 2 + 0 + (-3 + 2 - 1)*(3 - 1 + 0)). 6*r + 6 Expand 9*s**3 - 5*s**3 - s**3 + (2 + 2*s - 2)*(6*s**2 - 2*s**2 - 2*s**2) - 2*s**3 + 3*s**3 - 2*s**3. 6*s**3 Expand (-w - 5 + 5)*(-w**4 + 3*w**4 - 3*w**4) + 6*w + 13*w**5 - 6*w - 3*w**5 - 5*w + 5*w. 11*w**5 Expand (-3*k**2 + 7 - 7)*(2*k + 4 + 0*k - 1). -6*k**3 - 9*k**2 Expand (4 - 5 + 0)*(1 + 1 - 1)*(3*j**3 + 0*j**3 - 4*j**3). j**3 Expand ((0 - 1 + 0)*(-3 - 3 + 4) + 2 + 1 + 2)*(0*q - 6*q - q). -49*q Expand (3 + 2*y**3 - 3)*(-3 + 2 + 0)*(1 + 1 - 1). -2*y**3 Expand (2*i + 0*i - 4*i)*(5 - 4 + 2) + (6 - 5 + 1)*(-2 - i - 3*i + 0*i). -14*i - 4 Expand (-4*m**4 + 2*m**4 - 4*m**4)*(2 - 1 + 1). -12*m**4 Expand (-2*g**2 + 0*g**2 - g**2)*(-5 + 6 + 3). -12*g**2 Expand (0*y + 0*y - 2*y**4)*(5 - 2 - 1) - 2*y**2 + 2*y**2 + y**4 - 2*y**3 + y**4 + 2*y**3 + (4*y**4 - 5*y**4 + 2*y**4)*(3 + 2 - 1). 2*y**4 Expand (10 - 7*t**2 + 10*t**2 - 2*t**2)*(3*t + 0*t + 0*t). 3*t**3 + 30*t Expand -7*a**2 - 12*a**2 + 12*a**2 + (-2*a - a + 4*a)*(-5*a + 4*a - 2*a). -10*a**2 Expand (1 - 1 + o)*(0*o + 3*o - o) + 6*o**2 - o**2 - 4*o**2 + 3*o**2 + o**2 - o**2 + (o - 1 + 1)*(2 - 2 + 2*o). 8*o**2 Expand (1 - 2*z**2 - 4 + 5 + 3*z)*(5 - 5 - 3*z + 3*z + 0*z - 5*z - z + z + 2*z - 2 + 2 - 2*z + (3*z - 3*z + z)*(3 - 6 + 1) + 2*z - 2*z - 2*z). 18*z**3 - 27*z**2 - 18*z Expand -2*h**5 - 2*h**5 + 3*h**5 + (h**3 - 3*h**3 + h**3)*(-h**2 - h**2 + 3*h**2). -2*h**5 Expand -5*b**5 - b**5 + 3*b**5 + (-b**3 + b**3 + b**4)*(b + 2 - 2). -2*b**5 Expand -c + 3 - 3 + (-c - 5 + 5)*(-3 + 3 - 1)*(-1 + 2 - 4). -4*c Expand (1 + 2*d**2 - 1)*(7 - 7 - 13*d**3). -26*d**5 Expand (-d + d + 2*d)*(1 - 5 + 3)*(-3 + 3 + d)*(6*d**3 + 10*d**3 - 6*d**3). -20*d**5 Expand (3*t - 3*t + 3*t)*(32 + 3*t**4 - 32 - t**3). 9*t**5 - 3*t**4 Expand -38*n**5 + 14*n**5 - 46*n**5 + (-n**2 + 3*n**2 + 0*n**2)*(-5*n**3 - n**3 + 4*n**3). -74*n**5 Expand 3*t**2 - 4*t + 4*t - t**2 + 0*t**2 + 3*t**2 + (2 - 2*t - 2)*(0 + 0 + 4*t) + 2*t**2 + 3*t - 3*t. -t**2 Expand 3*u - u - 4*u + (-2 + 2 + u)*(-1 + 1 + 1) - 1 + 1 + 3*u + (0 + 2 + 0)*(-u + 0*u + 0*u + (-3 - 3 + 7)*(4*u - 4*u - u) - 3 + 3 - u - 4*u + 2*u - u). -10*u Expand (12 - 12 + 25)*(-2 - 2*u**4 + u**3 + 2). -50*u**4 + 25*u**3 Expand (-2*q + 3*q + 0*q)*((1 - 1 + 2)*(-1 + 7 - 1) + (-5 - 1 + 4)*(1 - 5 + 3) + 1 - 5 + 2). 10*q Expand (0 + 3 - 1)*(-1 + 1 - 2*a) - 2*a - 3*a - a - 2 + 3*a + 2. -7*a Expand (-a**4 - 3 + 3)*(2 - 10 + 10). -2*a**4 Expand -3*o**2 + 3*o - 3*o + 3*o**3 + (0*o - 3*o + 2*o)*(-6 + 6 - 6*o**2). 9*o**3 - 3*o**2 Expan

May 10: My favorite top lately...another Tiny Pocket Tank made from some mystery fabric (rayon?). It's super comfy and I can wear it with jeans and a cardigan! I love how the bias binding around the neck and armholes turned out. What have I learned so far from Me-Made-May? 1. I need to sew faster! It's only May 10 and I already feel like I'm running out of things to wear. This has definitely forced me to wear things that I'm not super excited about. Some of them should just be given away instead of sitting in the back of my closet. 2. My usual uniform is jeans, top and maybe a cardigan. I need to make more tops so my co-workers can see me in something other than my Tiny Pocket Tanks! 1 comment: Hi! Thanks for your comment over at my blog! I've been thinking about taking a sewing class at Fancy Tiger. I've been trying to teach myself, but when it comes to making clothes, I feel like I need extra guidance!

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How inbreeding and outbreeding influence the risk of extinction--a genetically explicit model. We have developed a stochastic model to explore the common effect which genetics and demography have on the extinction risk of endangered populations. The dynamics is formulated as a MARKOVian birth and death process (in continuous time), whereby selection acts through different mortalities of each genotype. With the help of this model we are able to show how inbreeding and outbreeding can influence the genetic variability and the survival of a population. Whether inbreeding or outbreeding takes place depends on the specific mating system. In our model we consider positive assortative as well as disassortative mating. In the case of additive fitness we show that inbreeding reduces the extinction risk and the genetic variability.

Richard Booth Professor of Law Martin G. McGuinn Chair in Business Law Biography Professor Booth joined the Villanova faculty in 2007 as the first Martin G. McGuinn Chair in Business Law. He came to Villanova from the University of Maryland School of Law where he was the Marbury Research Professor of Law. Professor Booth earned his A.B. from the University of Michigan and his J.D. from Yale Law School. Following law school, he practiced with Donovan Leisure Newton & Irvine, in the area of corporate and securities litigation. In 1982, he joined the faculty at Southern Methodist University, later moving to Case Western Reserve University, and then to the University of Maryland. In addition, he has taught at the University of Aberdeen (Scotland), George Washington University, the Wharton School, and the University of Pennsylvania Law School. He is the author or co-author of or a contributor to numerous books, including Financing the Corporation (West), Business Basics for Law Students (Aspen), Cases and Materials on Corporation Finance (West), and the West Blackletter on Corporation Law. He has published over sixty articles in scholarly journals as well as many pieces in the popular press including the Wall Street Journal, the New York Times, and Bloomberg’s SCOTUS Blog. Professor Booth’s recent scholarship has focused on securities fraud class actions, executive compensation, and business valuation, and in particular the implications of investor diversification for corporate finance and governance generally. Professor Booth is a member of the New York and Texas bars, as well as the bar of the United States Supreme Court. He is a life member of the American Law Institute and a member of Phi Beta Kappa. At Villanova, Booth teaches Corporate Finance, Business Planning & Venture Capital, and Securities Litigation Seminar. Education Villanova University Charles Widger School of Law is approved by the Council of the Section of Legal Education and Admissions to the Bar of the American Bar Association, 321 North Clark Street, Chicago, IL 60654, (312) 988-6738

Podcasts & RSS Feeds 4:08pm The Record Our Band Could Be Your Life, Still By Daoud Tyler-Ameen Minor Threat performs in 1980. The group is one of the bootstrapping indie bands profiled in the book Our Band Could Be Your Life. Susie Josephson HorganCourtesy of the artist My first summer out of college, I picked up a copy of a book that some friends had been trying to sell me on for years. It was all about these obscure rock bands from 30 years ago, and it was an uninviting 500 pages long. But halfway into the opening chapter about the California hardcore punk band Black Flag, I was hooked. Published in 2001, Our Band Could Be Your Life is a history lesson on the beginnings of indie rock, long before it was even called that. It tells the stories of a dozen bands who found modest but dedicated audiences without the aid of major record labels. Some, like Washington, D.C.'s Minor Threat, even started labels themselves. "It's funny — Our Band Could Be Your Life has been out for 10 years," says the book's author, New York writer and musician Michael Azerrrad. "But in the past maybe three years, I haven't been able to go out to an indie rock show without someone in the audience or someone in a band walking up to me and telling me they were inspired to do what they do by reading it." One of the people inspired by the book is Ronen Givony. He founded the new music series Wordless Music in New York City, and books classical and electronic performers at the downtown venue Le Poisson Rouge. "I've told everyone from the deans of Juilliard and the Manhattan School of Music I almost feel like it should be required freshman reading," says Givony. "The example of punk rock, of just being scrappy and being not content to let other people create your career for you – that is really what all young musicians need to learn." Patrick Stickles, frontman of the New Jersey punk band Titus Andronicus, says he learned a similar lesson when he read Our Band Could Be Your Life as a teenager. "That was a really empowering thing for a young kid like me — to learn that all these supposedly legendary groups were people just like me and my friends, who were just doing their thing in their own communities with their own resources," Stickles says. "It made me dream of someday getting to do that." One of the most relentlessly self-sufficient bands profiled in the book is Big Black, the Chicago industrial punk outfit helmed by Steve Albini. "I prided myself on the fact that my band never undertook a tour where we didn't make money," says Albini. "You don't get anywhere by pretending to be a big shot. You accomplish things by working within your means, and by keeping as much control of your own existence as possible." Albini says underground music and culture existed long before the period covered in the book – roughly 1981 to 1991 – and that it continues today. But, he says, there was certainly something about those years that caused people to take notice. "That was the period during which the independent bands and labels got their sea legs and established a solid foundation for all of the other independent artists to work with," he says. "You could have a comfortable existence completely outside of the mainstream music business." Years after Big Black disbanded, its music and ethics made an unlikely impression on Annie Clark, the woman behind the Brooklyn pop band St. Vincent. "One of the things that got me first was just the guitar playing," says Clark. "It's just this lacerating noise, this thing that kind of expresses, for lack of a better word, all of your suburban angst and rage. It's kind of physically painful in a really wonderful way." The extremely masculine hardcore punk that dominated the early sound of indie has given over to a broader range of musical styles in the years since. A concert in Manhattan last night, organized by Azerrad for the 10th anniversary of his book, was a testament to that change — and signaled that the strength of the spirit that held indie rock together in its early days has persisted to a new generation. Fourteen bands performed the songs of the indie icons profiled in Our Band, with lots of collaboration and cross-pollination. Titus Andronicus played songs by The Replacements, after an introduction by the Hold Steady's Craig Finn. St. Vincent played songs by Big Black before a moshing crowd of young men, many of whom were born after the earlier band broke up. One of the most significant changes in the last three decades is the role of women on stage at indie rock shows. Few women show up in Azerrad's book; most of them take peripheral roles. Of the fourteen bands on stage last night, all but three include a female member. Annie Clark says her career and those of her peers have been a lot easier thanks to the hard work of Albini and others. "You really had to be committed to the task. That's something that you sometimes don't see these days," says Clark. "You sometimes encounter an attitude of, 'Oh yeah, I'll be in a band for a few years, and then I'll do data entry or something.' But those guys really were lifers." Michael Azerrad concurs that emerging musicians today are dealing with a radically different playing field from their '80s forbears. "All the infrastructure is in place," he says. "There's a huge network of clubs perfectly set up to accommodate this kind of music. There's all kinds of media that publicize this music. There's all kinds of online music stores where people can buy this music instantly. A lot of it is just so much easier." But Steve Albini says it's important to remember that some things about being in a band — mainly, the inconveniences — will probably never change. "You need to find a place to practice. Someone has to come up with a van if you want to play out of town. This guy's girlfriend doesn't like the guitar player," Albini says. "I don't know that the experience of being in a band is ever really gonna change." Except that for the past 10 years, there's been a manual: a book that demonstrates how fledgling musicians can build a thriving community all by themselves. Copyright 2011 National Public Radio. To see more, visit http://www.npr.org/.

Vyziki Vyziki () is a mountain village in the municipal unit of Tropaia in Arcadia, Greece. It is located 85 km northwest of Tripoli, just off the midpoint of the national road EO74 which connects Tripoli to Pyrgos, and 2 km from Tropaia. The village is built at an altitude of 750 m. Most of its houses are traditional structures made of stone, and two of them are fortified Venetian tower-houses (see picture on the right). The two central "squares" (Greek: πλατείες), the folklore museum, and the St. Nicholas Cathedral stand out among the village buildings. It has been declared a traditional settlement. According to the latest census study (2011) the number of residents is 232, most of them over aged. Major productive activities in the Vyziki area involve livestock and dairy products, with the produced milk being forwarded to the nearest arcadian cheese plants. Antiquities The village's greater area has an elongated shape and runs from the east (near the Langadia mountains) to the west (reaching the river Ladon). On the eastern edge, there are ruins of the mighty Mattegriffon, the Frankish Castle of Akova[2] and every August - over the last 40 years - Vyziki holds cultural activities under the name "Akova Festival". In the same area, namely, inside the castle, under its foundation, and nearby, many items of antiquity have been found - columns, pottery pieces, etc. It has been suggested that it is very probable that Akova existed before the arrival of the Franks; It is located in a place inhabited in antiquity and whose occupation has had to be continuous[3]. This could possibly be the actual location of Ancient Teuthis, mentioned in Pausanias[4], as many 19th century famous travellers believed, but which nowadays has been claimed by the large village of Dimitsana.Finally, the west end of the Vyziki area enters the region of Ancient Thelpousa[5], a great city, which is still awaiting an official systematic excavation! References 2. Henry Fanshawe Tozer, "The Franks in the Peloponnese", ''J. Hell. Stud.'', 4 (1883) pp. 216–219 3 Antoine Bon, "La Morée franque", Paris, De Boccard 1969, p. 394 4 Pausanias ("Αρκαδικά"), 8.28(4) 5 Madeleine Jost, "Thelpousa d' Arcadie en 1938-1939", ''Bulletin de Correspondance Hellénique'', 110 (2) 1986, σελ. 633-645 External links ΒΥΖΙΚΙ ΑΡΚΑΔΙΑΣ (in Greek) Βυζίκι Category:Populated places in Peloponnese (region) Category:Arcadia Category:Populated places in Arcadia

Aside from R. Kelly, 'Walk It Out' was produced by Timbaland while 'I Can't Describe (The Way I Feel)' was crafted by proven hitmaker Pharrell Williams. Tracks like 'Dangerous' and 'He Ain't Goin Nowhere' prove that Hudson is more than just a ballad queen but can also sass things up with some fun in the same way the late Whitney Houston did on 'I Wanna Dance With Somebody' and 'So Emotional.'

Carbonic anhydrase XII promotes invasion and migration ability of MDA-MB-231 breast cancer cells through the p38 MAPK signaling pathway. Carbonic anhydrase (CA) XII, an extracellular enzyme involved in the regulation of the microenvironment acidity and tumor malignant phenotype, was originally identified as a protein overexpressed in some types of cancers, including breast cancer. However, the cellular function and mechanism of CAXII remained unclear. In this study, the effects of CAXII expression on invasion and migration of breast cancer cells was investigated. Gene knockdown of CAXII in the human breast cancer cell line MDA-MB-231 resulted in decreased invasion and migration by interfering with the p38 MAPK pathway. CAXII knockdown also decreased the expression of matrix metalloproteinase (MMP)-2, MMP-9, and urokinase-type plasminogen activator (u-PA), but increased tissue inhibitor of metalloproteinases (TIMP)-2 and plasminogen activator inhibitor (PAI)-1 expression. Furthermore, decreased invasive and migration ability of CAXII-knockdown cells were restored by an overexpression of CAXII. Results also showed that CAXII knockdown may decrease anchorage-independent growth and cell growth by inhibiting CDK6 and cyclin D1 expression. Furthermore, the impact of CAXII knockdown on invasion, migration and cell growth was further evidenced by effects on tumor size and metastasis of MDA-MB-231 cells in vivo. Taken together, these data suggested that CAXII may affect the capability of invasion and migration of MDA-MB-231 cells, which may be mediated through the p38 MAPK pathway.

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Evaluation of a program to promote diabetes care via existing agencies in African American communities. Some African Americans with (or at risk for) diabetes underutilize health care services. We report short-term results of a "training of trainers" workshop designed to address this problem. The training program includes culturally sensitive educational materials, including materials developed for the ADA's African American Program (AAP). Workshops were presented to a) the 1996 national meeting of the predominantly black National Missionary Baptist Convention's Nurses Guilds, b) a minority-owned, TN based managed care organization's "community outreach workers," and c) other interested community organizations. Evaluations were based on program satisfaction and an "intention to change" procedure that assessed participants' actions and the obstacles they faced 6 months later. Sixty-four group representatives from 13 states participated. They completed a satisfaction questionnaire and were asked to complete a form that asked them to check any of 12 diabetes-related actions (distributing ADA risk tests, offering AAP classes, etc.) they intended their church/community group to take within six months. Activities not listed could be added. Follow-up contact information was solicited. Satisfaction surveys were positive. 39 (61%) returned checklists with complete contact information. Intentions included: arrange for congregation/community group to take risk test (71% of respondents), distribute diabetes materials at community health fairs or church services (67%), present AAP modules (59%), promote healthy foods at pot luck suppers (56%) and arrange cooking or exercise classes (38%). Respondents were contacted by telephone 6 months post-workshop and asked whether they had fulfilled their intentions. Contact information for 6 (15%) was no longer valid, and we were unable to reach 7 others despite repeated attempts. Approximately 30% of intentions were fulfilled by nurses guild members, but less than 10% by other groups. Half of all fulfilled intentions occurred in a community served by an active ADA AAP Coalition. Barriers to fulfilling intentions included lack of time/support, group not ready to act or doing other programs, and failure to collaborate with the ADA or others for mutual assistance. Existing agencies, especially churches with nurses guilds, offer a means for promoting diabetes screening and awareness in African American communities. A training workshop was well received and influenced some participant groups' self-reported actions. Participants appear more likely to fulfill intentions to conduct diabetes-related programs when they collaborate with other churches, agencies and/or the ADA.

Gresha Schuilling Gresha Schulling (born 1983, Colombo) is a Sri Lankan musician. She has been hailed as a rising star from the world of popular music in South Asia. She comes from a distinguished musical family from Sri Lanka. Gresha was educated in Jeddah, Saudi Arabia at the Continental School, and in Sri Lanka. She took to music while she was in Saudi Arabia and even won a 'battle of the bands' contest in her school. When she returned to Sri Lanka she decided to follow her in popular music after she completed her GCE Advanced Levels. Cold Cold Night Christmas Single Gresha has recorded her debut Christmas single 'Cold Cold Night.' The Christmas song was written by father and son duo, the distinguished Sri Lanka-born singer/songwriter Nimal Mendis and his son Paul Marie Mendis. 'Cold Cold Night' was released by Media Eye Music in the United Kingdom in November 2007. The BBC said Gresha Schuilling was attempting to reach the No.1 spot in the Christmas Charts of 2007. Gresha Schuilling is recording her debut album in Colombo and it was scheduled to be released in the summer of 2008. Media Eye Music in London released her 'Cold Cold Night' Christmas single on download through the mediaeyeproductions website. She has also recorded the Nimal Mendis classic from the iconic Sri Lankan film, 'Ganga Addara.' The original song was recorded by the Sri Lankan film star, the late Vijaya Kumaratunga. 'Ganga Addara' will be released on her debut album. Sri Lanka's Ambassador for Autism Gresha Schuilling was appointed Sri Lanka's Ambassador for Autism by the Autism Awareness Campaign UK and Sri Lanka. She is determined to make a difference to the lives of all people with autism and Asperger syndrome in her island home. Gresha has been raising awareness of the condition - an estimated 39,000 children in Sri Lanka are on the autism spectrum. Gresha said: “Autistic is the adjective of a development disorder of the brain, NOT a definition of who a person is! Educate yourselves and others on this disorder more common that Down Syndrome or cystic fibrosis. It is four times more common in boys than in girls and is said to be as common as one in every 250 children born in recent years. It is our duty as people who have been given the talent and opportunity to do everything we can to increase awareness of Autism. Ignorance in this case is as far from bliss as it can get.” See also Sri Lanka Broadcasting Corporation Burgher People List of Sri Lankan musicians Nimal Mendis Autism Awareness Campaign UK References External links MediaEye Productions/MediaEye Music:Sri Lanka's Singing Sensation Gresha Schuilling World Music Central:Gresha Schuilling releases Christmas Single in London Top40-charts.com Gresha Schuilling releases Debut Christmas Single in London Blogger News Network:Gresha Schuilling releases UK Christmas Single Autism Action UK: Sri Lanka's Ambassador for Autism's UK Christmas Single UK Autism News: Gresha Schuilling 'Cold Cold Night' Christmas Single UK Release Asia Music News: Gresha Schuilling's 'Cold Cold Night' UK Christmas Single Autism Sri Lanka Blog: Gresha Schuilling Sri Lanka Ambassador for Autism's Christmas Single Category:1983 births Category:Burgher musicians Category:Living people Category:Sri Lankan female singers Category:21st-century women singers

The annulus of intervertebral discs develop defects or tears, and a portion of the nucleus pulposus can be squeezed toward, into or through the defect or tear. This creates pain and discomfort. Methods have been developed to repair these tears in the annulus. These methods include using sutures to close the tear. The sutures, however, can fail over time. Other methods use plugs that are inserted into the tear. These plugs, however, are typically inserted from the exterior of the annulus and require difficult positioning or enlarging of the tear to accommodate insertion of the plug. Some plugs only cover the external side of the annulus tear. Therefore, systems and methods are desired for annulus repair that cover both the internal and external surfaces of the annulus and that provide for access to the internal surface of the annulus for improved attachment of a repair device to the annulus.

Solar power generation is a renewable energy source of great potential. It can significantly reduce carbon emission and gets increasing attention. However, due to physical limitation the efficiency of solar power generation is still not desirable to date, and the cost of solar power generation remains very high. How to reduce the cost of solar power generation is a main focus of solar power development at present. FIG. 1 illustrates a conventional solar power producing equipment. It includes a glass substrate transporting apparatus 1 to load a quartz bracket 2 holding a glass substrate (not shown in the drawing) into a reactor 3 filled with high pressure fluorine. The glass substrate transporting apparatus 1 can load or unload a quartz bracket 2 into or from a plurality of reactors 3. The glass substrate transporting apparatus 1 includes an electric lift 4 and a holding rack 5 located on the electric lift 4. The electric lift 4 is slidable on a track 6 and movable by users on the track 6. The elevation of the electric lift 4 is adjustable to align the holding rack 5 with the reactors 3 so that the holding rack 5 can be moved into the reactors 3 to load or unload the quartz bracket 2. The aforesaid conventional glass substrate transporting apparatus 1 is moved manually on the track 6, hence operation speed is slow. Moreover, the number of the reactors 3 accessible by one glass substrate transporting apparatus 1 also is limited. Utilization of the glass substrate transporting apparatus 1 is not desirable, and production yield also is limited and cannot meet mass production requirement.

Let me clarify a point about the Performance Evaluation due dates: The written reviews are due in HR by January 2 (not signed off by employee) You ARE NOT required to have completed all your employee review meetings by January 2 You can start discussions as soon as you have completed the reviews, but the discussion should focus around performance. No variable pay or merit discussions should take place in the performance review meeting. Those will take place on or around January 18. Please call with any questions. Thanks and have a great holiday! Fran Fagan Sr. HR Rep Enron Transportation Services 713.853.5219 fran.fagan@enron.com

Are television channels overstepping the limits of good taste? Just when one thought one had seen it all -- from slush money changing hands, to hidden cameras in the bedroom -- on Indian television, came the attack on former information and broadcasting Ravi Shankar Prasad. But what followed as part of the television channels' coverage surely could not have been what the viewer would have anticipated. As Bharatiya Janata Party activists went berserk near lynching the assailant who shot at Prasad, news channels blithely telecast the macabre scenes of mob fury. Scenes guaranteed to churn one's stomach. There was no warning that the images could offend sensibilities, nor was there an advisory for the young and weak-hearted. Some self-regulation could surely have helped. Do you think the television networks should have shown those scenes unedited? What can the media do by way of keeping within viewer sensibilities?

--- abstract: 'The property of superadditivity of the quantum relative entropy states that, in a bipartite system ${\ensuremath{\mathcal H}}_{AB}={\ensuremath{\mathcal H}}_A \otimes {\ensuremath{\mathcal H}}_B$, for every density operator $\rho_{AB}$ one has ${D(\rho_{AB}||\sigma_A\otimes \sigma_B)} \ge {D(\rho_A||\sigma_A)}+{D(\rho_B||\sigma_B)}$. In this work, we provide an extension of this inequality for arbitrary density operators $\sigma_{AB}$. More specifically, we prove that $ \alpha (\sigma_{AB})\cdot {D(\rho_{AB}||\sigma_{AB})} \ge {D(\rho_A||\sigma_A)}+{D(\rho_B||\sigma_B)}$ holds for all bipartite states $\rho_{AB}$ and $\sigma_{AB}$, where $\alpha(\sigma_{AB})= 1+2 \norm{\sigma_A^{-1/2} \otimes \sigma_B^{-1/2} \, \sigma_{AB} \, \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} - {\ensuremath{\mathds{1}}}_{AB}}_\infty$.' address: - 'Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/ Nicolás Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spain' - 'QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark and NBIA, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark' - 'Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040 Madrid, Spain and Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/ Nicolás Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spain' author: - Ángela Capel - Angelo Lucia - 'David Pérez-García' title: Superadditivity of quantum relative entropy for general states --- Introduction and notation {#sec-1} ========================= The *quantum relative entropy* between two density operators $\rho$ and $\sigma$ in a finite dimensional Hilbert space, ${D(\rho||\sigma)}$, is given by $\tr[\rho (\log \rho - \log \sigma)]$ if $\text{supp}(\rho) \subseteq \text{supp}(\sigma) $ and by $+ \infty$ otherwise[^1]. It constitutes a measure of distinguishability between two quantum states and is a fundamental tool in quantum information theory [@libropetz], [@wilde]. The quantum relative entropy is the quantum analogue of the Kullback-Leibler divergence [@kld], the probabilistic relative entropy. Its origin lies in mathematical statistics, where it is used to measure how much two states differ in the sense of statistical distinguishability. The larger the relative entropy of two states is, the more information for discriminating between the hypotheses associated to them can be obtained from an observation. One of the main properties of quantum relative entropy is *superadditivity*, which states that in a bipartite system ${\ensuremath{\mathcal H}}_{AB}={\ensuremath{\mathcal H}}_A \otimes {\ensuremath{\mathcal H}}_B$ one has: $$\label{superadditivity} {D(\rho_{AB}||\sigma_A\otimes \sigma_B)} \ge {D(\rho_A||\sigma_A)}+{D(\rho_B||\sigma_B)}$$ for all $\rho_{AB}$, where we use the standard notation $\rho_A=\tr_B[\rho_{AB}]$ and $\tr_B$ is the partial trace. Since (Proposition \[prop:sigmaprod\]) $${D(\rho_{AB}||\sigma_A\otimes \sigma_B)} -{D(\rho_A||\sigma_A)}-{D(\rho_B||\sigma_B)}= {D(\rho_{AB}||\rho_A\otimes \rho_B)},$$ is equivalent to the fact that the mutual information $I_\rho(A:B):= {D(\rho_{AB}||\rho_A\otimes \rho_B)}$ is always non-negative, a fact that appears ubiquitously in quantum information theory. In the form , superadditivity of the quantum relative entropy has found applications in e.g. quantum thermodynamics [@gallego], statistical physics [@libropetz Chapter 13] or hypothesis testing [@petz]. Indeed, as proven recently in [@axcharRE] (building on results from [@matsumoto]), the property of superadditivity, along with the properties of *continuity* with respect to the first variable, *monotonicity* and *additivity* (Proposition \[prop:REprop\]), characterizes axiomatically the quantum relative entropy. The main aim of this work is to provide a quantitative extension of (\[superadditivity\]) for an arbitrary density operator $\sigma_{AB}$. Note that for all $\rho_{AB}$ and $\sigma_{AB}$, as a consequence of monotonicity of the quantum relative entropy for the partial trace, the following holds: $$\label{monotonicity} 2{D(\rho_{AB}||\sigma_{AB})}\ge {D(\rho_A||\sigma_A)} + {D(\rho_B||\sigma_B)}.$$ Therefore we aim to give a constant $\alpha (\sigma_{AB}) \in [1,2]$ at the LHS of (\[superadditivity\]) that measures how far $\sigma_{AB}$ is from $\sigma_A \otimes \sigma_B$. Following [@clasico] we will consider as $\alpha(\sigma_{AB})-1$ the distance from ${\ensuremath{\mathds{1}}}$ to “$\sigma_{AB}$ [*multiplied by the inverse of $\sigma_A\otimes\sigma_B$*]{}”. In the case in which $\sigma_{AB}$ and $\sigma_A\otimes\sigma_B$ commute there is a unique way to define this: $\sigma_{AB} \,(\sigma_A^{-1}\otimes \sigma_B^{-1})$. In the non-commutative case, however, there are many possible ways to define the multiplication by the inverse. The one we will take in the result below is a symmetric analogue of the commutative case, $(\sigma_A^{-1/2}\otimes \sigma_B^{-1/2}) \, \sigma_{AB} \, (\sigma_A^{-1/2}\otimes \sigma_B^{-1/2})$. Another one that will appear in the proof of this result is the derivative of the matrix logarithm on $\sigma_A\otimes \sigma_B$ evaluated on $\sigma_{AB}$, $\mathcal{T}_{\sigma_A\otimes \sigma_B}(\sigma_{AB})$, whose explicit equivalent expressions shown in [@lieb] and [@sutter] will be presented later. \[thm:quasifactorizationAB\] For any bipartite states $\rho_{AB},\sigma_{AB}$: $$\label{eq:superadditivity} (1+2\|H(\sigma_{AB})\|_{\infty}){D(\rho_{AB}||\sigma_{AB})}\ge {D(\rho_A||\sigma_A)} + {D(\rho_B||\sigma_B)},$$ where $H(\sigma_{AB}) = \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} \, \sigma_{AB} \, \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} - {\ensuremath{\mathds{1}}}_{AB}$, and ${\ensuremath{\mathds{1}}}_{AB} $ denotes the identity operator in ${\ensuremath{\mathcal H}}_{AB}$. Note that $H(\sigma_{AB})=0$ if $\sigma_{AB}=\sigma_A\otimes \sigma_B$. This result constitutes an improvement to (\[monotonicity\]) whenever $ \|H (\sigma_{AB}) \|_{\infty} \leq 1/2 $ (and, hence, $ 1+2\|H (\sigma_{AB}) \|_{\infty} \leq 2 $). Then, it is likely to be relevant for situations where it is natural to assume $\sigma_{AB} \sim \sigma_A \otimes \sigma_B$. This is the case of (quantum) many body systems where such property is expected to hold for spatially separated regions $A, B$ in the Gibbs state above the critical temperature. Indeed, a classical version of Theorem \[thm:quasifactorizationAB\] proven by Cesi [@cesi] and Dai Pra, Paganoni and Posta [@clasico], was the key step to provide the arguably simplest proof of the seminal result of Martinelli and Olivieri [@marti-oliv] connecting the decay of correlations in the Gibbs state of a classical spin model with the mixing time of the associated Glauber dynamics, via a bound on the log-Sobolev constant. Notation -------- We consider a finite dimensional Hilbert space ${\mathcal{H}}$. We denote the set of bounded linear operators acting on ${\mathcal{H}}$ by ${\mathcal{B}}= {\mathcal{B}}({\ensuremath{\mathcal H}})$ (whose elements we denote by lowercase Latin letters: $f,g$...), and its subset of Hermitian operators by ${\mathcal{A}}\subseteq {\mathcal{B}}$ (whose elements we call *observables*). The set of positive semidefinite Hermitian operators is denoted by ${\mathcal{A}}^+$. We also denote the set of density operators by ${\mathcal{S}}= \qty{f \in {\mathcal{A}}^+ \, : \, \tr[f]=1} $ (whose elements we also call *states* and denote by lowercase Greek letters: $\sigma, \rho$...). A linear map $\mathcal{T}: {\mathcal{B}}\rightarrow {\mathcal{B}}$ is called a *superoperator*. We say that a superoperator $\mathcal{T}$ is *positive* if it maps positive operators to positive operators. Moreover, we denote $\mathcal{T}$ as *completely positive* if $\mathcal{T} \otimes {\ensuremath{\mathds{1}}}: \mathcal{B} \otimes \mathcal{M}_n \rightarrow \mathcal{B} \otimes \mathcal{M}_n $ is positive for every $n \in {\mathbb{N}}$, where $ \mathcal{M}_n $ is the space of complex $n \times n$ matrices. We also say that $\mathcal{T}$ is *trace preserving* if $\tr[\mathcal{T}(f)]= \tr[f]$ for all $f \in {\mathcal{B}}$. Finally, if $\mathcal{T}$ verifies all these properties, i.e., is a completely positive and trace preserving map, it is called a *quantum channel* (for more information on this topic, see [@wolf]). We denote by $\norm{\cdot}_1$ the trace norm $\left( \norm{f}_1 = \tr[\sqrt{f^* f}] \right)$ and by $\norm{\cdot}_\infty$ the operator norm ${\displaystyle}\left( \norm{f}_\infty = \text{sup} \qty{ \norm{f(x)}_{{\ensuremath{\mathcal H}}} : \norm{x}_{{\ensuremath{\mathcal H}}}=1 }\right)$. In the following section, we will make use of this (Hölder) inequality [@bhatia]: $$\norm{f g}_1 \leq \norm{f}_1 \norm{g}_\infty \text{ \phantom{asfs} for every }f, g \in {\mathcal{B}}.$$ In most of the paper, we consider a bipartite finite dimensional Hilbert space ${\mathcal{H}}_{AB}= {\mathcal{H}}_A \otimes {\mathcal{H}}_B$. When this is the case, we use the previous notation placing the subindex ${AB}$ (resp. $A$, $B$) in each of the previous sets to denote that the operators considered act on ${\ensuremath{\mathcal H}}_{AB}$ (resp. ${\ensuremath{\mathcal H}}_A$, ${\ensuremath{\mathcal H}}_B$). There is a natural inclusion of ${\mathcal{A}}_{A}$ in ${\mathcal{A}}_{AB}$ by identifying ${\mathcal{A}}_{A} = {\mathcal{A}}_{A} \otimes {\ensuremath{\mathds{1}}}_{B} $. Relative entropy ---------------- Let ${\ensuremath{\mathcal H}}$ be a finite dimensional Hilbert space, $f,g\in {\mathcal{A}}^+$, $f$ verifying $\tr[f] \neq 0$. The *quantum relative entropy* of $f$ and $g$ is defined by [@umegaki]: $${D(f||g)} = \frac{1}{\tr[ f]}\tr \left[ f (\log f - \log g) \right].$$ In most of the paper we only consider density matrices (with trace $1$). Let $\rho , \sigma \in {\mathcal{S}}$. In this case, the quantum relative entropy is given by: $${D(\rho||\sigma)} = \tr \left[ {\rho} (\log \rho - \log \sigma) \right].$$ In the following proposition, we collect some well-known properties of the relative entropy, which will be of use in the following section. \[prop:REprop\]\ Let ${\ensuremath{\mathcal H}}_{AB}$ be a bipartite finite dimensional Hilbert space, ${\ensuremath{\mathcal H}}_{AB} = {\ensuremath{\mathcal H}}_A \otimes {\ensuremath{\mathcal H}}_B$. Let $\rho_{AB}, \sigma_{AB} \in {\mathcal{S}}_{AB}$. The following properties hold: 1. **Non-negativity.** ${D(\rho_{AB}||\sigma_{AB})} \geq 0$ and ${D(\rho_{AB}||\sigma_{AB})}=0 \Leftrightarrow {\rho_{AB}}=\sigma_{AB}$. 2. **Finiteness.** ${D(\rho_{AB}||\sigma_{AB})} < \infty $ if, and only if, $\text{supp}(\rho_{AB}) \subseteq \text{supp}(\sigma_{AB})$, where supp stands for support. 3. **Monotonicity.** ${D(\rho_{AB}||\sigma_{AB})} \geq {D(T(\rho_{AB})||T(\sigma_{AB}))}$ for every quantum channel $T$. 4. **Additivity.** ${D(\rho_A \otimes \rho_B||\sigma_A \otimes \sigma_B)}= {D(\rho_A||\sigma_A)} + {D(\rho_B||\sigma_B)}$. These properties, especially the property of non-negativity, allow to consider the relative entropy as a measure of separation of two states, even though, technically, it is not a distance (with its usual meaning), since it is not symmetric and lacks a triangle inequality. Let us prove now the property of superadditivity, whenever $\sigma_{AB}= \sigma_A \otimes \sigma_B$. \[prop:sigmaprod\] Let ${\ensuremath{\mathcal H}}_{AB}={\ensuremath{\mathcal H}}_A \otimes {\ensuremath{\mathcal H}}_B$ and $\rho_{AB}, \sigma_{AB} \in {\mathcal{S}}_{AB}$. If $\sigma_{AB}= \sigma_A \otimes \sigma_B$, then ${D(\rho_{AB}||\sigma_{AB})} = I_\rho(A:B) + {D(\rho_A||\sigma_A)}+{D(\rho_B||\sigma_B)}$, where $I_\rho(A:B)={D(\rho_{AB}||\rho_A \otimes \rho_B)}$ is the mutual information [@shannon]. As a consequence, ${\displaystyle}{D(\rho_{AB}||\sigma_{A}\otimes \sigma_B )} \geq {D(\rho_A||\sigma_A)}+{D(\rho_B||\sigma_B)}$. Since $\sigma_{AB}= \sigma_A \otimes \sigma_B$, we have $$\begin{aligned} {D(\rho_{AB}||\sigma_{A} \otimes \sigma_{B})} &=& \tr[\rho_{AB}(\log \rho_{AB} - \log \sigma_{A} \otimes \sigma_{B})]\\ &=& \tr[\rho_{AB} (\log \rho_{AB} - \log \rho_{A} \otimes \rho_{B} +\log \rho_{A} \otimes \rho_{B} - \log \sigma_{A} \otimes \sigma_{B})] \\ &=& {D(\rho_{AB}||\rho_A \otimes \rho_B)} + {D(\rho_A \otimes \rho_B|| \sigma_{A} \otimes \sigma_{B})}\\ &=& I_\rho (A:B) + {D(\rho_A||\sigma_A)} + {D(\rho_B||\sigma_B)}.\end{aligned}$$ Now, since $I_\rho (A:B)$ is a relative entropy, it is greater or equal than zero (property 1 of Proposition \[prop:REprop\]), so ${\displaystyle}{D(\rho_{AB}||\sigma_{A}\otimes \sigma_B )} \geq {D(\rho_A||\sigma_A)}+{D(\rho_B||\sigma_B)}$. We prove now a lemma for observables (non necessarily of trace $1$) which yields a relation between the relative entropy of two observables and the relative entropy of some dilations of each of them. In particular, it is a useful tool to express the relative entropy of two observables in terms of the relative entropy of their normalizations (i.e., the quotient of each of them by their trace). \[lemma:rel-entropy-homogeneity\] Let ${\ensuremath{\mathcal H}}$ be a finite dimensional Hilbert space and let $f,g\in {\mathcal{A}}^+$ such that $\tr[f] \neq 0$. For all positive real numbers $a$ and $b$, we have: $${D(af||bg)} = {D(f||g)} + \log \frac{a}{b}.$$ $$\begin{aligned} {D(af||bg)} &=& \frac{1}{a \tr f} \left(a \tr \left[ f \left( \log a f - \log bg \right) \right] \right) \\ &=& \frac{1}{\tr f} \qty(\tr[ f \log a] + \tr[ f \log f] - \tr[ f \log b] - \tr[f \log g]) \\ &=&\frac{1}{\tr f} (\tr[ f \left( \log f - \log g \right) ] ) + \log a - \log b \\ &=& {D(f||g)} + \log \frac{a}{b},\end{aligned}$$ where, in the first and third equality, we are using the linearity of the trace, and we are denoting $\log a {\ensuremath{\mathds{1}}}$ by $\log a$ for every $a\geq 0$. Since the relative entropy of two density matrices is non-negative (property 1 of Proposition \[prop:REprop\]), we have the following corollary: \[lemma:cond-ent\] Let ${\ensuremath{\mathcal H}}$ be a finite dimensional Hilbert space and let $f,g\in {\mathcal{A}}^+$ such that $\tr[f] \neq 0$ and $\tr[g] \neq 0$. Then, the following inequality holds: $$\label{eq:cond-ent} {D(f||g)} \ge - \log \frac{\tr[ g]}{\tr[ f]}.$$ Since $f/\tr[f]$ and $g / \tr[g]$ are density matrices, we have that ${D(f/\tr[f] \,||\, g / \tr[g])} \ge 0 $, and we can apply Lemma \[lemma:rel-entropy-homogeneity\]: ${\displaystyle}0 \leq {D(f/\tr[f] \,||\, g / \tr[g])} = {D(f||g)} + \log \frac{\tr[g]}{\tr[f]} $. Proof of main result {#sec:results} ==================== We divide the proof of Theorem \[thm:quasifactorizationAB\] in four steps. In the first step, we provide a lower bound for the relative entropy of $\rho_{AB}$ and $\sigma_{AB}$ in terms of ${D(\rho_{A}||\sigma_{A})}$, ${D(\rho_{B}||\sigma_{B})} $ and an error term, which we will further bound in the following steps. \[step:1\] For density matrices $\rho_{AB}, \sigma_{AB} \in {\mathcal{S}}_{AB}$, it holds that $$\label{eq:step-1} {D(\rho_{AB}||\sigma_{AB})} \ge {D(\rho_{A}||\sigma_{A})} + {D(\rho_{B}||\sigma_{B})} - \log \tr M,$$ where $ M = \exp \bqty{ \log \sigma_{AB} - \log \sigma_{A} \otimes \sigma_{B} +\log \rho_{A} \otimes \rho_{B} } $ and equality holds (both sides being equal to zero) if $\rho_{AB} = \sigma_{AB}$.\ Moreover, if $\sigma_{AB} = \sigma_A \otimes \sigma_B$, then $\log \tr M =0$. It holds that: $$\begin{aligned} {\displaystyle}{D(\rho_{AB}||\sigma_{AB})} & - & \left[ {D(\rho_{A}||\sigma_{A})} + {D(\rho_{B}||\sigma_{B})} \right]=\\ &=& {D(\rho_{AB}||\sigma_{AB})} - {D(\rho_{A} \otimes \rho_{B}||\sigma_{A} \otimes \sigma_{B})} \\ &=& \tr \left[ {\rho_{AB}} \left( \log {\rho_{AB}} - \underbrace{ \left( \, \log {\sigma_{AB}} - \log \sigma_{A} \otimes \sigma_{B} +\log \rho_{A} \otimes \rho_{B} \, \right) }_{\log M} \right) \right] \\ & =& {D(\rho_{AB}||M)},\end{aligned}$$ where $M$ is defined as in the statement of the step and in the first equality we have used the fourth property of Proposition \[prop:REprop\]. We can now apply Corollary \[lemma:cond-ent\] to obtain that ${D(\rho_{AB}||M)} = \tr[ {\rho_{AB}} (\log {\rho_{AB}} - \log M) ] \ge - \log \tr M$. It is easy to check, given the definition of $M$, that $M=\sigma_{AB}$ if $\rho_{AB} = \sigma_{AB}$, so both sides are equal to zero in this case. Also, if $\sigma_{AB}= \sigma_A \otimes \sigma_B$, $M$ is equal to $\rho_A\otimes \rho_B$. In both cases we have $\log \tr M = 0$. Our target now is to bound the error term, $\log \tr M$, in terms of the relative entropy of $\rho_{AB}$ and $\sigma_{AB}$ times a constant which depends only on $A$, $B$ and $\sigma_{AB}$, and represents how far $\sigma_{AB}$ is from being a tensor product. In the second step of the proof, we will bound this term by the trace of the product of a term which contains this ‘distance’ between $\sigma_{AB}$ and $\sigma_{A} \otimes \sigma_{B}$ and another term which depends on $\rho_{AB}$ and not on $\sigma_{AB}$. However, before that, we need to introduce some concepts and results. First, we recall the Golden-Thompson inequality, proven independently in [@golden] and [@thompson] (and extended to the infinite dimensional case in [@GTinfdim] and [@GTinfdim2]), which says that for Hermitian operators $f$ and $g$, $$\tr[e^{f+g}] \leq \tr[e^f e^g],$$ where we denote by $e^f$ the exponential of $f$, given by ${\displaystyle}e^f:= \ov{\infty}{\un{k=0}{\sum}} \frac{f^k}{k!} $. The trivial generalization of the Golden-Thompson inequality to three operators instead of two in the form $\tr[e^{f+g+h}] \leq \tr[e^f e^g e^h]$ is false, as Lieb mentioned in [@lieb]. However, in the same paper, he provides a correct generalization of this inequality for three operators. This result has recently been extended by Sutter et al. in [@sutter] via de so-called multivariate trace inequalities (see also the subsequent paper by Wilde [@multioperator2], where similar inequalities are derived following the statements of [@multioperator1]). \[thm:Lieb\] Let $f, g$ be positive semidefinite operators, and recall the definition of $\mathcal{T}_g$: $$\mathcal T_g(f) = \int_0^\infty \dd{t} (g+t)^{-1} f (g+t)^{-1} .$$ $\mathcal T_g$ is positive semidefinite if $g$ is. We have that $$\tr[ \exp(-f+g+h)] \le \tr[ e^h \mathcal T_{e^f}(e^g)].$$ This superoperator $\mathcal T_g$ provides a pseudo-inversion of the operator $g$ with respect to the operator where it is evaluated. In particular, if $f$ and $g$ commute, it is exactly the standard inversion, as we can see in the following corollary. If $f$ and $g$ commute, then $$\mathcal T_g(f) = f \int_0^\infty \dd{t} (g + t)^{-2} = f g^{-1},$$ and therefore $$\tr[ \exp(-f+g+h) ]\le \tr[ e^h e^{-f} e^g ] = \tr[ e^h e^{-f+g}] .$$ This shows that Lieb’s theorem is really a generalization of Golden-Thompson inequality. We use an alternative definition of this superoperator to obtain a necessary tool for the proof of Step \[step:2\]. In [@sutter Lemma 3.4], Sutter, Berta and Tomamichel prove the following result: For ${f} $ a positive semidefinite operator and $g $ a Hermitian operator the following holds: ${\displaystyle}\mathcal{T}_g(f)= \int_{-\infty}^\infty dt \, \beta_0 (t) \, g^{\frac{-1-it}{2}} \,f \, g^{\frac{-1+it}{2}} ,$ with ${\displaystyle}\beta_0(t)= \frac{\pi}{2} (\cosh(\pi t)+ 1)^{-1} $. Using this expression for $\mathcal{T}_{\sigma_A \otimes \sigma_B} (\sigma_{AB})$, we can prove the following result, which is a quantum version of a result used in [@clasico]. \[lemma:remark\] For every operator $O_{A} \in {\mathcal{B}}_A$ and $O_{B} \in {\mathcal{B}}_B$ the following holds: ${\displaystyle}\tr[ L(\sigma_{AB}) \, \sigma_{A} \otimes O_{B} ]= \tr[ L(\sigma_{AB}) \, O_{A} \otimes \sigma_{B} ]= 0,$ where ${\displaystyle}L(\sigma_{AB}) = \mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) - {\ensuremath{\mathds{1}}}_{AB}$. We only prove $\tr[ L(\sigma_{AB}) \, \sigma_{A} \otimes O_{B} ]=0$, since the other equality is completely analogous. $$\begin{aligned} & \tr[ L(\sigma_{AB}) \, \sigma_{A} \otimes O_{B} ]=\\ &\phantom{asdad} = \tr[ \left( \mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) - {\ensuremath{\mathds{1}}}_{AB}\right) \sigma_{A} \otimes O_{B}]\\ &\phantom{asdad}=\tr[ \mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) \sigma_{A} \otimes O_{B}]- \tr[\sigma_{A} \otimes O_{B}] \\ &\phantom{asdad}= \tr[\int_{- \infty}^\infty dt \, \beta_0 (t) \left( \sigma_A \otimes \sigma_B \right)^{\frac{-1-it}{2}} \, \sigma_{AB} \, \left( \sigma_A \otimes \sigma_B \right)^{\frac{-1+it}{2}} \, \sigma_{A} \otimes O_{B}] - \tr[O_B]\\ &\phantom{asdad}= \int_{- \infty}^\infty dt \, \beta_0 (t) \tr[ \sigma_A^{\frac{-1-it}{2}} \otimes \sigma_B^{\frac{-1-it}{2}} \, \sigma_{AB} \, \sigma_A^{\frac{-1+it}{2}} \otimes \sigma_B^{\frac{-1+it}{2}} \, \sigma_{A} \otimes O_{B}]-\tr[O_{B}],\end{aligned}$$ because $\tr[\sigma_A]=1$, the integral commutes with the trace, $\beta_0(t)$ is a scalar for every $t \in {\mathbb{R}}$ and the exponent in the power of a tensor product can be split into both terms. Now, since the trace is cyclic and using the fact that any operator in ${\ensuremath{\mathcal H}}_B$ commutes with every operator in ${\ensuremath{\mathcal H}}_A$, we have: $$\begin{aligned} & \tr[ L(\sigma_{AB}) \, \sigma_{A} \otimes O_{B} ]=\\ &\phantom{asdad}= \int_{- \infty}^\infty dt \, \beta_0 (t) \tr[ \sigma_{AB} \, \sigma_A^{\frac{-1+it}{2}} \otimes \sigma_B^{\frac{-1+it}{2}} \, \sigma_{A} \otimes O_{B} \, \sigma_A^{\frac{-1-it}{2}} \otimes \sigma_B^{\frac{-1-it}{2}}]-\tr[O_{B}]\\ &\phantom{asdad}= \int_{- \infty}^\infty dt \, \beta_0 (t) \tr[ \sigma_{AB} \, \left( \sigma_A^{\frac{-1+it}{2}} \sigma_{A} \, \sigma_A^{\frac{-1-it}{2}} \right) \otimes \left( \sigma_B^{\frac{-1+it}{2}} O_{B} \,\sigma_B^{\frac{-1-it}{2}} \right)]-\tr[O_{B}]\\ &\phantom{asdad}= \int_{- \infty}^\infty dt \, \beta_0 (t) \tr[ \sigma_{AB} \, {\ensuremath{\mathds{1}}}_A \otimes \left( \sigma_B^{\frac{-1+it}{2}} O_{B} \,\sigma_B^{\frac{-1-it}{2}} \right)]-\tr[O_{B}]\\ &\phantom{asdad}= \int_{- \infty}^\infty dt \, \beta_0 (t) \tr[ \sigma_{B} \, \sigma_B^{\frac{-1+it}{2}} O_{B} \,\sigma_B^{\frac{-1-it}{2}} ]-\tr[O_{B}]\\ &\phantom{asdad}= \int_{- \infty}^\infty dt \, \beta_0 (t) \tr[ \sigma_B^{\frac{-1-it}{2}} \, \sigma_{B} \, \sigma_B^{\frac{-1+it}{2}} O_{B} ]-\tr[O_{B}]\\ &\phantom{asdad} = \tr[ O_{B}] \int_{- \infty}^\infty dt \, \beta_0 (t) -\tr[O_{B}]\\ &\phantom{asdad}= 0,\end{aligned}$$ where we have used ${\displaystyle}\int_{- \infty}^\infty dt \, \beta_0 (t)=1 $, and the fact that, for every $f_A \in {\mathcal{B}}_A$ and $g_{AB} \in {\mathcal{S}}_{AB}$, the following holds: ${\displaystyle}\tr[f_A \otimes {\ensuremath{\mathds{1}}}_B \, g_{AB}] = \tr[f_A \, g_A] $. We are now in position to develop the second step of the proof. \[step:2\] With the same notation of \[step:1\], we have that $$\log \tr M \le \tr[L(\sigma_{AB}) \left( \rho_{A} - \sigma_A \right) \otimes \left( \rho_{B} - \sigma_B \right) ],$$ where ${\displaystyle}L(\sigma_{AB})= \mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) - {\ensuremath{\mathds{1}}}_{AB}$. We apply Lieb’s theorem to the error term of inequality (\[eq:step-1\]): $$\begin{aligned} \tr M &=& \tr\left[ \exp( \underbrace{\log \sigma_{AB} }_{g} - \underbrace{\log \sigma_{A} \otimes \sigma_{B} }_{f} + \underbrace{\log \rho_{A} \otimes \rho_{B} }_{h} ) \right] \\ &\leq & \tr[ \rho_{A} \otimes \rho_{B} \mathcal T_{\sigma_{A} \otimes \sigma_{B}} ( \sigma_{AB})]\\ &=& \tr[ \rho_{A} \otimes \rho_{B} \underbrace{\left( \mathcal T_{\sigma_{A} \otimes \sigma_{B}} (\sigma_{AB}) -{\ensuremath{\mathds{1}}}_{AB} \right)}_{L(\sigma_{AB})}] + \underbrace{\tr[ \rho_{A} \otimes \rho_{B}]}_1,\end{aligned}$$ where we are adding and substracting $\rho_A \otimes \rho_B$ inside the trace in the last equality. Now, using the fact $\log(x)\le x-1$, we have $\log \tr M \leq \tr M - 1 \leq \tr[ L(\sigma_{AB}) \, \rho_{A} \otimes \rho_{B} ]$. Finally, in virtue of Lemma \[lemma:remark\], it is clear that $ {\displaystyle}\tr[ L(\sigma_{AB}) \, \rho_{A} \otimes \rho_{B} ]= \tr[L(\sigma_{AB}) \left( \rho_{A} - \sigma_A \right) \otimes \left( \rho_{B} - \sigma_B \right) ] $. Therefore, $ \log \tr M \leq \tr[L(\sigma_{AB}) \left( \rho_{A} - \sigma_A \right) \otimes \left( \rho_{B} - \sigma_B \right) ]$. Notice that if $\sigma_{AB}= \sigma_A \otimes \sigma_B$, then $\mathcal T_{\sigma_{A} \otimes \sigma_{B}} ( \sigma_{AB})= \left( \sigma_{A} \otimes \sigma_{B} \right)^{-1} \sigma_{A} \otimes \sigma_{B}= {\ensuremath{\mathds{1}}}_{AB} $, so $L(\sigma_{AB})=0$. In the third step of the proof, we need to bound $\tr[L(\sigma_{AB}) \left( \rho_{A} - \sigma_A \right) \otimes \left( \rho_{B} - \sigma_B \right) ]$ in terms of the relative entropy of $\rho_{AB}$ and $\sigma_{AB}$ times a constant depending only on $L(\sigma_{AB})$ (since $L(\sigma_{AB})$ represents how entangled $\sigma_{AB}$ is between the regions $A$ and $B$). The first well-known result we will use in this step is Pinsker’s inequality [@csiszar; @pinsker]. \[thm:pinsker\] For $\rho_{AB}$ and $\sigma_{AB}$ density matrices, it holds that $$\norm{\rho_{AB}-\sigma_{AB}}_1^2 \le 2 {D(\rho_{AB}||\sigma_{AB})}.$$ This result will be of use at the end of the proof to finally obtain the relative entropy in the right-hand side of the desired inequality. However, it is important to notice the different scales of the ${\mathbb{L}}^1$-norm of the difference between $\rho_{AB}$ and $\sigma_{AB}$ and the relative entropy of $\rho_{AB}$ and $\sigma_{AB}$ in Pinsker’s inequality. Since we are interested in obtaining the relative entropy with exponent one, we will need to increase the degree of the term with the trace we already have and from which we will construct an ${\mathbb{L}}^1$-norm (since, for the moment, its degree is one). We will see later that the fact that in $ \tr[L (\sigma_{AB}) \left( \rho_{A} - \sigma_A \right) \otimes \left( \rho_{B} - \sigma_B \right) ]$ we have $ \left( \rho_{A} - \sigma_A \right) \otimes \left( \rho_{B} - \sigma_B \right) $ split into two regions, the multiplicativity of the trace with respect to tensor products and the monotonicity of the relative entropy play a decisive role in the proof. Another important fact that we notice in the left-hand side of Pinsker’s inequality is that there is a difference between two states (in fact, the ones appearing in the relative entropy). This justifies the use of Lemma \[lemma:remark\] at the end of Step \[step:2\], to obtain something similar to the difference between $\rho_{AB}$ and $\sigma_{AB}$. We are now ready to prove the third step in the proof of Theorem \[thm:quasifactorizationAB\]. \[step:3\] With the notation of Theorem \[thm:quasifactorizationAB\], $$\tr[L (\sigma_{AB}) \left( \rho_{A} - \sigma_A \right) \otimes \left( \rho_{B} - \sigma_B \right) ]\le 2 \norm{L(\sigma_{AB})}_\infty {D(\rho_{AB}||\sigma_{AB})}.$$ We use the multiplicativity with respect to tensor products of the trace norm and Hölder’s inequality between the trace norm and the operator norm. Thus, $$\begin{aligned} \tr[ L(\sigma_{AB}) \, (\rho_{A} - \sigma_{A})\otimes(\rho_{B}-\sigma_{B}) ]&\le & \norm{L(\sigma_{AB})}_\infty \norm{ \left( \rho_{A} - \sigma_{A} \right) \otimes \left( \rho_{B}-\sigma_{B} \right) }_1\\ &=& \norm{L(\sigma_{AB})}_\infty \norm{\rho_{A} - \sigma_{A}}_1 \norm{\rho_{B}-\sigma_{B}}_1.\end{aligned}$$ Finally, Pinsker’s inequality (Theorem \[thm:pinsker\]) implies that ${\displaystyle}\norm{\rho_{A} - \sigma_{A}}_1 \leq \sqrt{ 2 {D(\rho_{A}||\sigma_{A})} }, \phantom{sadda} \norm{\rho_{B} - \sigma_{B}}_1 \leq \sqrt{ 2 {D(\rho_{B}||\sigma_{B})} } $. Therefore, ${\displaystyle}\norm{\rho_{A} - \sigma_{A}}_1 \norm{\rho_{B}-\sigma_{B}}_1\le 2 \sqrt{{D(\rho_{A}||\sigma_{A})} {D(\rho_{B}||\sigma_{B})}} \le 2 {D(\rho_{AB}||\sigma_{AB})},$ where in the last inequality we have used monotonicity of the relative entropy with respect to the partial trace (Proposition \[prop:REprop\]). If we now put together Steps \[step:1\], \[step:2\] and \[step:3\], we obtain the following expression $$(1 + 2 \norm{L(\sigma_{AB})}_\infty) {D(\rho_{AB}||\sigma_{AB})}\ge {D(\rho_A||\sigma_A)} + {D(\rho_B||\sigma_B)},$$ with ${\displaystyle}L(\sigma_{AB}) = \mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) - {\ensuremath{\mathds{1}}}_{AB}$. This inequality already constitutes a quantitative extension of (\[superadditivity\]) for arbitrary density operators $\sigma_{AB}$ in the sense that if $\sigma_{AB}$ is a tensor product between $A$ and $B$, we recover the usual superadditivity, and in general $\norm{L(\sigma_{AB})}_\infty$ measures how far $\sigma_{AB}$ is from $\sigma_A \otimes \sigma_B$. In the fourth and final step of the proof, we bound $\norm{L(\sigma_{AB})}_\infty$ by $\norm{ \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} \, \sigma_{AB} \, \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} - {\ensuremath{\mathds{1}}}_{AB}}_\infty$, a quantity from which the closeness to $0$ whenever $\sigma_{AB}$ is near from being a tensor product is directly deduced. It also has some physical interpretation in quantum many body systems that will be discussed after proving Step \[step:4\]. First, we need to introduce the setting of non-commutative ${\mathbb{L}}^p$ spaces with a $\rho$-weighted norm [@kosaki]. The central property of these non-commutative ${{\mathbb{L}}}^p$ spaces is that they are equipped with a *weighted norm* which, for a full rank state $\rho\in {\mathcal{S}}_{AB}$, is given by ${\displaystyle}\norm{f}_{{\mathbb{L}}^p(\rho)} := \tr[\abs{\rho^{1/2p} f \rho^{1/2p}}^p]^{1/p} \, $ for every $\,f \in {\mathcal{A}}_{AB}$. Analogously, the $\rho$-*weighted inner product* is given by ${\displaystyle}\left\langle f, g \right\rangle_\rho := \tr[\sqrt{\rho} f \sqrt{\rho} g] \, $ for every $\,f,g \in {\mathcal{A}}_{AB}$. Some fundamental properties of these spaces are collected in the following proposition. \[prop:noncomLp\] Let $\rho \in {\mathcal{S}}_{AB}$. The following properties hold for $\rho$-weighted norms: 1. **Order.** $\forall p,q \in [1, \infty)$, with $p \leq q$, we have $\norm{f}_{{\mathbb{L}}^p(\rho)} \leq \norm{f}_{{\mathbb{L}}^q(\rho)} \, \forall f \in {\mathcal{A}}_{AB}$. 2. **Duality.** $\forall f \in {\mathcal{A}}_{AB}$, we have $\norm{f}_{{\mathbb{L}}^p(\rho)} = \sup \qty{ \left\langle g, f \right\rangle_\rho, g \in {\mathcal{A}}_{AB}, \norm{g}_{{\mathbb{L}}^q(\rho)} \leq 1 }$ for $1/p+1/q=1$. 3. **Operator norm.** $\forall f \in {\mathcal{A}}_{AB}$, we have $\norm{f}_{{\mathbb{L}}^\infty(\rho)} = \norm{f}_\infty$, the usual operator norm. Another tool we will use in the proof of Step \[step:4\] is the following result. \[lemma:contract\] Consider $\rho \in {\mathcal{S}}_{AB}$ and let $T$ be a quantum channel verifying $T^*(\rho)=\rho$, where $T^*$ denotes the dual of $T$ with respect to the Hilbert-Schmidt scalar product. Then, $T$ is contractive between ${\mathbb{L}}^1(\rho)$ and ${\mathbb{L}}^1(\rho)$, i.e., the following inequality holds for every $X \in {\mathcal{B}}_{AB}$: $$\norm{T(X)}_{{\mathbb{L}}^1(\rho)} \leq \norm{X}_{{\mathbb{L}}^1(\rho)}.$$ Using the property of duality for the $\rho$-weighted norms of ${\mathbb{L}}^p$-spaces (property 2 of Proposition \[prop:noncomLp\]), we can write: $$\begin{aligned} {\displaystyle}\norm{T(X)}_{{\mathbb{L}}^1(\rho)} &= \un{\norm{Y}_{{\mathbb{L}}^\infty(\rho)} \leq 1}{\text{sup}} \tr[T(X)\,\rho^{1/2} \, Y \, \rho^{1/2}] \\ &= \un{\norm{Y}_\infty \leq 1}{\text{sup}} \tr[T(X)\,\rho^{1/2} \, Y \, \rho^{1/2}] \\ &= \un{- {\ensuremath{\mathds{1}}}\leq Y \leq {\ensuremath{\mathds{1}}}}{\text{sup}} \tr[T(X)\, \rho^{1/2} \, Y \, \rho^{1/2}],\end{aligned}$$ where in the first step we have used the fact that, for every $\rho \in {\mathcal{S}}_{AB}$, $\norm{\cdot}_{{\mathbb{L}}^\infty(\rho)}$ coincides with the operator norm. Recalling now that $T^*$ is the dual of $T$ with respect to the Hilbert-Schmidt scalar product, we have: $$\begin{aligned} {\displaystyle}\tr[T(X)\, \rho^{1/2} \, Y \, \rho^{1/2}] &= \tr[X\, T^*( \rho^{1/2} \, Y \, \rho^{1/2})] \\ &= \tr[X\, \rho^{1/2} \, \rho^{-1/2} \, T^*( \rho^{1/2} \, Y \, \rho^{1/2}) \, \rho^{-1/2} \, \rho^{1/2} ].\end{aligned}$$ Since we are considering the supremum over the observables verifying $-{\ensuremath{\mathds{1}}}\leq Y \leq {\ensuremath{\mathds{1}}}$, if we apply to these inequalitites $T^*(\rho^{1/2} \cdot \rho^{1/2})$, we have $-\rho \leq T^*(\rho^{1/2} \, Y \, \rho^{1/2}) \leq \rho $ (because of the assumption $T^*(\rho)= \rho$). Hence, if we denote $Z= \rho^{-1/2} \, T^*( \rho^{1/2} \, Y \, \rho^{1/2}) \, \rho^{-1/2}$, it is clear that whenever $ -{\ensuremath{\mathds{1}}}\leq Y \leq {\ensuremath{\mathds{1}}}$, also $-{\ensuremath{\mathds{1}}}\leq Z \leq {\ensuremath{\mathds{1}}}$. Therefore, $$\begin{aligned} \norm{T(X)}_{{\mathbb{L}}^1(\rho)} &= \un{- {\ensuremath{\mathds{1}}}\leq Y \leq {\ensuremath{\mathds{1}}}}{\text{sup}} \tr[T(X)\, \rho^{1/2} \, Y \, \rho^{1/2}] \\ &= \un{- {\ensuremath{\mathds{1}}}\leq Y \leq {\ensuremath{\mathds{1}}}}{\text{sup}} \tr[X\, \rho^{1/2} \, \rho^{-1/2} \, T^*( \rho^{1/2} \, Y \, \rho^{1/2}) \, \rho^{-1/2} \, \rho^{1/2} ] \\ &\leq \un{- {\ensuremath{\mathds{1}}}\leq Z \leq {\ensuremath{\mathds{1}}}}{\text{sup}} \tr[X\, \rho^{1/2} \, Z \, \rho^{1/2} ] \\ &= \norm{X}_{{\mathbb{L}}^1(\rho)} ,\end{aligned}$$ where the last equality comes again from the property of duality of weighted ${\mathbb{L}}^p$-norms. In the proof of the previous lemma we have made strong use of the property of duality of ${\mathbb{L}}^p(\rho)$. Indeed, considering the ${\mathbb{L}}^1(\rho)$-norm as dual of the operator norm, has been essential to obtain the desired result. Using similar tools, we can now prove the last step in the proof of Theorem \[thm:quasifactorizationAB\]. \[step:4\] With the notation of the previous steps, we have $$\norm{L(\sigma_{AB})}_\infty \leq \norm{ \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} \, \sigma_{AB} \, \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} - {\ensuremath{\mathds{1}}}_{AB}}_\infty.$$ The strategy we follow in this proof is the opposite to the one used in the previous lemma, i.e., we study now the ${\mathbb{L}}^\infty(\sigma_A \otimes \sigma_B)$-norm as the dual of the ${\mathbb{L}}^1(\sigma_A \otimes \sigma_B)$-norm. Since $\norm{\cdot}_{{\mathbb{L}}^\infty (\rho_{AB})}$ coincides with the usual $\infty$-norm (operator norm) for every $\rho_{AB} \in {\mathcal{S}}_{AB}$, we can write ${\displaystyle}\norm{L(\sigma_{AB})}_\infty = \norm{\mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) - {\ensuremath{\mathds{1}}}_{AB}}_{{\mathbb{L}}^\infty(\sigma_A \otimes \sigma_B)}$. Using the aforementioned property of duality for the $\sigma_A \otimes \sigma_B$-weighted norms of ${\mathbb{L}}^p$-spaces, we have: $$\begin{aligned} & \norm{\mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) - {\ensuremath{\mathds{1}}}_{AB}}_{{\mathbb{L}}^\infty(\sigma_A \otimes \sigma_B)} = \\ & \phantom{asdasddad}=\un{\norm{O_{AB}}_{{\mathbb{L}}^1(\sigma_A \otimes \sigma_B)} \leq 1}{\text{sup}} \left\langle O_{AB} , \mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) - {\ensuremath{\mathds{1}}}_{AB} \right\rangle_{\sigma_A \otimes \sigma_B} \\ &\phantom{asdasddad} =\un{\norm{O_{AB}}_{{\mathbb{L}}^1(\sigma_A \otimes \sigma_B)} \leq 1}{\text{sup}} \tr[ (\sigma_A \otimes \sigma_B)^{1/2} \, O_{AB}\, (\sigma_A \otimes \sigma_B)^{1/2} \left( \mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) - {\ensuremath{\mathds{1}}}_{AB} \right)] \\ &\phantom{asdasddad} =\un{\norm{O_{AB}}_{{\mathbb{L}}^1(\sigma_A \otimes \sigma_B)} \leq 1}{\text{sup}} \left( \underbrace{\tr[ \sigma_A^{1/2} \otimes \sigma_B^{1/2} \, O_{AB}\, \sigma_A^{1/2} \otimes \sigma_B^{1/2} \mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) ]}_{R} \right. \\ & \left. \phantom{asdasddadaszxczzxdasdasdads} - \underbrace{\tr[\sigma_A^{1/2} \otimes \sigma_B^{1/2} \, O_{AB}\, \sigma_A^{1/2} \otimes \sigma_B^{1/2} ] }_{S} \right).\end{aligned}$$ Let us analyze the terms $R$ and $S$ separately. For $R$, we have: $$\begin{aligned} R&= \tr[ \sigma_A^{1/2} \otimes \sigma_B^{1/2} \, O_{AB}\, \sigma_A^{1/2} \otimes \sigma_B^{1/2} \mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) ] \\ & = \tr[ (\sigma_A \otimes \sigma_B)^{1/2} \, O_{AB}\, (\sigma_A \otimes \sigma_B)^{1/2} \int_{- \infty}^\infty dt \, \beta_0 (t) \left( \sigma_A \otimes \sigma_B \right)^{\frac{-1-it}{2}} \, \sigma_{AB} \, \left( \sigma_A \otimes \sigma_B \right)^{\frac{-1+it}{2}} ] \\ & = \tr[ O_{AB}\, \int_{- \infty}^\infty dt \, \beta_0 (t) \left( \sigma_A \otimes \sigma_B \right)^{\frac{-it}{2}} \, \sigma_{AB} \, \left( \sigma_A \otimes \sigma_B \right)^{\frac{it}{2}} ] \\ & =\int_{- \infty}^\infty dt \, \beta_0 (t) \tr[O_{AB} \, \left( \sigma_A \otimes \sigma_B \right)^{\frac{-it}{2}} \, \sigma_{AB} \, \left( \sigma_A \otimes \sigma_B \right)^{\frac{it}{2}}] \\ & =\int_{- \infty}^\infty dt \, \beta_0 (t) \tr[\left( \sigma_A \otimes \sigma_B \right)^{\frac{it}{2}} \; O_{AB} \, \left( \sigma_A \otimes \sigma_B \right)^{\frac{-it}{2}} \, \sigma_{AB} ] \\ & = \tr[ \sigma_{AB} \underbrace{ \int_{- \infty}^\infty dt \, \beta_0 (t) \left( \sigma_A \otimes \sigma_B \right)^{\frac{it}{2}} \, O_{AB} \, \left( \sigma_A \otimes \sigma_B \right)^{\frac{-it}{2}}}_{\widetilde{O}_{AB}} ],\end{aligned}$$ where in the third and last equality we have used the fact that the integral and the trace commute, and the fourth equality is due to the cyclicity of the trace. We have also defined: ${\displaystyle}\widetilde{O}_{AB}:= \int_{- \infty}^\infty dt \, \beta_0 (t) \left( \sigma_A \otimes \sigma_B \right)^{\frac{it}{2}} \, O_{AB} \, \left( \sigma_A \otimes \sigma_B \right)^{\frac{-it}{2}}$. If we were able to express $S$ in terms of $\widetilde{O}_{AB}$, we could simplify the expression that appears in the supremum above. We can do that in the following way: $$\begin{aligned} S& = \tr[\sigma_A^{1/2} \otimes \sigma_B^{1/2} \, O_{AB}\, \sigma_A^{1/2} \otimes \sigma_B^{1/2} ] \\ &= \tr[\sigma_A^{1/2} \otimes \sigma_B^{1/2} \, O_{AB}\, \sigma_A^{1/2} \otimes \sigma_B^{1/2} \int_{- \infty}^\infty dt \, \beta_0 (t) ] \\ &= \int_{- \infty}^\infty dt \, \beta_0 (t) \tr[\sigma_A^{1/2} \otimes \sigma_B^{1/2} \, O_{AB}\, \sigma_A^{1/2} \otimes \sigma_B^{1/2} ] \\ &= \int_{- \infty}^\infty dt \, \beta_0 (t) \tr[(\sigma_A\otimes \sigma_B) \, (\sigma_A\otimes \sigma_B)^{\frac{it}{2}} \, O_{AB}\, (\sigma_A\otimes \sigma_B)^{\frac{-it}{2}} ] \\ &= \tr[(\sigma_A\otimes \sigma_B) \int_{- \infty}^\infty dt \, \beta_0 (t) (\sigma_A\otimes \sigma_B)^{\frac{it}{2}} \, O_{AB}\, (\sigma_A\otimes \sigma_B)^{\frac{-it}{2}} ] \\ &= \tr[(\sigma_A\otimes \sigma_B) \, \widetilde{O}_{AB} ],\end{aligned}$$ where we have used again the properties of cyclicity of the trace and commutativity of the integral and the trace. Placing now the values for $R$ and $S$ that we have just computed in the supremum of the first part of the proof, we have: $$\begin{aligned} \norm{\mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) - {\ensuremath{\mathds{1}}}_{AB}}_{{\mathbb{L}}^\infty(\sigma_A \otimes \sigma_B)} &= \un{\norm{O_{AB}}_{L^1(\sigma_A \otimes \sigma_B)} \leq 1}{\text{sup}} \left( \tr[\sigma_{AB} \, \widetilde{O}_{AB} ] - \tr[\sigma_A \otimes \sigma_B \, \widetilde{O}_{AB} ] \right) \\ &= \un{\norm{O_{AB}}_{L^1(\sigma_A \otimes \sigma_B)} \leq 1}{\text{sup}} \tr[\, \widetilde{O}_{AB} \left( \sigma_{AB} - \sigma_A \otimes \sigma_B \right) ] .\end{aligned}$$ This expression looks much simpler than the one we had before. However, we need to prove that $\norm{\widetilde{O}_{AB}}_{L^1(\sigma_A \otimes \sigma_B)} \leq 1$ in order to see $\widetilde{O}_{AB}$ as one of the terms where the supremum is taken. Indeed, if we consider the map $T: {\mathcal{A}}_{AB} \rightarrow {\mathcal{A}}_{AB}$ given by ${\displaystyle}O_{AB} \mapsto \int_{- \infty}^\infty dt \, \beta_0 (t) (\sigma_A\otimes \sigma_B)^{\frac{it}{2}} \, O_{AB}\, (\sigma_A\otimes \sigma_B)^{\frac{-it}{2}} $, it is clearly a quantum channel and also verifies $T^*(\sigma_A \otimes \sigma_B)= \sigma_A \otimes \sigma_B$. Hence, in virtue of Lemma \[lemma:contract\], we have ${\displaystyle}\norm{\widetilde{O}_{AB}}_{{\mathbb{L}}^1(\sigma_A \otimes \sigma_B)} \leq \norm{O_{AB}}_{{\mathbb{L}}^1(\sigma_A \otimes \sigma_B)} $, and, therefore, ${\displaystyle}\un{\norm{O_{AB}}_{L^1(\sigma_A \otimes \sigma_B)} \leq 1}{\text{sup}} \tr[\widetilde{O}_{AB} \left( \sigma_{AB} - \sigma_A \otimes \sigma_B \right) ] \leq \un{\norm{\Omega_{AB}}_{L^1(\sigma_A \otimes \sigma_B)} \leq 1}{\text{sup}} \tr[\Omega_{AB} \left( \sigma_{AB} - \sigma_A \otimes \sigma_B \right) ]. $ In this last supremum over elements of $1$-norm, we can undo the previous transformations in order to obtain again an $\infty$-norm. First, we need to write the term in the supremum as a $\sigma_A \otimes \sigma_B$-product of two terms: $$\begin{aligned} & \tr[\Omega_{AB} \left( \sigma_{AB} - \sigma_A \otimes \sigma_B \right) ] = \\ & \phantom{asdasdd} = \tr[ (\sigma_A \otimes \sigma_B)^{1/2} \, (\sigma_A \otimes \sigma_B)^{-1/2} \, \sigma_{AB} \, (\sigma_A \otimes \sigma_B)^{-1/2} \, (\sigma_A \otimes \sigma_B)^{1/2} \, \Omega_{AB} ] \\ & \phantom{asdasddas} - \tr[ (\sigma_A \otimes \sigma_B)^{1/2} \, \Omega_{AB} \, (\sigma_A \otimes \sigma_B)^{1/2} ] \\ & \phantom{asdasdd} = \left\langle \Omega_{AB} , (\sigma_A \otimes \sigma_B)^{-1/2} \, \sigma_{AB} \, (\sigma_A \otimes \sigma_B)^{-1/2} \right\rangle_{\sigma_A \otimes \sigma_B} -\left\langle \Omega_{AB} , {\ensuremath{\mathds{1}}}_{AB} \right\rangle_{\sigma_A \otimes \sigma_B}\\ & \phantom{asdasdd} = \left\langle \Omega_{AB} , (\sigma_A \otimes \sigma_B)^{-1/2} \, \sigma_{AB} \, (\sigma_A \otimes \sigma_B)^{-1/2} -{\ensuremath{\mathds{1}}}_{AB} \right\rangle_{\sigma_A \otimes \sigma_B}. \end{aligned}$$ Finally, using again the property of duality for the norms of ${\mathbb{L}}^1(\sigma_A \otimes \sigma_B)$ and $ {\mathbb{L}}^\infty(\sigma_A \otimes \sigma_B)$, we have: $$\begin{aligned} & \un{\norm{\Omega_{AB}}_{L^1(\sigma_A \otimes \sigma_B)} \leq 1}{\text{sup}} \tr[\Omega_{AB} \left( \sigma_{AB} - \sigma_A \otimes \sigma_B \right) ] \\ & \phantom{asdasdad} = \un{\norm{\Omega_{AB}}_{L^1(\sigma_A \otimes \sigma_B)} \leq 1}{\text{sup}} \left\langle \Omega_{AB} , (\sigma_A \otimes \sigma_B)^{-1/2} \, \sigma_{AB} \, (\sigma_A \otimes \sigma_B)^{-1/2} - {\ensuremath{\mathds{1}}}_{AB} \right\rangle_{\sigma_A \otimes \sigma_B} \\ & \phantom{asdasdad} = \norm{\sigma_A^{-1/2} \otimes \sigma_B^{-1/2} \, \sigma_{AB} \, \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} - {\ensuremath{\mathds{1}}}_{AB}}_{{\mathbb{L}}^{\infty}(\sigma_A \otimes \sigma_B)} \\ & \phantom{asdasdad}= \norm{\sigma_A^{-1/2} \otimes \sigma_B^{-1/2} \, \sigma_{AB} \, \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} - {\ensuremath{\mathds{1}}}_{AB}}_{\infty},\end{aligned}$$ where we have used again the fact that $\norm{\cdot}_{{\mathbb{L}}^\infty (\rho_{AB})}$ coincides with the usual $\infty$-norm for every $\rho_{AB} \in {\mathcal{S}}_{AB}$. In conclusion, ${\displaystyle}\norm{\mathcal{T}_{\sigma_{A} \otimes \sigma_{B}} \left(\sigma_{AB} \right) - {\ensuremath{\mathds{1}}}_{AB}}_{\infty} \leq \norm{\sigma_A^{-1/2} \otimes \sigma_B^{-1/2} \, \sigma_{AB} \, \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} - {\ensuremath{\mathds{1}}}_{AB}}_{\infty}. $ By putting together Step \[step:1\], Step \[step:2\], Step \[step:3\] and Step \[step:4\], we conclude the proof of Theorem \[thm:quasifactorizationAB\]. This result constitutes an extension of the superadditivity property, i.e., the constant $H(\sigma_{AB})$ that appears in the statement of the main theorem is $0$ when $\sigma_{AB}=\sigma_A \otimes \sigma_B$ and is small whenever $\sigma_{AB} \sim \sigma_A \otimes \sigma_B$. A trivial upper bound can be found with respect to the trace distance as follows, $$\begin{aligned} & \norm{\sigma_A^{-1/2} \otimes \sigma_B^{-1/2} \, \sigma_{AB} \, \sigma_A^{-1/2} \otimes \sigma_B^{-1/2} - {\ensuremath{\mathds{1}}}_{AB}}_{\infty} = \\ & \phantom{asdasdaad} = \norm{\sigma_A^{-1/2} \otimes \sigma_B^{-1/2} ( \sigma_{AB} - \sigma_A \otimes \sigma_B) \sigma_A^{-1/2} \otimes \sigma_B^{-1/2}}_\infty \\ & \phantom{asdasdaad} \leq \norm{\sigma_A^{-1/2} \otimes \sigma_B^{-1/2} ( \sigma_{AB} - \sigma_A \otimes \sigma_B) \sigma_A^{-1/2} \otimes \sigma_B^{-1/2}}_1 \\ & \phantom{asdasdaad} \leq \norm{\sigma_A^{-1/2} \otimes \sigma_B^{-1/2} }_\infty \norm{\sigma_{AB} - \sigma_A \otimes \sigma_B}_1 \norm{\sigma_A^{-1/2} \otimes \sigma_B^{-1/2} }_\infty \\ & \phantom{asdasdaad} \leq \sigma_{\text{min}}^{-2} \, \norm{\sigma_{AB} - \sigma_A \otimes \sigma_B}_1 .\end{aligned}$$ The term $\norm{H(\sigma_{AB})}_\infty$ is also closely related to certain forms of *decay of correlations* of states that have already appeared in quantum many body systems, such as *LTQO (Local Topological Quantum Order)* [@spiros], or the concept of *local indistinguishability* as a strengthened form of *weak clustering* in [@kast-brand]. Let us suppose that $\norm{H(\sigma_{AB})}_\infty \leq \lambda(\ell)$ for a certain small scalar $\lambda(\ell)$ that decays sufficiently fast as a function of the distance $\ell$ between regions $A$ and $B$ in a many body system, and denote by $\left\langle f \right\rangle_\varphi$ the expected value of an observable $f \in {\mathcal{A}}_{AB} $ with respect to a state $\varphi$ (usually the ground or thermal state of the system). Then, for every observable of the form $O_A \otimes O_B \geq 0$, if the reduced density matrix on $AB$ of $\varphi$ is $\sigma_{AB}$, the previous condition can be rewritten as ${\displaystyle}\abs{\left\langle O_A O_B \right\rangle_\varphi - \left\langle O_A \right\rangle_\varphi \left\langle O_B \right\rangle_\varphi } \leq \lambda \left\langle O_A \right\rangle_\varphi \left\langle O_B \right\rangle_\varphi $. One can now compare this expression with the definition of decay of correlations ${\displaystyle}\abs{\left\langle O_A O_B \right\rangle_\varphi - \left\langle O_A \right\rangle_\varphi \left\langle O_B \right\rangle_\varphi } \leq \lambda(\ell) \norm{O_A}_\infty \norm{O_B}_\infty$, or LTQO ${\displaystyle}\abs{\left\langle O_A O_B \right\rangle_\varphi - \left\langle O_A \right\rangle_\varphi \left\langle O_B \right\rangle_\varphi } \leq \lambda(\ell) \left\langle O_A \right\rangle_\varphi \norm{O_B}_\infty$. Conclusion ========== In this work, we have proven an extension of the property of superadditivity of the quantum relative entropy for general states. Our result constitutes an improvement to the usual lower bound for the relative entropy of two bipartite states, given by the property of monotonicity, in terms of the relative entropies in the two constituent spaces, whenever the second state is near to be a tensor product. Therefore, it might be relevant for situations where this property is expected to hold, such as quantum many body systems, in which it is likely that the Gibbs state satisfies this property in spatially separated systems. In [@kast-brand], Kastoryano and Brandao proved, for certain Gibbs samplers, the existence of a positive spectral gap for the dissipative dynamics, via a quasi-factorization result of the variance. This provides a bound for the mixing time of the evolution of the semigroup that drives the system to thermalization which is polynomial in the system size. We leave for future work the possibility of using the result of the present paper to obtain a quasi-factorization of the relative entropy in quantum many body systems, which could allow us to prove, under some conditions of decay of correlations on the Gibbs state, the existence of a positive log-Sobolev constant, obtaining an exponential improvement in the bound for the mixing time obtained in [@kast-brand]. Acknowledgment {#acknowledgment .unnumbered} ============== We are very grateful to D. Sutter and M. Tomamichel, who detected an error in a previous version of the paper. We also thank M. Junge for fruitful discussions. AC and DPG acknowledge support from MINECO (grant MTM2014-54240-P), from Comunidad de Madrid (grant QUITEMAD+- CM, ref. S2013/ICE-2801), and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648913). AC is partially supported by a La Caixa-Severo Ochoa grant (ICMAT Severo Ochoa project SEV-2011-0087, MINECO). AL acknowledges financial support from the European Research Council (ERC Grant Agreement no 337603), the Danish Council for Independent Research (Sapere Aude) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). This work has been partially supported by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO). 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--- abstract: 'Phase field crystal (PFC) theory is extensively used for modelling the phase behaviour, structure, thermodynamics and other related properties of solids. PFC theory can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Here, we carefully identify all of these approximations and explain the consequences of each. One approximation that is made in standard derivations is to neglect a term of form $\nabla\cdot[n\nabla{{\cal L}}n]$, where $n$ is the scaled density profile and ${{\cal L}}$ is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and that dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in $n$. However, the consequences of doing this are: (i) the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and (ii) other periodic structures, such as stripes, are erroneously predicted to be thermodynamic equilibrium structures. In general, ${{\cal L}}$ consists of a non-local convolution involving the pair direct correlation function. A second approximation sometimes made in deriving PFC theory is to replace ${{\cal L}}$ by a gradient expansion involving derivatives. We show that this leads to the possibility of the density going to zero, with its logarithm going to $-\infty$ whilst being balanced by the fourth derivative of the density going to $+\infty$. This subtle singularity leads to solutions failing to exist above a certain value of the average density. We illustrate all of these conclusions with results for a particularly simple model two-dimensional fluid, the generalised exponential model of index 4 (GEM-4), chosen because a DDFT is known to be accurate for this model. The consequences of the subsequent PFC approximations can then be examined. These include the phase diagram being both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal phase. Thus, although PFC models are very successful as phenomenological models of crystallisation, we find it impossible to derive the PFC as a theory for the (scaled) density distribution when starting from an accurate DDFT, without introducing spurious artefacts. However, we find that making a simple one-mode approximation for the *logarithm* of the density distribution $\log\rho({{\bm x}})$ (rather than for $\rho({{\bm x}})$), is surprisingly accurate. This approach gives a tantalising hint that accurate PFC-type theories may instead be derived as theories for the field $\log\rho({{\bm x}})$, rather than for the density profile itself.' author: - 'Andrew J. Archer' - 'Daniel J. Ratliff' - 'Alastair M. Rucklidge' - Priya Subramanian title: 'Deriving phase field crystal theory from dynamical density functional theory: consequences of the approximations' --- Introduction ============ The phase field crystal (PFC) theory for matter is widely used and has been successfully applied to describe a broad range of phenomena, including, for example, grain boundary dynamics [@Elder2002; @Elder2004], crystal nucleation [@Backofen2010; @Toth2010], crystal growth [@Archer2012], glass formation [@Berry2008a], crack propagation [@Elder2004] and many other properties of condensed matter. For more background and examples of situations to which the PFC theory has been applied, see the excellent review [@Emmerich2012]. The PFC theory was originally proposed, in the spirit of ‘regular’ phase field theory (PFT), as a diffuse-interface theory for the time evolution of an order parameter field [@Elder2002]. The equations of PFT are obtained via symmetry, thermodynamic and other arguments and the result is a theory that is widely used in materials science and other disciplines to model the structure of materials. For more background on PFT see for example Ref. [@Boettinger2002] and references therein. The central and original idea in extending PFT to arrive at PFC theory is to incorporate aspects of the microscopic structure of the material into the model [@Elder2002]. The result is a theory that operates on atomic length scales and diffusive time scales [@Emmerich2012]. By this we mean that PFC theory is a theory for a field that exhibits numerous maxima, each of which is identified as the average location of the atoms (or more generally ‘particles’) in the system. This idea is powerful because, by building into the theory more information about the underlying material structure, it enables the inclusion of much more of the physics coming from particle correlations to be incorporated. Over the years several variants of PFC theory have been developed that are able to describe a range different crystalline (and even quasicrystalline) structures [@Jaatinen2009; @Pisutha-Arnond2013b; @Wu2010; @Barkan2011; @Achim2014; @Subramanian2016; @Jiang2017; @Savitz2018]. Thus, the original PFC [@Elder2002] may be viewed as the simplest partial differential equation model one can conceive of for a conserved order parameter exhibiting peaks arranged with crystalline ordering. It is obtained from a (scaled) free energy ${{\cal F_{\text{$\alpha$}}}}$ that is a functional of the dimensionless order parameter $n$: $${{\cal F_{\text{$\alpha$}}}}[n] = \int \left(\frac{1}{2}n\left((k_s^2+\nabla^2)^2 - r\right)n + \frac{1}{4}n^4\right){{{\rm d}}{{\bm x}}}, \label{eq:PFCFalpha}$$ where $n({{\bm x}},t)$ is a field that depends on position in space ${{\bm x}}$ and on time $t$, and $k_s$ is an inverse length scale that determines the lattice spacing of the crystal. The parameter $r$ defines how near the system is to freezing. The time evolution of the conserved field $n$ is given by the dynamics $$\frac{\partial n}{\partial t} = \nabla^2 \left(\frac{{{\delta{\mkern-1mu}{{\cal F_{\text{$\alpha$}}}}}}}{{{\delta{\mkern-1mu}n}}}\right) = - \nabla^2 \left(rn - (k_s^2+\nabla^2)^2n - n^3\right), \label{eq:PFCalphadynamics}$$ where $\frac{{{\delta{\mkern-1mu}{{\cal F_{\text{$\alpha$}}}}}}}{{{\delta{\mkern-1mu}n}}}$ is the functional derivative of ${{\cal F_{\text{$\alpha$}}}}$ with respect to $n({{\bm x}})$. Given the ingredients in the model, it is therefore no surprise that PFC theory is closely related to the Swift–Hohenberg equation [@Swift1977]: $$\frac{\partial n}{\partial t} = - \frac{{{\delta{\mkern-1mu}{{\cal F_{\text{$\alpha$}}}}}}}{{{\delta{\mkern-1mu}n}}} = rn - (k_s^2+\nabla^2)^2n - n^3, \label{eq:SwiftHohenberg}$$ which is one of the archetypal equations in pattern formation theory. As one can see above, both the Swift–Hohenberg equation and PFC theory can be expressed as a different type of dynamics based on the same free energy functional [@Emmerich2012]. The Swift–Hohenberg equation (\[eq:SwiftHohenberg\]) is based on an underlying dynamics that seeks to minimise the free energy over time, whilst the PFC dynamics (\[eq:PFCalphadynamics\]), which also decreases the free energy over time, in addition enforces a conservation of the average value of the order parameter in the system. Thus, the PFC equation (\[eq:PFCalphadynamics\]) is sometimes referred to as the conserved Swift–Hohenberg equation [@Thiele2013; @Sagui1994; @Knobloch2015; @Matthews2000; @Emmerich2012]. In the years after PFC theory was originally proposed it was realised that it could be derived from classical dynamical density functional theory (DDFT) [@Elder2007; @Teeffelen2009; @Huang2010a; @Emmerich2012; @Archer2012], via a sequence of several different approximations. Below, we say much more on what these approximations are. DDFT is a theory for the time evolution of the ensemble average one-body (number) density profile $\rho({{\bm x}},t)$, for a non-equilibrium system of interacting classical particles. DDFT is based on equilibrium density functional theory (DFT) [@Evans1979a; @Evans1992; @Hansen2013] and for an equilibrium system, DDFT is equivalent to DFT. DDFT was originally developed as a theory for Brownian particles with over-damped stochastic equations of motion [@Marconi1999; @Marconi2000; @Archer2004; @Archer2004a], but it has also been extended to describe the dynamics of under-damped systems and atomic or molecular systems where the particle dynamics is governed by Newton’s equations of motion [@Archer2006; @Archer2009; @Goddard2012; @Goddard2013; @DuranOlivencia2017; @Schmidt2018]. This body of work shows that when such systems are not too far from equilibrium, then the dynamics predicted by the original DDFT is still often correct in the long-time limit where the particle dynamics is dominated by diffusive processes. This is because DDFT corresponds to a dynamics given by the continuity equation $$\frac{\partial \rho}{\partial t} = -\nabla \cdot \mathbf{j}, \label{eq:continuity}$$ where the current $\mathbf{j}\propto-\nabla \mu({{\bm x}},t)$, with $\mu({{\bm x}},t)$ a local (non-equilibrium) chemical potential [@Marconi1999; @Marconi2000; @Archer2004; @Archer2004a]. Eq. (\[eq:continuity\]) is of course expected since the total number of particles in the system $N=\int \rho({{\bm x}},t){{{\rm d}}{{\bm x}}}$ is a conserved quantity. Refs. [@Elder2007; @Teeffelen2009; @Huang2010a; @Emmerich2012; @Archer2012] give various different derivations of the PFC model, starting from DFT and/or DDFT. Here, starting from DDFT, we systematically show how all the various different theories are related and we identify and highlight the significance of each of the approximations that are made in the derivation of PFC theory. We show that there is a particular term of the form $\nabla\cdot[n\nabla{{\cal L}}n]$, where ${{\cal L}}$ is a linear operator, that is almost universally neglected because it is ‘of higher order’ [@Huang2010a], but this term is actually important for stabilizing crystalline structures: its contribution is of the same order as some of the terms that are retained. As we explain in detail, neglecting this term essentially forces one to make the Taylor expansion of the ideal gas logarithmic term in the free energy in order to recover something physically reasonable. We show that neglecting this term, as is done in PFC theory, [and the subsequent replacement of the logarithm by its Taylor series,]{} leads to the spurious appearance in the phase diagram of an extra spinodal and also alters the relative stabilities of the crystal state compared to a stripe phase and also other phases, leading in two dimensions (2D) to the stripe phase becoming the global free energy minimum state for certain parameter values. Essentially, all this behaviour originates because the function $\log(1+n)$ has one root, but when it is replaced by a truncated Taylor expansion, the resulting polynomial generally has two roots. Our arguments also directly apply in three dimensions to explain why lamellar phases occur as equilibrium phases in PFC theory. Recall that most PFC theories predict that as one moves in the phase diagram away from the region where there is coexistence between the liquid and the crystal, moving deeper into the crystalline portion of the phase diagram, such stripe/lamellar phases appear as equilibrium structures and are global minima of the free energy [@Emmerich2012]. Of course, particles with isotropic pair interactions generally never ‘freeze’ to form striped phases, unless they have an unusual and special form for the pair potential between the particles [@Imperio2004; @Imperio2006; @Archer2007]. DDFT, taken together with a reliable approximation for the Helmholtz free energy functional of course does not predict such stripe phases for crystallisation from simple liquids. The linear operator ${{\cal L}}$ has the form of a non-local convolution involving the pair direct correlation function plus another simpler term (see Eq.  below). Another approximation that is often made in deriving PFC theories is to approximate ${{\cal L}}$ by a gradient expansion involving derivatives. We show below that if one makes this approximation whilst simultaneously retaining the logarithmic term from the ideal gas free energy, this results in a theory that still predicts reasonably accurately the freezing transition, but as one increases the average density, moving deeper into the region of the phase diagram where the crystal phase occurs, there is a point where $\rho({{\bm x}})\to0$ at the points in space ${{\bm x}}$ between the density peaks, where the density is a minimum. On increasing the average density beyond this point in the phase diagram, there is no solution to the theory. We analyse in detail this singular behaviour. As $\rho({{\bm x}})\to0$ we have $\log\rho({{\bm x}})\to-\infty$, of course. In the equation for the equilibrium density profile this divergence is initially balanced by the term involving the fourth derivative, $\partial^4\rho/\partial x^4\to+\infty$. However, when the average density in the system is increased beyond the value at which this divergence occurs, we find there is no solution. We illustrate these conclusions by finding the predicted structures and phase diagram for the 2D version of the GEM-4 (Generalised Exponential Model of index 4) [@Mladek2006; @Prestipino2014], chosen because DDFT based on a simple approximation (the so-called random phase approximation (RPA) [@Likos2001]) for the Helmholtz free energy functional can be very accurate for predicting the equilibrium structures formed in this model and also the thermodynamics [@Mladek2007; @Prestipino2014; @Archer2014]. At higher temperatures, the 2D GEM-4 system exhibits just a single fluid phase and at higher densities a single crystal phase. At lower temperatures, where the RPA DDFT is no longer accurate, there is a hexatic phase and multiple crystalline phases as the density is increased [@Prestipino2014]. Here we do not consider this regime, restricting ourselves to the regime where there is just one fluid and one crystal phase, which are predicted accurately by the RPA DDFT. This enables us to investigate the effect of making subsequent approximations to the DDFT, including those made to derive PFC theory. We find that the PFC type theories spuriously predict three additional phases that are in reality not present in the phase diagram (i.e., are not thermodynamically stable). These are (i) a stripe phase, (ii) what we refer to as ‘down hexagons’ (in contrast to the true crystal structure, which we refer to as ‘up hexagons’) and then at even higher densities a melting to form (iii) another uniform liquid phase. We show how the approximations made in deriving the PFC result in these structures being predicted. The final contribution of this paper is to show that there is a very simple and accurate ansatz one can make for the form of the equilibrium crystal density profile in DDFT (and so also for DFT, of course). The ansatz is $\rho({{\bm x}})=\rho_0e^{\phi({{\bm x}})}$, where $\rho_0$ is a constant and the field $\phi({{\bm x}})$ is approximated by a sinusoid of the form $\phi({{\bm x}})\approx\phi_0+\phi_1e^{i{{\bm k}}\cdot{{\bm x}}}+$complex conjugate (in one dimension), plus other similar terms (in higher dimensions), where $\phi_0$ and $\phi_1$ are constants. The results presented here are for the GEM-4 model and show why this approximation is unexpectedly accurate: the approximation is able to replicate almost exactly the numerical solution to the DDFT problem, from small to arbitrarily large amplitude density variations. We expect this ansatz also to be reliable for other systems. This form of one-mode theory gives a hint for future directions to develop accurate PFC-type theories, since using a one-mode approximation in PFC is often fairly accurate. This paper is structured as follows: In Sec. \[sec:2\] we present our systematic step-by-step derivation of PFC, starting from DDFT. After each approximation, we carefully state the model, i.e., we give the corresponding free energy functional and also the expression for the chemical potential, which is a quantity that is a constant at all points in space for equilibrium states. In order to keep track of the different orders in which the approximations can be made, we give each model a name, starting with  for the original PFC model in Eq. (\[eq:PFCalphadynamics\]) above, and with DDFT-0 for the original formulation of DDFT below. The different DDFT approximations result in five different versions, DDFT-1 to DDFT-5. Similarly, we explain the various different approximations that can be made to each of these, leading to a corresponding PFC theory, which we name  to . Note that the criterion we use here for distinguishing between whether we refer to a theory as a DDFT or a PFC is based on whether the free energy which is minimised by the dynamical equations (i.e., the Lyapunov functional) has the logarithmic ideal gas term or not: if it does not have the logarithm, we refer to it as a . [Table \[tab:DDFTvsPFC\] below is there to help the reader navigate the various models and the approximations made in each one.]{} Sec. \[sec:2\] concludes with a summarising discussion. In Sec. \[sec:3\] we present results for the GEM-4 system comparing predictions for the density profiles and thermodynamics of equilibria for two of the different DDFT theories and also two of the PFC theories. In this section we also present the phase diagrams for the GEM-4 system predicted by these different DDFT and PFC theories. By comparing all of these we are able to assess the accuracy of the different theories and the validity of the various approximations. In Sec. \[sec:4\] we discuss the implications of the main two approximations and analyse the singular behaviour displayed by some models. In Sec. \[sec:5\] we introduce the ansatz $\rho({{\bm x}})=\rho_0e^{\phi({{\bm x}})}$ and derive the new one-mode approximation for . We draw our conclusions in Sec. \[sec:6\]. The paper includes two appendices in which we describe the numerical (continuation) methods we use to calculate the density profiles. Derivation of the Phase Field Crystal model from DDFT {#sec:2} ===================================================== In this section we progress from the original formulation of DDFT (which we call DDFT-0) through a series of approximations (DDFT-1, …, DDFT-5), as listed in Table \[tab:DDFTvsPFC\]. Our main starting point is DDFT-2. From this point, there are three main approximations that can be made (or not made): (i) the Ramakrishan–Yussouff (RY) or the random phase approximation (RPA) for the free energy; (ii) the gradient expansion of the convolution term; and (iii) the Taylor expansion of the logarithmic term. Making (or not making) the first two of these approximations results in DDFT-3, DDFT-4 and DDFT-5. Then, making the third approximation from DDFT-2 results in , from DDFT-3 results in , and so on up to . The  model can be rescaled to recover the original version of , , see Eqs.  and . The various models are summarised in Table \[tab:DDFTvsPFC\]. Amongst the models we present below, DDFT-5 is equivalent to the model derived by Huang  [@Huang2010a] and advocated by van Teeffelen  [@Teeffelen2009] (named PFC1 in that paper), and DDFT-3 and  are equivalent to the models named DDFT and PFC2 by van Teeffelen  [@Teeffelen2009]. ---------------- ----- ----- ----- ----- ----- ------------------------------------------------ --------------------------- -------------------------------------- ------------------- Name Dynamics $Q$, $C$, $R$ \[2pt\] Yes N/A Yes Yes Yes (\[eq:PFCalphadynamics\]) (\[eq:PFCFalpha\]) — \[4pt\] DDFT-0 (\[eq:DDFT0dynamics\]) (\[eq:separatedF\]) (\[eq:chempotdefn\]) — \[4pt\] DDFT-1 Yes (\[eq:DDFT1dynamics\]), (\[eq:DDFT1deltaF\]) (\[eq:DDFTF1\]) (\[eq:ChemPot\]), (\[eq:betadFdn1\]) — \[4pt\] DDFT-2 Yes Yes (\[eq:DDFT2dynamics\]) (\[eq:DDFTF2\]) (\[eq:betadFdn2\]) (\[eq:qandc\]) \[4pt\] DDFT-3 Yes N/A Yes (\[eq:DDFT3dynamics\]) (\[eq:DDFTF3\]) (\[eq:betadFdn3\]) \[4pt\] DDFT-4 Yes N/A Yes (\[eq:DDFT4dynamics\]) as (\[eq:DDFTF2\]) as (\[eq:betadFdn2\]) (\[eq:qandc\]) \[4pt\] DDFT-5 Yes N/A Yes Yes (\[eq:DDFT5dynamics\]) (\[eq:DDFTF5\]) (\[eq:betadFdn5\]) \[4pt\] Yes Yes Yes (\[eq:PFCdynamics\]), (\[eq:PFCbetadynamics\]) (\[eq:DDFTFPFCbeta\]) (\[eq:betadFdnbeta\]) (\[eq:PFCqandc\]) \[4pt\] Yes N/A Yes Yes (\[eq:PFCgammadynamics\]) (\[eq:DDFTFPFCgamma\]) (\[eq:betadFdngamma\]) \[4pt\] Yes N/A Yes Yes as (\[eq:PFCbetadynamics\]) as (\[eq:DDFTFPFCbeta\]) as (\[eq:betadFdnbeta\]) (\[eq:PFCqandc\]) \[4pt\] Yes N/A Yes Yes Yes (\[eq:PFCepsilondynamics\]) as (\[eq:DDFTFPFCgamma\]) as (\[eq:betadFdngamma\]) \[4pt\] ---------------- ----- ----- ----- ----- ----- ------------------------------------------------ --------------------------- -------------------------------------- ------------------- Dynamic Density Functional Theory: DDFT-0 ----------------------------------------- The starting point for all of our derivations is the key DDFT equation [@Marconi1999; @Marconi2000; @Archer2004; @Archer2004a]: $$\frac{\partial \rho}{\partial t} = \nabla \cdot \left[ \beta M(\rho) \nabla \frac{{{\delta{\mkern-1mu}{{\cal F}}}}}{{{\delta{\mkern-2mu}\rho}}} \right], \label{eq:DDFT0dynamics}$$ where $\beta=(k_BT)^{-1}$ (with $k_B$ being Boltzmann’s constant and $T$ being temperature), $M(\rho)$ is the positive $\rho$-dependent mobility. The Helmholtz free energy ${{\cal F}}[\rho]$ depends on the density profile $\rho({{\bm x}},t)$ integrated over space; hence ${{\cal F}}[\rho]$ depends on time but not on position [@Archer2004]. The expression ${{{\delta{\mkern-1mu}{{\cal F}}}}}/{{{\delta{\mkern-2mu}\rho}}}$ is the functional derivative of ${{\cal F}}$ with respect to $\rho({{\bm x}},t)$, which therefore depends on both time and on position. DDFT usually takes $M(\rho)=D \rho$, i.e., the mobility is proportional to density [@Marconi1999; @Marconi2000; @Archer2004; @Archer2004a], where $D$ is the diffusion coefficient. We henceforth scale time so that $D=1$. With boundary conditions that do not allow material to enter or leave the system, $N=\int\!\rho({{\bm x}}){{{\rm d}}{{\bm x}}}$ (or equivalently, the mean density) is a constant of the motion and is the total number of particles in the system. With suitable boundary conditions, one can readily show that the Helmholtz free energy decreases monotonically with time: $$\frac{{{\rm d}}{{\cal F}}}{{{{\rm d}}{t}}} = - \int \! \beta M(\rho) \left| \nabla \frac{{{\delta{\mkern-1mu}{{\cal F}}}}}{{{\delta{\mkern-2mu}\rho}}} \right|^2 {{{\rm d}}{{\bm x}}}\leq 0, \label{eq:Fdecreases}$$ so (assuming that ${{\cal F}}[\rho]$ is bounded below) the system typically evolves to a (local) minimum of ${{\cal F}}$, which is a dynamically stable equilibrium of (\[eq:DDFT0dynamics\]). Here, ‘dynamically stable’ means that small perturbations away from the equilibrium decay, and ‘equilibrium’ means that ${\partial \rho}/{\partial t} =0$ and ${{{\rm d}}{{\cal F}}}/{{{{\rm d}}{t}}}=0$. Owing to the dynamics being governed by a continuity equation , such perturbations cannot change the mean density. Local minima of ${{\cal F}}$ that are not the global minimum are thermodynamically metastable. The system can also have dynamically unstable equilibria, for which ${{\cal F}}$ is a saddle or maximum. From (\[eq:Fdecreases\]), we see that all equilibria of (\[eq:DDFT0dynamics\]) satisfy $\nabla({{\delta{\mkern-1mu}{{\cal F}}}}/{{\delta{\mkern-2mu}\rho}})=0$, so $$\frac{{{\delta{\mkern-1mu}{{\cal F}}}}}{{{\delta{\mkern-2mu}\rho}}} = \text{constant} = \mu, \label{eq:chempotdefn}$$ where $\mu$ is the chemical potential of the equilibrium. This is of course the Euler–Lagrange equation for the problem of finding stationary points of the functional ${{\cal F}}[\rho]$, subject to the constraint of fixed mean density. Note however that when evolving (\[eq:DDFT0dynamics\]) forward in time from an arbitrary initial condition, $\mu$ is not necessarily known *a priori*. The theory can also be cast in terms of the grand potential (also called the Landau free energy) functional [@Evans1979a; @Evans1992; @Hansen2013]: $$\Omega[\rho] = {{\cal F}}[\rho] - \mu N = {{\cal F}}[\rho] - \mu \int \! \rho({{\bm x}}){{{\rm d}}{{\bm x}}}. \label{eq:omegadefn}$$ From this it follows that the functional derivative of $\Omega$ is $$\frac{{{\delta{\mkern-1mu}\Omega}}}{{{\delta{\mkern-2mu}\rho}}} = \frac{{{\delta{\mkern-1mu}{{\cal F}}}}}{{{\delta{\mkern-2mu}\rho}}} - \mu, \label{eq:dOmega}$$ and that this is zero at equilibrium: equilibria are extreme values of $\Omega$. Like the Helmholtz free energy, the grand potential decreases monotonically with time, since Eq.  is also true if one replaces ${{\cal F}}$ by $\Omega$. Therefore, for two phases to coexist, they must have the same specific grand potential (i.e., the same pressure) and the same chemical potential. Thus, the global minimum of $\Omega$ for a given $\mu$ and $T$ is the thermodynamic equilibrium state of the system [@Evans1979a; @Evans1992; @Hansen2013]. Following the usual approach in DFT, we separate the Helmholtz free energy into three parts: the ‘ideal gas’ contribution, which is proportional to the temperature but takes no account of particle interactions, an excess (${{\cal F_{\text{ex}}}}$) over the ideal gas contribution arising from the particle interactions, and the contribution due to an external potential ${U_{\text{ext}}}({{\bm x}})$, as follows [@Evans1979a; @Evans1992; @Hansen2013]: $${{\cal F}}[\rho] = k_BT \int \rho\left(\log(\Lambda^d\rho) - 1\right){{{\rm d}}{{\bm x}}}+ {{\cal F_{\text{ex}}}}[\rho] + \int\rho{U_{\text{ext}}}{{{\rm d}}{{\bm x}}}, \label{eq:separatedF}$$ where the integral is taken over the volume $V$ in three dimensions ($d=3$) (or the area in 2D, $d=2$) and where $\Lambda$ is the thermal de Broglie wavelength. Since for our purposes here the value of $\Lambda$ is irrelevant (changing $\Lambda$ will shift the values of ${{\cal F}}$ and $\mu$ by constants), we henceforth set $\Lambda=1$. We also consider bulk systems and so we assume that ${U_{\text{ext}}}=0$. With the separation in Eq. (\[eq:separatedF\]), we have $$\beta\frac{{{\delta{\mkern-1mu}{{\cal F}}}}}{{{\delta{\mkern-2mu}\rho}}} = \log\rho + \beta\frac{{{\delta{\mkern-1mu}{{\cal F_{\text{ex}}}}}}}{{{\delta{\mkern-2mu}\rho}}}, \label{eq:separatedFdrho}$$ which gives $$\beta\nabla\frac{{{\delta{\mkern-1mu}{{\cal F}}}}}{{{\delta{\mkern-2mu}\rho}}}=\frac{1}{\rho}\nabla\rho+\dots, \label{eq:rhoinverse}$$ where on the right hand side we only explicitly write the contribution from the ideal gas part of the free energy. Inserting this into Eq. (\[eq:DDFT0dynamics\]) with $M=D\rho$ we obtain $$\frac{\partial \rho}{\partial t} = \nabla^2 \rho+\dots, \label{eq:rhodiffusion}$$ in which the coefficient in front of the term $\nabla^2 \rho$ is $D$, but our choice of time scaling has $D=1$. Note that this term is *linear* in $\rho$, in spite of it originating from a *nonlinear* logarithmic contribution to the free energy. We refer to the model up to this point as DDFT-0. Expansion of ${{\cal F_{\text{ex}}}}$: DDFT-1 --------------------------------------------- To proceed, we must have an expression for the excess Helmholtz free energy functional ${{\cal F_{\text{ex}}}}[\rho]$. We use a functional Taylor expansion, which is also that used in all derivations of PFC theory. This gives the free energy functional of the system of interest in terms of properties of a reference system, which are assumed to be known. The reference system that is chosen is a uniform liquid, with constant density $\rho_0$. The density profile of the system of interest may be varying in space and with an average density that may be different from $\rho_0$. The functional Taylor series expansion of the excess free energy can be written in terms of the density difference ${{\Delta{\mkern-1.0mu}\rho}}({{\bm x}},t)=\rho({{\bm x}},t)-\rho_0$ as follows [@Evans1992; @Hansen2013]: $$\begin{split} {{\cal F_{\text{ex}}}}[\rho] &= {{\cal F_{\text{ex}}}}[\rho_0] - k_BT \!\! \int \! c^{(1)}({{\bm x}}_1){{\Delta{\mkern-1.0mu}\rho}}({{\bm x}}_1){{{\rm d}}{{\bm x}}}_1 \\ & \quad{}-\frac{k_BT}{2!} \!\! \int \! c^{(2)}({{\bm x}}_1,{{\bm x}}_2) {{\Delta{\mkern-1.0mu}\rho}}({{\bm x}}_1) {{\Delta{\mkern-1.0mu}\rho}}({{\bm x}}_2) {{{\rm d}}{{\bm x}}}_1{{{\rm d}}{{\bm x}}}_2 \\ & \quad{}-\frac{k_BT}{3!} \!\! \int \! c^{(3)}({{\bm x}}_1,{{\bm x}}_2,{{\bm x}}_3) \times{}\\ &\qquad\qquad\qquad {{\Delta{\mkern-1.0mu}\rho}}({{\bm x}}_1) {{\Delta{\mkern-1.0mu}\rho}}({{\bm x}}_2) {{\Delta{\mkern-1.0mu}\rho}}({{\bm x}}_3) {{{\rm d}}{{\bm x}}}_1{{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3\\ & \quad{} + \text{similar fourth order term} + \dots. \end{split} \label{eq:expandFex}$$ The expressions $c^{(n)}$ in the equation above are proportional to the first and higher functional derivatives of ${{\cal F_{\text{ex}}}}$ with respect to density, all evaluated at $\rho=\rho_0$: $$c^{(n)}({{\bm x}}_1,\dots,{{\bm x}}_n) = -\beta \frac{\delta^n\!{{\cal F_{\text{ex}}}}}{{{\delta{\mkern-2mu}\rho}}({{\bm x}}_1)\dots{{\delta{\mkern-2mu}\rho}}({{\bm x}}_n)}[\rho_0]. \label{eq:defncn}$$ These functions $c^{(n)}$ are known as [*direct correlation functions*]{} [@Evans1979a; @Evans1992; @Hansen2013], and are related to $n$-point density correlation functions. In the two-point case, $c^{(2)}$ is the pair direct correlation function and is related to the pair correlation function (i.e., the radial distribution function) through the Ornstein–Zernike equation [@Evans1979a; @Evans1992; @Hansen2013]. These direct correlation functions depend on our choice of $\rho_0$ and depend directly on temperature through the linear factor of $\beta$ in the definition and also indirectly via the fact that the correlations in a liquid change with temperature. Note also that $c^{(1)}[\rho_0]$ is a constant when $\rho_0$ is a constant. For a homogeneous liquid with distant (or periodic) boundaries, these functions depend on displacements but not on absolute position, so (through a slight abuse of notation) we also write $$c^{(n)}({{\bm x}}_1,\dots,{{\bm x}}_n) = c^{(n)}({{\Delta{\mkern-1.0mu}{{\bm x}}}}_2,{{\Delta{\mkern-1.0mu}{{\bm x}}}}_3\dots,{{\Delta{\mkern-1.0mu}{{\bm x}}}}_n), \label{eq:shiftorigin}$$ where ${{\Delta{\mkern-1.0mu}{{\bm x}}}}_j={{\bm x}}_j-{{\bm x}}_1$ [@Hansen2013]. We also take the liquid to be isotropic. We are considering density perturbations away from the liquid state, so it is convenient to write $$\label{eq:rho_sub} \rho({{\bm x}},t)=\rho_0 (1+n({{\bm x}},t)).$$ We do not assume that $n$ is small, but it is often the case that the average of $n({{\bm x}},t)$ over the whole system is small. Note also that $\rho({{\bm x}},t)=\rho_0$ is a stationary solution of (\[eq:DDFT0dynamics\]). Substituting Eq. (\[eq:expandFex\]) into Eq. (\[eq:DDFT0dynamics\]) and writing only the terms up to $c^{(1)}$, we get: $$\frac{\partial n}{\partial t} = \nabla^2 n - \nabla^2 c^{(1)} - \nabla \cdot \left[ n \nabla c^{(1)} \right] + \dots \\ \label{eq:DDFT1dynamicsc1}$$ That the uniform liquid state is an equilibrium of (\[eq:DDFT0dynamics\]) implies that $n=0$ is a solution of equation (\[eq:DDFT1dynamicsc1\]): all terms not written down involve ${{\Delta{\mkern-1.0mu}\rho}}$ and so are zero for the uniform liquid with density $\rho_0$. Recall that $c^{(1)}[\rho_0]$ is a constant, which means terms involving gradients of this can be dropped. [Whilst this constant term does not influence the structure (density profile) both in and out of equilibrium, it does affect the thermodynamics (i.e., free energy value) and so also mechanical properties [@Wang2018b].]{} With this, we can write the equation for the time evolution of $n({{\bm x}},t)$ (up to ${{\cal O}}(c^{(4)})$) as: $$\begin{split} \frac{\partial n}{\partial t} =& \nabla^2 n - \rho_0 \nabla^2 \!\!\int\! c^{(2)}({{\bm x}},{{\bm x}}_2)n({{\bm x}}_2){{{\rm d}}{{\bm x}}}_2 \\ &{}- \rho_0 \nabla \cdot \left[ n \nabla \!\!\int\! c^{(2)}({{\bm x}},{{\bm x}}_2)n({{\bm x}}_2){{{\rm d}}{{\bm x}}}_2 \right]\\ &{}- \frac{\rho_0^2}{2} \nabla^2 \!\!\int\! c^{(3)}({{\bm x}},{{\bm x}}_2,{{\bm x}}_3)n({{\bm x}}_2)n({{\bm x}}_3){{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3\\ &{}- \frac{\rho_0^2}{2} \nabla \cdot \left[ n \nabla \!\!\int\! c^{(3)}({{\bm x}},{{\bm x}}_2,{{\bm x}}_3)n({{\bm x}}_2)n({{\bm x}}_3){{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3 \right]\\ &{}- \frac{\rho_0^3}{6} \nabla^2 \!\!\int\! c^{(4)}({{\bm x}},{{\bm x}}_2,{{\bm x}}_3,{{\bm x}}_4) \times{}\\ & \qquad\qquad\qquad n({{\bm x}}_2)n({{\bm x}}_3)n({{\bm x}}_4){{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3{{{\rm d}}{{\bm x}}}_4 \\ &{}- \frac{\rho_0^3}{6} \nabla \cdot \bigg[ n \nabla \!\!\int\! c^{(4)}({{\bm x}},{{\bm x}}_2,{{\bm x}}_3,{{\bm x}}_4) \times{}\\ & \qquad\qquad\qquad n({{\bm x}}_2)n({{\bm x}}_3)n({{\bm x}}_4){{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3{{{\rm d}}{{\bm x}}}_4 \bigg] + \dots\\ \end{split} \label{eq:DDFT1dynamicspretruncate}$$ where we have suppressed writing the time dependence of $n$ throughout and the ${{\bm x}}$ dependence of $n$ when it is not inside an integral. We have written this equation so that the first line is linear in $n$, the next two lines are quadratic in $n$, the fourth and fifth lines are cubic in $n$, and the last line is quartic in $n$. Since the first line is linear in $n$, and both terms involve a Laplacian, we can write the linearised version of (\[eq:DDFT1dynamicspretruncate\]) in terms of the negative Laplacian of a linear operator ${{\cal L}}$: $$\frac{\partial n}{\partial t} = - \nabla^2 {{\cal L}}n, \label{eq:DDFTlinear}$$ where $${{\cal L}}n({{\bm x}}) = - n({{\bm x}}) + \rho_0 \!\!\int\! c^{(2)}({{\bm x}},{{\bm x}}_2)n({{\bm x}}_2){{{\rm d}}{{\bm x}}}_2. \label{eq:defnL}$$ The non-local operator ${{\cal L}}$ is most conveniently considered in terms of its Fourier transform, or equivalently, in terms of how it acts on modes of the form $\exp(i{{\bm k}}\cdot{{\bm x}})$. If $${{\cal L}}e^{i{{\bm k}}\cdot{{\bm x}}} = \sigma({{\bm k}}) e^{i{{\bm k}}\cdot{{\bm x}}}, \label{eq:defnsigma}$$ then $\sigma({{\bm k}})$ is the eigenvalue of ${{\cal L}}$ with eigenfunction $\exp(i{{\bm k}}\cdot{{\bm x}})$. With this, the linear equation (\[eq:DDFTlinear\]) can readily be solved in terms of linear combinations of functions like $\exp\left({k^2\sigma({{\bm k}}) t + i{{\bm k}}\cdot{{\bm x}}}\right)$, where $k^2\sigma({{\bm k}})$ is the growth rate for a mode with wavevector ${{\bm k}}$, and $k=|{{\bm k}}|$. If $\sigma({{\bm k}})$ is negative for all ${{\bm k}}$, all small amplitude density modulations decay to zero, and the liquid state is dynamically stable. ![Illustrative example of the growth rate $k^2\sigma(k)$ as a function of wavenumber $k$. Small amplitude modes with $k^2\sigma(k)<0$ decay exponentially in time, while those with $k^2\sigma(k)>0$ grow exponentially. Throughout we scale lengths so that the maximum growth rate occurs at $k=1$.[]{data-label="fig:growthrate"}](fig_growthrate) Recall that for a bulk liquid, $c^{(2)}({{\bm x}},{{\bm x}}_2)=c^{(2)}({{\Delta{\mkern-1.0mu}{{\bm x}}}}_2)$, with ${{\Delta{\mkern-1.0mu}{{\bm x}}}}_2={{\bm x}}_2-{{\bm x}}$, and for spherically symmetric (isotropic) particles, $c^{(2)}({{\bm x}},{{\bm x}}_2)=c^{(2)}(|{{\Delta{\mkern-1.0mu}{{\bm x}}}}_2|)$. Therefore, in this case $\sigma({{\bm k}})=\sigma(k)$, i.e., $\sigma$ depends only on the wavenumber $k=|{{\bm k}}|$. The eigenvalue $\sigma(k)$ can be expressed as: $$\begin{split} \sigma(k) &= - 1 + \rho_0 \!\!\int\! c^{(2)}(|{{\bm x}}_2-{{\bm x}}|) e^{i{{\bm k}}\cdot({{\bm x}}_2-{{\bm x}})}{{{\rm d}}{{\bm x}}}_2\\ &= -1 + \rho_0 {\hat c}^{(2)}(k), \label{eq:sigmactwo} \end{split}$$ where ${\hat c}^{(2)}$ is the Fourier transform of $c^{(2)}$. Recall from (\[eq:defncn\]) that $c^{(2)}$ is proportional to $\beta$, so if ${\hat c}^{(2)}$ has any positive Fourier components, decreasing the temperature (increasing $\beta$) can be expected to lead to a range of wavenumbers with positive growth rates, and the liquid being dynamically unstable to modes with wavenumbers centered around the maximum of $k^2\sigma(k)$, see Fig. \[fig:growthrate\]. We have scaled lengths so that the maximum growth rate occurs at wavenumber $k=1$. This argument, of course, assumes that the product $\beta^{-1}c^{(2)}$ is independent of temperature. This is not true in general, but for some systems it is a good approximation (at least over a limited range of temperatures) – see Ref. [@Somerville2018] for a recent discussion on this for a particular colloidal system. Recall too that for an equilibrium liquid the static structure factor $S(k)=[1-\rho_0{\hat c}^{(2)}(k)]^{-1}$. $S(k)$ is proportional to the Fourier transform of the radial distribution function [@Hansen2013]. So, for the stable uniform liquid, we have $\sigma(k)=-1/S(k)$. We refer to the state point at which the uniform liquid becomes linearly unstable to density modulations with wavenumber $k\neq0$ as the *spinodal point*, in keeping with the terminology of [@Trudu2006]. The more common usage of the term ‘spinodal’ relates to the onset of the zero-wavenumber phase separation instability of liquid–liquid or gas–liquid phase separation [@Hansen2013; @Archer2004]. At the spinodal point, the density and temperature are such that the liquid is dynamically marginally stable, that is, the maximum of $k^2\sigma(k)$ is zero. Therefore, at higher temperatures, small amplitude density modulations decay, and at lower temperatures, small amplitude density modulations grow. For a given fixed value of $\rho_0$, the spinodal temperature is $T_s$, with a corresponding $\beta_s=(k_BT_s)^{-1}$. Similarly, either increasing the density $\rho_0$ of the liquid or increasing the chemical potential $\mu$ can also lead to crossing the spinodal. With (\[eq:defnL\]), we can eliminate $c^{(2)}$ in favour of ${{\cal L}}$ in (\[eq:DDFT1dynamicspretruncate\]), and obtain (truncating at ${{{\cal O}}}(c^{(4)})$): $$\begin{split} \frac{\partial n}{\partial t} =& - \nabla^2 \left({{\cal L}}n + \tfrac{1}{2}n^2\right) - \nabla \cdot \left[ n \nabla {{\cal L}}n \right]\\ &{}- \frac{\rho_0^2}{2} \nabla^2 \!\!\int\! c^{(3)}({{\bm x}},{{\bm x}}_2,{{\bm x}}_3)n({{\bm x}}_2)n({{\bm x}}_3){{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3\\ &{}- \frac{\rho_0^2}{2} \nabla \cdot \left[ n \nabla \!\!\int\! c^{(3)}({{\bm x}},{{\bm x}}_2,{{\bm x}}_3)n({{\bm x}}_2)n({{\bm x}}_3){{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3 \right]\\ &{}- \frac{\rho_0^3}{6} \nabla^2 \!\!\int\! c^{(4)}({{\bm x}},{{\bm x}}_2,{{\bm x}}_3,{{\bm x}}_4)\times{}\\ &\qquad\qquad\qquad n({{\bm x}}_2)n({{\bm x}}_3)n({{\bm x}}_4){{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3{{{\rm d}}{{\bm x}}}_4\\ &{}- \frac{\rho_0^3}{6} \nabla \cdot \bigg[ n \nabla \!\!\int\! c^{(4)}({{\bm x}},{{\bm x}}_2,{{\bm x}}_3,{{\bm x}}_4) \times{}\\ & \qquad\qquad\qquad n({{\bm x}}_2)n({{\bm x}}_3)n({{\bm x}}_4){{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3{{{\rm d}}{{\bm x}}}_4 \bigg]\\\ \end{split} \label{eq:DDFT1dynamics}$$ where we have used the result $\nabla \cdot \left[n \nabla n\right] = \frac{1}{2}\nabla^2n^2$. [For an ideal gas, with ${{\cal L}}n=-n$ and $c^{(2)}=c^{(3)}=c^{(4)}=0$, the first line of the equation above reduces to the diffusion equation, $\frac{\partial n}{\partial t}=\nabla^2n$, similar to (\[eq:rhodiffusion\]).]{} At this point, we have made no approximations beyond expanding the free energy in Eq. (\[eq:expandFex\]) and truncating at ${{{\cal O}}}(c^{(4)})$. We refer to the model at this stage, truncated in this way, as . In the new variables, and incorporating $c^{(2)}$ into ${{\cal L}}$, the Helmholtz free energy ${{\cal F}}$ can be expressed (up to fourth order) in terms of a scaled free energy ${{{\cal F}_{1}}}={{\cal F}}/\rho_0$, where $$\begin{split} \beta{{{\cal F}_{1}}}[n] &= \int \! \big([1+n({{\bm x}}_1)] \log [1+n({{\bm x}}_1)] - n({{\bm x}}_1)\big){{{\rm d}}{{\bm x}}}_1\\ & \quad{}- \frac{1}{2} \! \int \! \left(n^2({{\bm x}}_1) + n({{\bm x}}_1){{\cal L}}n({{\bm x}}_1) \right){{{\rm d}}{{\bm x}}}_1 \\ & \quad{}-\frac{\rho_0^2}{6} \!\! \int \! c^{(3)}({{\bm x}}_1,{{\bm x}}_2,{{\bm x}}_3) \times{}\\ &\qquad\qquad\quad n ({{\bm x}}_1) n ({{\bm x}}_2) n ({{\bm x}}_3) {{{\rm d}}{{\bm x}}}_1{{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3\\ & \quad{}-\frac{\rho_0^3}{24} \! \int \! c^{(4)}({{\bm x}}_1,{{\bm x}}_2,{{\bm x}}_3,{{\bm x}}_4) \times{}\\ &\qquad\qquad\quad n ({{\bm x}}_1) n ({{\bm x}}_2) n ({{\bm x}}_3) n ({{\bm x}}_4) {{{\rm d}}{{\bm x}}}_1{{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3{{{\rm d}}{{\bm x}}}_4, \end{split} \label{eq:DDFTF1}$$ and where we have also dropped terms that do not contribute to (\[eq:DDFT1dynamics\]). In these variables, the DDFT that leads to the dynamics (\[eq:DDFT1dynamics\]) is $$\frac{\partial n}{\partial t} = \nabla \cdot \left[ \beta (1+n) \nabla \frac{{{\delta{\mkern-1mu}{{{\cal F}_{1}}}}}}{{{\delta{\mkern-1mu}n}}} \right]. \label{eq:DDFT1deltaF}$$ Note that, because of the $\log(1+n)$ term in (\[eq:DDFTF1\]), $n$ is constrained so that $1+n$ is always non-negative. Also, because of Eq. , we have $$\frac{{{\delta{\mkern-1mu}{{\cal F}}}}}{\delta\rho}=\frac{{{\delta{\mkern-1mu}{{{\cal F}_{1}}}}}}{{{\delta{\mkern-1mu}n}}}. \label{eq:derivs_eq}$$ Moreover, states that satisfy $$\frac{{{\delta{\mkern-1mu}{{{\cal F}_{1}}}}}}{{{\delta{\mkern-1mu}n}}} = {{\Delta{\mkern-0.5mu}\mu}}, \label{eq:ChemPot}$$ where ${{\Delta{\mkern-0.5mu}\mu}}=\mu-\mu_0$ and where \[see (\[eq:chempotdefn\]), (\[eq:separatedF\]) and (\[eq:expandFex\])\] $$\mu_0=k_BT\log\Lambda^d\rho_0-k_BTc^{(1)}[\rho_0], \label{eq:mu_rho_0}$$ are equilibrium solutions of (\[eq:DDFT1dynamics\]), or equivalently, extrema of ${{{\cal F}_{1}}}$. Henceforth, we redefine $\mu$ to be $\beta{{\Delta{\mkern-0.5mu}\mu}}/\rho_0$, which is a shifted and rescaled chemical potential. For the free energy in (\[eq:DDFTF1\]), we have $$\begin{split} \beta\frac{{{\delta{\mkern-1mu}{{{\cal F}_{1}}}}}}{{{\delta{\mkern-1mu}n}}} &= \log\left(1+n({{\bm x}})\right) - n({{\bm x}}) - {{\cal L}}n({{\bm x}}) \\ & \quad{}-\frac{\rho_0^2}{2} \!\! \int \! c^{(3)}({{\bm x}},{{\bm x}}_2,{{\bm x}}_3) n ({{\bm x}}_2) n ({{\bm x}}_3) {{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3\\ & \quad{}-\frac{\rho_0^3}{6} \! \int \! c^{(4)}({{\bm x}},{{\bm x}}_2,{{\bm x}}_3,{{\bm x}}_4) \times{}\\ &\qquad\qquad\quad n ({{\bm x}}_2) n ({{\bm x}}_3) n ({{\bm x}}_4) {{{\rm d}}{{\bm x}}}_2{{{\rm d}}{{\bm x}}}_3{{{\rm d}}{{\bm x}}}_4. \end{split} \label{eq:betadFdn1}$$ At equilibrium, this expression (the rescaled chemical potential $\mu$) does not vary in space. The reference liquid $n=0$ has ${{{\cal F}_{1}}}=0$ and $\mu=0$. The zero value for ${{{\cal F}_{1}}}$ arises (in part) from dropping ${{\cal F_{\text{ex}}}}[\rho_0]$ from (\[eq:expandFex\]), while the zero value for the rescaled chemical potential is a consequence of , which is equivalent to dropping $c^{(1)}$ from (\[eq:expandFex\]) and setting $\Lambda=1$ in . Simplification of $c^{(3)}$ and $c^{(4)}$: DDFT-2 ------------------------------------------------- As the next step, Huang  [@Huang2010a] kept only the zero-wavenumber components of $c^{(3)}$ and $c^{(4)}$, or equivalently, they took $$\begin{split} c^{(3)}({{\bm x}},{{\bm x}}_2,{{\bm x}}_3)&=c^{(3)}_0 \delta({{\bm x}}-{{\bm x}}_2)\delta({{\bm x}}-{{\bm x}}_3),\\ c^{(4)}({{\bm x}},{{\bm x}}_2,{{\bm x}}_3,{{\bm x}}_4)&=c^{(4)}_0 \delta({{\bm x}}-{{\bm x}}_2)\delta({{\bm x}}-{{\bm x}}_3)\delta({{\bm x}}-{{\bm x}}_4), \label{eq:definecs} \end{split}$$ where $c^{(3)}_0$ and $c^{(4)}_0$ are constants (our sign convention is opposite to that of [@Huang2010a]). This is equivalent to making a local density approximation (LDA) [@Evans1992] for these terms in the free energy. We could in principle include terms involving $c^{(5)}$ and higher as well, treated in the same way: these would contribute a more general function of $n$ in the free energy, treated with the . However, since we are investigating the effect of approximations that have not yet been discussed, we keep as simple a free energy as possible at this point, consistent with truncating at ${{\cal O}}(c^{(4)})$. With this, the free energy in (\[eq:DDFTF1\]) becomes $$\begin{split} \beta{{{\cal F}_{2}}}[n] &= \int \! \big([1+n({{\bm x}})] \log [1+n({{\bm x}})] - n({{\bm x}})\big){{{\rm d}}{{\bm x}}}\\ & \quad{}+ \! \int \! \bigg( - \frac{1}{2} \left(n^2({{\bm x}}) + n{{\cal L}}n({{\bm x}}) \right) \\ & \qquad\qquad{}-\frac{\rho_0^2}{6}c^{(3)}_0 n^3 ({{\bm x}}) -\frac{\rho_0^3}{24}c^{(4)}_0 n^4 ({{\bm x}}) \bigg){{{\rm d}}{{\bm x}}},\\ \end{split} \label{eq:DDFTF2}$$ and the four terms involving $c^{(3)}$ and $c^{(4)}$ in (\[eq:DDFT1dynamics\]) become $$\begin{split} &- \frac{\rho_0^2}{2}c^{(3)}_0 \nabla^2 n^2,\quad - \frac{\rho_0^2}{2}c^{(3)}_0 \nabla \cdot \left[ n \nabla n^2 \right],\\ &- \frac{\rho_0^3}{6}c^{(4)}_0 \nabla^2 n^3 \quad\text{and}\quad - \frac{\rho_0^3}{6}c^{(4)}_0 \nabla \cdot \left[ n \nabla n^3 \right]. \end{split} \label{eq:cthreedelta}$$ Using $\nabla \cdot \left[ n \nabla n^2\right]=\frac{2}{3}\nabla^2n^3$ and $\nabla \cdot \left[ n \nabla n^3\right]=\frac{3}{4}\nabla^2n^4$, Huang  [@Huang2010a] combined (\[eq:cthreedelta\]) and (\[eq:DDFT1dynamics\]) to get $$\frac{\partial n}{\partial t} = - \nabla^2 \left({{\cal L}}n + Qn^2 + Cn^3 + Rn^4\right) - \nabla \cdot \left[ n \nabla {{\cal L}}n \right] \label{eq:DDFT2dynamics}$$ where $$Q = \frac{1}{2} + \frac{\rho_0^2}{2}c^{(3)}_0 ,\quad C = \frac{\rho_0^2}{3}c^{(3)}_0 + \frac{\rho_0^3}{6}c^{(4)}_0 \quad\text{and}\quad R = \frac{\rho_0^3}{8}c^{(4)}_0. \label{eq:qandc}$$ We also have a chemical potential $$\begin{split} \mu=\beta\frac{{{\delta{\mkern-1mu}{{{\cal F}_{2}}}}}}{{{\delta{\mkern-1mu}n}}} &= \log\left(1+n({{\bm x}})\right) - n({{\bm x}}) - {{\cal L}}n({{\bm x}})\\ & \quad\quad{}- \frac{\rho_0^2}{2} c^{(3)}_0 n^2 ({{\bm x}}) - \frac{\rho_0^3}{6} c^{(4)}_0 n^3 ({{\bm x}}), \end{split} \label{eq:betadFdn2}$$ which does not vary in space at equilibrium. Up to this point, we refer to the model as . Here, we retain the $n^4$ term (as did Huang  [@Huang2010a]), because otherwise the dynamics in (\[eq:DDFT2dynamics\]) would not be consistent with the free energy (\[eq:DDFTF2\]) and the DDFT dynamics (\[eq:DDFT1deltaF\]) (with ${{\cal F}}_2$ instead of ${{\cal F}}_1$). The next three models involve making (or not making) two approximations: (i) assuming the Ramakrishan–Yussouff or random phase approximation, which leads to a quadratic excess Helmholtz free energy functional, and (ii) making a gradient expansion of the linear operator ${{\cal L}}$. Quadratic excess free energy: DDFT-3 ------------------------------------ Often, the free energy functional in (\[eq:expandFex\]) is truncated at ${{\cal O}}({{\Delta{\mkern-1.0mu}\rho}}^2)$. This is known as the Ramakrishan–Yussouff (RY) approximation [@Ramakrishnan1979; @Teeffelen2009; @Emmerich2012], which effectively sets $c^{(3)}=c^{(4)}=0$. A mathematically equivalent approximation arises in the treatment of soft purely repulsive particles modelling soft matter, namely the RPA [@Likos2001]. Here, two soft isotropic particles at ${{\bm x}}_1$ and ${{\bm x}}_2$ separated by a distance $x_{12}=|{{\bm x}}_1-{{\bm x}}_2|$ interact through a potential energy $u(x_{12})$, which depends only on the magnitude of the distance and is finite for all values of $x_{12}$. The excess free energy \[c.f. Eq. (\[eq:expandFex\])\] is then $${{\cal F_{\text{ex}}}}[\rho] = \frac{1}{2} \! \int \! u(|{{\bm x}}_1-{{\bm x}}_2|) \rho({{\bm x}}_1) \rho({{\bm x}}_2) {{{\rm d}}{{\bm x}}}_1{{{\rm d}}{{\bm x}}}_2. \label{eq:RPAfreeenergy}$$ This amounts to setting $c^{(3)}=c^{(4)}=0$ and $$c^{(2)}({{\bm x}}_1,{{\bm x}}_2) = - \beta u(|{{\bm x}}_1-{{\bm x}}_2|) \label{eq:RPAc2u}$$ in . The eigenvalues $\sigma(k)$ can thus be related to the Fourier transform of $u$ through (\[eq:sigmactwo\]) [@Likos2001; @Archer2012]: $$\begin{split} \sigma(k) &= - 1 - \rho_0 \beta \!\!\int\! u(|{{\bm x}}-{{\bm x}}_2|) e^{i{{\bm k}}\cdot({{\bm x}}_2-{{\bm x}})}{{{\rm d}}{{\bm x}}}_2\\ &= - 1 - \rho_0 \beta {\hat u}(k). \label{eq:sigmauhat} \end{split}$$ Setting $c^{(3)}=c^{(4)}=0$ implies from (\[eq:qandc\]) that $Q=\frac{1}{2}$, $C=0$ and $R=0$, and results in a free energy $$\begin{split} \beta{{{\cal F}_{3}}}[n] &= \int \! \big((1+n({{\bm x}}_1)) \log (1+n({{\bm x}}_1)) - n({{\bm x}}_1)\big){{{\rm d}}{{\bm x}}}_1\\ & \quad{}- \frac{1}{2} \! \int \! \left(n^2({{\bm x}}_1) + n({{\bm x}}_1){{\cal L}}n({{\bm x}}_1) \right){{{\rm d}}{{\bm x}}}_1. \end{split} \label{eq:DDFTF3}$$ With this choice of free energy, the dynamics in (\[eq:DDFT2dynamics\]) becomes: $$\frac{\partial n}{\partial t} = - \nabla^2 \left({{\cal L}}n + \tfrac{1}{2}n^2\right) - \nabla \cdot \left[ n \nabla {{\cal L}}n \right], \label{eq:DDFT3dynamics}$$ along with an analogous version of (\[eq:betadFdn2\]), for the chemical potential: $$\mu=\beta\frac{{{\delta{\mkern-1mu}{{{\cal F}_{3}}}}}}{{{\delta{\mkern-1mu}n}}} = \log\left(1+n({{\bm x}})\right) - n({{\bm x}}) - {{\cal L}}n({{\bm x}}) \label{eq:betadFdn3}$$ We refer to this model as ; it is equivalent to DDFT-1 with the RY approximation, and to DDFT-0 with ${{\cal F_{\text{ex}}}}$ given by the RPA approximation. Before moving on to make further approximations, it is worth noting a useful property that DDFT-3 and the subsequent theories derived from it possess. If the pair potential $u(x_{12})$ in Eq.  can be written as $u(x_{12})=\epsilon\psi(x_{12})$, where $\epsilon$ is a parameter that controls the overall strength of the potential, then from Eqs. , and  we obtain: $$\mu= \log\left(1+n({{\bm x}})\right) +\rho_0\beta\epsilon \!\!\int\! \psi(|{{\bm x}}-{{\bm x}}_2|) n({{\bm x}}_2){{{\rm d}}{{\bm x}}}_2. \label{eq:convol_form}$$ The consequence of this is that for a given $\psi$, the behaviour of the model depends only on the combination of parameters $\rho_0\beta\epsilon$ and the value of $\mu$. If one changes the value of the reference density $\rho_0$ to some other value, then this is entirely equivalent to solving the system with the original reference density $\rho_0$ at a different value of $\beta\epsilon$. We should emphasize that this is only true if $\psi$ does not change with density, which in general is not true, but is approximately the case for some systems. Gradient expansion of the linear term: DDFT-4 --------------------------------------------- Returning to DDFT-2, Huang  [@Huang2010a] (following [@Elder2004]) expanded ${{\cal L}}$ in powers of the gradient operator $\nabla$, replacing ${{\cal L}}$ by the simplest linear operator that allows a positive growth rate for modes with a wavenumber $k_s$. Scaling lengths so that $k_s=1$ results in: $${{{\cal L}_{\text{grad}}}}n = r n - \gamma(1+\nabla^2)^2 n, \label{eq:cLgraddefn}$$ so $\sigma(k)=r - \gamma(1-k^2)^2$ from (\[eq:defnsigma\]). This approximation is equivalent (within scaling) to a local gradient expansion of (\[eq:defnL\]), expanding the Fourier transform of $c^{(2)}$ about its maximum: $$\rho_0 {\hat c}^{(2)}(k) = 1 + r - \gamma(1-k^2)^2, \label{eq:defnGradientExpansion}$$ where the function $\rho_0{\hat c}^{(2)}(k)$ and its second derivative evaluated at $k=1$ are $1+r$ and $-8\gamma$, respectively. Here, $r$ is a parameter, notionally increasing with $\beta$ (and with $\rho_0$) and equal to zero at the spinodal point, when $\beta=\beta_s$. This parameter controls the growth rate of waves with wavenumber $1$: effectively, $r$ is the height of the maximum at $k=1$ in the growth rate curve in Fig. \[fig:growthrate\]. The second parameter $\gamma$ can be used to fit the curvature of ${\hat c}^{(2)}(k)$ at $k=1$. With this gradient expansion, the dynamics is $$\begin{split} \frac{\partial n}{\partial t} &= - \nabla^2 \left({{{\cal L}_{\text{grad}}}}n + Qn^2 + Cn^3 + Rn^4\right) \\ &\qquad {} - \nabla \cdot \left[ n \nabla {{{\cal L}_{\text{grad}}}}n \right]. \end{split} \label{eq:DDFT4dynamics}$$ We refer to this model as : ${{{\cal L}_{\text{grad}}}}$ is now a (local) partial differential operator and (\[eq:DDFT4dynamics\]) is a partial differential equation. The free energy and chemical potential can be found from (\[eq:DDFTF2\]) and (\[eq:betadFdn2\]), setting ${{\cal L}}={{{\cal L}_{\text{grad}}}}$. The lower bound $n\geq-1$ is still respected. This model is equivalent to that written down by [@Huang2010a]. Higher powers (or other functions) of the Laplacian can be retained in ${{{\cal L}_{\text{grad}}}}$, to improve the accuracy of the match between the eigenvalues of ${{\cal L}}$ and ${{{\cal L}_{\text{grad}}}}$, as done for example by [@Jaatinen2009; @Pisutha-Arnond2013b], or to introduce additional unstable length scales, as done for example by [@Wu2010; @Barkan2011; @Achim2014; @Subramanian2016] and others. See also Eq.  below and the associated discussion. RY and gradient expansion: DDFT-5 --------------------------------- Finally, we can make both the RY/RPA approximation ($c^{(3)}_0=c^{(4)}=0$) and replace the linear operator ${{\cal L}}$ by ${{{\cal L}_{\text{grad}}}}$ to get the model advocated in Ref. [@Teeffelen2009]. The free energy and evolution equation are $$\begin{split} \beta{{{\cal F}_{5}}}[n] &= \int \! \big((1+n({{\bm x}}_1)) \log (1+n({{\bm x}}_1)) - n({{\bm x}}_1)\big){{{\rm d}}{{\bm x}}}_1\\ & \quad{}- \frac{1}{2} \! \int \! \left(n^2({{\bm x}}_1) + n({{\bm x}}_1){{{\cal L}_{\text{grad}}}}n({{\bm x}}_1) \right){{{\rm d}}{{\bm x}}}_1, \end{split} \label{eq:DDFTF5}$$ and $$\frac{\partial n}{\partial t} = - \nabla^2 \left({{{\cal L}_{\text{grad}}}}n + \tfrac{1}{2}n^2\right) - \nabla \cdot \left[ n \nabla {{{\cal L}_{\text{grad}}}}n \right], \label{eq:DDFT5dynamics}$$ along with an analogous version of Eq. (\[eq:betadFdn3\]) for the chemical potential: $$\mu=\beta\frac{{{\delta{\mkern-1mu}{{{\cal F}_{5}}}}}}{{{\delta{\mkern-1mu}n}}} = \log\left(1+n({{\bm x}})\right) - n({{\bm x}}) - {{{\cal L}_{\text{grad}}}}n({{\bm x}}) \label{eq:betadFdn5}$$ This model is named PFC1 in [@Teeffelen2009], but here we call it DDFT-5 for consistency. [PFC models]{} -------------- The final simplification that can be made (or not made) is to discard the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ (or $\nabla\cdot\left[n\nabla{{{\cal L}_{\text{grad}}}}{n}\right]$) term from the dynamical equations for the four DDFT models DDFT-2, …, DDFT-5, resulting in four PFC models , …, . Huang  [@Huang2010a] justify making this simplification on the grounds that this term is not truly quadratic in $n$: the presence of ${{\cal L}}{n}$ in the expression means that it is effectively of higher order. However, we show below that this term does in fact make an important contribution to the free energy: at least as important as the $c^{(3)}$ term. In addition, dropping this term implies significant changes to the DDFT dynamics, the mobility and the nonlinear terms in the free energy. In fact, the $(1+n)$ factor in the mobility in (\[eq:DDFT1deltaF\]), the logarithm in the ideal gas free energy in (\[eq:separatedF\]) and the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term in (\[eq:DDFT1dynamics\]) are inextricably linked. This can be seen in the progression from (\[eq:separatedF\]) to (\[eq:rhodiffusion\]): the functional derivative of the ideal gas term in (\[eq:separatedF\]) (the first term on the right hand side) leads to the $\log\rho$ term in (\[eq:separatedFdrho\]), the gradient of this leads to $\rho^{-1}\nabla\rho$ in (\[eq:rhoinverse\]), and the mobility being $M=D\rho$ cancels the $\rho^{-1}$, leading to a diffusion equation in (\[eq:rhodiffusion\]). [If the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term is dropped from (\[eq:DDFT2dynamics\]), the equation for $n$ becomes of the form $\frac{\partial n}{\partial t} = \nabla^2\frac{{{\delta{\mkern-1mu}{{\cal G}}}}}{{{\delta{\mkern-1mu}n}}}$ for some functional ${{\cal G}}[n]$. We can see the implications of this by returning to (\[eq:DDFT0dynamics\]) and taking the steps needed to get to this modified version of (\[eq:DDFT2dynamics\]). Clearly the mobility in (\[eq:DDFT0dynamics\]) has been taken to be constant. If we now think of the ideal gas part of the free energy in (\[eq:separatedF\]) and (\[eq:separatedFdrho\]), but with a constant mobility in the dynamical equation, we end up with the ideal gas term contribution to the equation for $\rho$ being the form $$\frac{\partial\rho}{\partial t} = \frac{1}{\rho}\nabla^2\rho - \frac{1}{\rho^2}\left|\nabla\rho\right|^2$$ instead of the diffusion equation (\[eq:rhodiffusion\]). This unlikely equation can be avoided, and the diffusion equation recovered at leading order, by expanding the logarithm in (\[eq:DDFTF2\]) in a Taylor series. Thus, dropping the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term is equivalent to taking constant mobility and expanding the logarithm.]{} It is because of these substantial changes that we opt to use the term ‘DDFT’ for all models based on free energies that have the logarithmic ideal gas term, the non-constant mobility and the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term retained. In contrast, we use the term ‘PFC’ for models based on expanding the logarithm, having a constant mobility and the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term dropped. One consequence of expanding the logarithm [up to ${{\cal O}}(n^4)$, as is done in most PFC derivations [@Emmerich2012], is that the ideal gas part of the free energy contributes cubic and quartic (as well as quadratic) terms to the free energy,]{} so going from DDFT-2 to  turns out not to be just a matter of dropping the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term. So, a consistent free energy–dynamics derivation [@Elder2004; @Teeffelen2009] involves going back to DDFT-2 and replacing the logarithm in (\[eq:DDFTF2\]) by: $$(1+n)\log(1+n)={n}+\tfrac{1}{2}n^2-\tfrac{1}{6}n^3+\tfrac{1}{12}n^4, \label{eq:expandlogarithm}$$ resulting in a free energy $$\begin{split} \beta{{\cal F_{\text{$\beta$}}}}[n] &= \int \! \Big( - \tfrac{1}{2}n{{\cal L}}n - \tfrac{1}{6}n^3 + \tfrac{1}{12}n^4\\ & \qquad\quad {} - \frac{\rho_0^2}{6} c^{(3)}_0 n^3 - \frac{\rho_0^3}{24} c^{(4)}_0 n^4\Big){{{\rm d}}{{\bm x}}}_1, \label{eq:DDFTFPFCbeta} \end{split}$$ where we have suppressed writing the ${{\bm x}}_1$ dependency of $n({{\bm x}}_1)$. Taking the mobility $M(\rho)$ in (\[eq:DDFT0dynamics\]) to be a constant ($M=D\rho_0$) implies (after scaling) $$\frac{\partial n}{\partial t} = \nabla^2 \left[\beta\frac{{{\delta{\mkern-1mu}{{\cal F_{\text{$\beta$}}}}}}}{{{\delta{\mkern-1mu}n}}} \right], \label{eq:PFCdynamics}$$ similar to (\[eq:PFCalphadynamics\]). This leads to the PFC dynamical equation: $$\frac{\partial n}{\partial t} = - \nabla^2 \left[ {{\cal L}}{n} + Qn^2 + Cn^3 \right] \label{eq:PFCbetadynamics}$$ and to a chemical potential $$\mu=\beta\frac{{{\delta{\mkern-1mu}{{\cal F_{\text{$\beta$}}}}}}}{{{\delta{\mkern-1mu}n}}} = - {{\cal L}}{n} - Qn^2 - Cn^3, \label{eq:betadFdnbeta}$$ where $Q$ is as in (\[eq:qandc\]) but $C$ is different: $$Q = \frac{1}{2} + \frac{\rho_0^2}{2}c^{(3)}_0 \quad\text{and}\quad C = - \frac{1}{3} + \frac{\rho_0^3}{6}c^{(4)}_0. \label{eq:PFCqandc}$$ We refer to this model as , and recall that the factor of $\beta$ in front of ${{\cal F_{\text{$\beta$}}}}$ is the inverse temperature. The end result here is that  (\[eq:PFCbetadynamics\]) is not the same as DDFT-2 (\[eq:DDFT2dynamics\]) with the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term removed: the cubic coefficient $C$ is different and the quartic contribution $Rn^4$ in (\[eq:DDFT2dynamics\]) is absent. For the cubic coefficient, the contribution proportional to $c^{(3)}_0$ in (\[eq:qandc\]) comes from the non-constant mobility, while the $-\frac{1}{3}$ term in (\[eq:PFCqandc\]) comes from expanding the logarithm. The contribution to $C$ proportional to $c^{(4)}_0$ is the same. Moreover, the $\frac{1}{2}$ in $Q$ in (\[eq:qandc\]) and (\[eq:PFCqandc\]), while having the same numerical value, arises for two different reasons: non-constant mobility versus expanding the logarithm. An additional difference between the DDFT and PFC models is that in the PFC models, the constraint that $n\geq-1$ (i.e., $\rho\geq0$) is not enforced. As in the DDFT derivations, we can now make (or not make) the RY/RPA approximation and the gradient expansion. We consider first the RY/RPA approximation, setting $c^{(3)}=c^{(4)}=0$ in . The free energy is $$\beta{{\cal F_{\text{$\gamma$}}}}[n] = \int \! \left( - \tfrac{1}{2}n{{\cal L}}n - \tfrac{1}{6}n^3 + \tfrac{1}{12}n^4 \right){{{\rm d}}{{\bm x}}}_1, \label{eq:DDFTFPFCgamma}$$ the dynamics is $$\frac{\partial n}{\partial t} = - \nabla^2 \left[ {{\cal L}}{n} + \tfrac{1}{2}n^2 - \tfrac{1}{3}n^3 \right] \label{eq:PFCgammadynamics}$$ and the chemical potential is $$\mu=\beta\frac{{{\delta{\mkern-1mu}{{\cal F_{\text{$\gamma$}}}}}}}{{{\delta{\mkern-1mu}n}}} = - {{\cal L}}{n} - \tfrac{1}{2}n^2 + \tfrac{1}{3}n^3. \label{eq:betadFdngamma}$$ We refer to this model as , and it is effectively the same as  but with $Q=\frac{1}{2}$ and $C=-\frac{1}{3}$. Finally, the gradient expansion can be made, replacing ${{\cal L}}$ by ${{{\cal L}_{\text{grad}}}}$ in all expressions in this subsection, resulting in  (without RY/RPA) and  (with RY/RPA). We refer to these models collectively as the PFC models, and have chosen the names   to distinguish these from the PFC1 and PFC2 models of Ref. [@Teeffelen2009]. The quadratic term in the dynamics ($Qn^2$) can be removed (provided $C\neq0$) by adding a constant to $n({{\bm x}})$, but we choose not to do this as it implies a change to what was meant by $\rho_0$ in the reference liquid. In addition, a negative $C$ can be scaled to $-1$. With these changes,  is equivalent to the original model (\[eq:PFCalphadynamics\]) of [@Elder2002; @Elder2004]: $$\frac{\partial n}{\partial t} = - \nabla^2 \left(r n - \gamma(1+\nabla^2)^2n + Qn^2 + Cn^3\right), \label{eq:PFCepsilondynamics}$$ where we have written out ${{{\cal L}_{\text{grad}}}}$ explicitly, and $Q=\frac{1}{2}$ and $C=-\frac{1}{3}$ (or $Q=0$ and $C=-1$ after scaling and adding a constant to $n$ – returning to the conserved Swift–Hohenberg equation). The implication of dropping the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term in the dynamics (\[eq:DDFT3dynamics\]) for DDFT-3 is now apparent: without this term, Eq. (\[eq:DDFT3dynamics\]) reduces to (\[eq:PFCgammadynamics\]) but with the cubic term removed. The absence of the cubic term here implies a free energy as in (\[eq:DDFTFPFCgamma\]) that is not bounded below, i.e., a free energy that is non-physical, and so the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term can have the effect of stabilizing patterns. In addition, dropping the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term is consistent only with a theory with a constant mobility. Summary ------- To summarise, we have carefully laid out the various approximations made in the progression from the DDFT-0 starting point (\[eq:DDFT0dynamics\]) to the final PFC (\[eq:PFCalphadynamics\],\[eq:PFCgammadynamics\]) written down in [@Elder2002; @Elder2004]. We have largely followed earlier derivations [@Teeffelen2009; @Huang2010a; @Emmerich2012], seeking to clarify the approximations that are made. Along the way, we have identified four intermediate versions of DDFT, listed for clarity in Table \[tab:DDFTvsPFC\]. The change in name from DDFT to PFC could be made at any point in this progression, but we prefer to make the name change at the point where the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term is dropped (along with all the other changes that are implied by this), since removing this term marks a considerable alteration to the free energy expression and to the dynamics. The PFC model (\[eq:PFCbetadynamics\]) is appealing in its simplicity, and it gives insight into a variety of crystallisation phenomena, but the derivations of the model from DDFT presented here, as well as the derivation from Ref. [@Huang2010a], are both problematic. Just dropping the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term, as done by Huang  [@Huang2010a], means that the dynamics is not equivalent to a DDFT with mobility proportional to $\rho$. On the other hand, the alternative is to expand the logarithm up to ${{\cal O}}(n^4)$ in (\[eq:expandlogarithm\]) in order to provide a *nonlinear* stabilizing term ($\frac{1}{12}n^4$) in the free energy (\[eq:DDFTFPFCgamma\]). However, in the original formulation, the logarithm comes from the ideal gas term in (\[eq:separatedF\]), and leads to a *linear* diffusive term in the dynamics. The stabilizing nonlinear terms in (\[eq:DDFT2dynamics\]) are provided by $c^{(3)}_0$, $c^{(4)}_0$ (in DDFT-2) and by the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term – these are all absent in . Indeed, all these models only make physical sense if their free energies are bounded below. The free energies for DDFT-0 and DDFT-1 are too general to make any comment, but that for DDFT-2 (\[eq:DDFTF2\]) can be discussed. The $(1+n) \log (1+n) - n$ term is bounded below by zero, and the $-\tfrac{1}{2}\int{n}{{\cal L}}{n}{{{\rm d}}{{\bm x}}}$ term is bounded below because the eigenvalues of ${{\cal L}}$ are bounded above: $$- \int \! n({{\bm x}}){{\cal L}}n({{\bm x}}) {{{\rm d}}{{\bm x}}}\geq - {\sigma_{\text{max}}}\int \! n^2({{\bm x}}) {{{\rm d}}{{\bm x}}},$$ where ${\sigma_{\text{max}}}$ is the maximum over $k$ of $\sigma(k)$ (we have in mind a $\sigma(k)$ as in Fig. \[fig:growthrate\]). In any case, this term, along with the other quadratic and cubic terms, is dominated by the quartic in $n$, which is bounded below provided $c^{(4)}_0<0$. If $c^{(4)}_0=0$, then $c^{(3)}_0<0$ will do, recalling that $n\geq-1$. For DDFT-3, with the RY/RPA approximation $c^{(3)}_0=c^{(4)}=0$, the boundedness of the free energy (\[eq:DDFTF3\]) depends on the $n^2 + n{{\cal L}}{n}$ combination. From (\[eq:defnL\]), the relevant term is $$\begin{split} -\int \! & \left(n^2({{\bm x}}_1) + n({{\bm x}}_1){{\cal L}}n({{\bm x}}_1) \right){{{\rm d}}{{\bm x}}}_1 = \\ & \qquad - \rho_0 \!\!\int\! n({{\bm x}}_1)c^{(2)}({{\bm x}}_1,{{\bm x}}_2)n({{\bm x}}_2){{{\rm d}}{{\bm x}}}_1{{{\rm d}}{{\bm x}}}_2. \end{split}$$ In general, this is not bounded below, but it is in certain circumstances. For example, it is if ${\sigma_{\text{max}}}<-1$, and it is if $c^{(2)}({{\bm x}}_1,{{\bm x}}_2)\leq0$ (or $u(|{{\bm x}}_1-{{\bm x}}_2|)\geq0$ for RPA) for all ${{\bm x}}_1$ and ${{\bm x}}_2$, which is the case in the numerical examples below. The PFC models are not constrained to have $n\geq-1$, but ${{\cal F_{\text{$\beta$}}}}$ (\[eq:DDFTFPFCbeta\]) is bounded by the $n^4$ term as long as its coefficient is positive; ${{\cal F_{\text{$\gamma$}}}}$ (\[eq:DDFTFPFCgamma\]) is always bounded below, because the expansion of the logarithm in (\[eq:expandlogarithm\]) was truncated after an even powered term. Throughout we have made the simplest choices in the approximations, but other authors have made many other choices. For example, the original PFC paper [@Elder2002], as well as later papers [@Elder2007; @Huang2010a; @Robbins2012a; @Alster2017a; @Elder2017a], included a two-component (binary) version of the PFC model. Recently, Wang  [@Wang2018c] took a much closer look at $c^{(3)}$ and $c^{(4)}$, expressing these in terms of isotropic tensors and so allowing these functions to introduce bond angle dependence into the free energy. Some choices of $c^{(3)}$ and $c^{(4)}$ lead to nonlinear terms that include gradients, which can affect the selection of the final stable crystal [@Wu2010a]. The gradient expansion approach has been generalised in two ways: (i) higher order terms or rational functions were considered by [@Jaatinen2009; @Pisutha-Arnond2013b] in order to improve the fit between the functional form and the Fourier transform of $c^{(2)}$, and (ii) PFC models with two unstable length scales have been put forward by several authors [@Pisutha-Arnond2013b; @Wu2010; @Barkan2011; @Achim2014; @Subramanian2016; @Jiang2017], since these allow more complex crystals (face-centered cubic, icosahedral quasicrystals, …) to be stabilized. We discuss the model of [@Jaatinen2009] in more detail below. Alternative approaches involving weighted densities are also possible [@Jaatinen2010a]. Comparison of DDFT and PFC {#sec:3} ========================== We are interested in the effects of the approximations made in going from DDFT to . A full assessment of the validity of the RY/RPA approximation for ${{\cal F_{\text{ex}}}}$, which in itself constitutes a major simplification, is beyond the scope of the present study. The general conclusion on the validity of the RY/RPA approximation is that it depends on the nature of the interactions between the particles; there are examples in the literature where this approximation is reliable and others where it works badly – see for example the discussion in Refs [@Evans2009; @Lowen2009] and references therein. Here, we consider one particular system where the RPA is accurate and then we focus on the effects of approximating ${{\cal L}}$ by ${{{\cal L}_{\text{grad}}}}$ and of making the suite of other approximations inherent in going from DDFT to PFC: expanding the logarithm, assuming constant mobility and dropping the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term. To this end, we start with DDFT-3 and solve (\[eq:betadFdn3\]), rewritten here as: $$\text{DDFT-3:}\quad \log\left(1+n({{\bm x}})\right) - n({{\bm x}}) - {{\cal L}}n({{\bm x}}) = \mu. \label{eq:betadFdn3_rewritten}$$ The system that we consider is particles interacting via the GEM-4 [@Mladek2006; @Prestipino2014] potential: this is a model for soft-matter particles and in particular for dendrimers and other polymers in suspension, treating the polymers via an effective pair potential between their centers of mass. This potential is soft, i.e., finite for all values of $x_{12}$ [@Mladek2006; @Prestipino2014; @Likos2001; @Likos2006; @Lenz2012], and is $$u(x_{12}) = \epsilon e^{-\left(x_{12}/R\right)^4}, \label{eq:GEM4Potential}$$ where the parameter $\epsilon$ controls the strength of the potential and $R$ controls its spatial range. We consider here the system in 2D [@Prestipino2014; @Archer2014]. As long as the temperature and density are high enough that the particle cores regularly overlap (the regime in which the system freezes), the RPA approximation is known to be rather accurate for the GEM-4 system and gives a good account of the phase diagram and the structure of the liquid and solid phases [@Mladek2007; @Prestipino2014; @Archer2014]. From Eqs. (\[eq:defnL\]), (\[eq:RPAc2u\]) and we obtain the linear operator ${{\cal L}}$: $${{\cal L}}n({{\bm x}}) = - n({{\bm x}}) - \rho_0\beta\epsilon \!\!\int\! e^{-|{{\bm x}}-{{\bm x}}_2|^4/R^4} n({{\bm x}}_2){{{\rm d}}{{\bm x}}}_2. \label{eq:defnL_GEM4}$$ Recall from (\[eq:defnsigma\]) that ${{\cal L}}$ has eigenvalues $\sigma(k)$ with eigenfunctions $e^{i{{\bm k}}\cdot{{\bm x}}}$. We can choose the combined parameter $\rho_0\beta\epsilon$ and soft-particle radius $R$ so that the maximum in $\sigma(k)$ occurs at $k=1$ when the system is at the linear stability threshold, i.e., this is a maximum with $\sigma(1)=0$, similar to Fig. \[fig:growthrate\]. In 2D, to satisfy this condition we must have $\rho_0\beta\epsilon=0.2455$ and $R=5.0962$ – see Appendix \[app:LinearGEM4\] for details. ![The eigenvalue $\sigma(k)$ of ${{\cal L}}$ plotted as a function of wavenumber $k$ for the GEM-4 potential (solid line) for $R=5.0962$ and $\rho_0\beta\epsilon=0.2455$, which is at the threshold where the system becomes linearly unstable. This has $\sigma(0)=-18.75$. We also display $\sigma(k)$ from the gradient expansion of ${{\cal L}}$ (dashed line), i.e., a Taylor expansion in Fourier space around $k=1$, which is the PFC relation for ${{{\cal L}_{\text{grad}}}}$, $\sigma(k)=-\gamma(1-k^2)^2$, with $\gamma=4.37$. Recall that the growth rate of Fourier modes with wavenumber $k$ is is $k^2\sigma(k)$. The dotted line labelled ${{{\cal L}_{\text{grad-8}}}}$ (EOF) is the curve for (\[eq:cLgradJaatinen\]), the eighth-order fitting proposed in [@Jaatinen2009]: it nearly coincides with the GEM-4 curve for $k\leq1$.[]{data-label="fig:dispersionGEM4"}](dispersion_relation){width="8.6truecm"} With this choice of parameters, the eigenvalue $\sigma(k)$ is shown as a solid line in Fig. \[fig:dispersionGEM4\]. The figure also shows (dashed line) the eigenvalue for the gradient expansion of ${{\cal L}}$ around $k=1$: $${{{\cal L}_{\text{grad}}}}n({{\bm x}}) = - \gamma(1+\nabla^2)^2 n({{\bm x}}), \label{eq:cLgradGEM4}$$ where $\gamma=4.37$ is chosen to match the second derivative $\frac{{{\rm d}}^2\sigma}{{{\rm d}}{k}^2}$ at $k=1$, as done for example in Refs. [@Elder2004; @Wu2007; @Jaatinen2009]. The dotted line in Fig. \[fig:dispersionGEM4\] is the eigenvalue for (\[eq:cLgradJaatinen\]), the eighth-order fitting model proposed in [@Jaatinen2009] and discussed in more detail below. ![(a) Liquid density ($1+{n_{\text{liq}}}$) and (b) specific grand potential ${\Omega_{\text{liq}}}/A$ as a function of the scaled chemical potential $\mu$, for DDFT-3 (solid black line), DDFT-5 (dashed black line),  (indistinguishable from DDFT-3) and  (dashed magenta line). []{data-label="fig:liquiddensity"}](liquid_density){width="8.6truecm"} ![(a) Liquid density ($1+{n_{\text{liq}}}$) and (b) specific grand potential ${\Omega_{\text{liq}}}/A$ as a function of the scaled chemical potential $\mu$, for DDFT-3 (solid black line), DDFT-5 (dashed black line),  (indistinguishable from DDFT-3) and  (dashed magenta line). []{data-label="fig:liquiddensity"}](liquid_omega){width="8.6truecm"} In what follows we compare solutions of (\[eq:betadFdn3\_rewritten\]) for DDFT-3 with solutions of the analogous equations for DDFT-5,  and : $$\begin{aligned} &\text{DDFT-5:}& \log\left(1+n\right) - n - {{{\cal L}_{\text{grad}}}}n &= \mu, \label{eq:betadFdn5_rewritten}\\ &\text{{\hbox{PFC-$\gamma$}}:}& -\tfrac{1}{2}n^2 + \tfrac{1}{3} n^3 - {{\cal L}}n &= \mu, \label{eq:betadFdngamma_rewritten}\\ &\text{{\hbox{PFC-$\epsilon$}}:}& -\tfrac{1}{2}n^2 + \tfrac{1}{3} n^3 - {{{\cal L}_{\text{grad}}}}n &= \mu. \label{eq:betadFdnepsilon_rewritten} \end{aligned}$$ See Appendix \[app:continuation\] for details of the pseudo-arclength continuation numerical method we use for solving these equations. Also, in the supplementary material we include a <span style="font-variant:small-caps;">Matlab</span> code for solving DDFT-3. Note that throughout what follows, we refer to the quantity $1+n({{\bm x}})$ as the ‘density’. Since the DDFT-5,  and  represent different forms of Taylor expansion around the reference state with density $\rho_0$, there are a variety of ways comparison between solutions can be made. Here, we opt to fix ${{\cal L}}$ and ${{{\cal L}_{\text{grad}}}}$ as in (\[eq:defnL\_GEM4\]) and (\[eq:cLgradGEM4\]) with the specified values of $\rho_0\beta\epsilon$, $R$ and $\gamma$. This implies that at $\mu=0$ the reference state with $n=0$ is at the spinodal point and is marginally unstable to modes with wavenumber $k=1$. We then vary $\mu$ starting from $\mu=0$ and follow the liquid, stripe and hexagonal solutions of (\[eq:betadFdn3\_rewritten\]) and (\[eq:betadFdn5\_rewritten\])–(\[eq:betadFdnepsilon\_rewritten\]) in appropriately sized two-dimensional domains. For a given value of $\mu$ the different solutions have different values for the mean density $1+\bar{n}=1+\frac{1}{A}\!\!\int\! n({{\bm x}}){{{\rm d}}{{\bm x}}}$, where $A$ is the area of the domain. For each state we calculate the specific grand potential: $$\frac{\Omega[n]}{A} = \frac{{{\cal F}}[n]}{A} - \mu (1+\bar{n}), \label{eq:specificgrandpotential}$$ where ${{\cal F}}$ is ${{{\cal F}_{3}}}$, ${{{\cal F}_{5}}}$, ${{\cal F_{\text{$\gamma$}}}}$ or ${{\cal F_{\text{$\epsilon$}}}}$, as appropriate. We also minimise $\Omega/A$ with respect to the domain size $A$ by applying the approach described in Appendix \[app:continuation\]. For a given value of the chemical potential $\mu$ and the combined parameter $\rho_0\beta\epsilon$, the thermodynamic equilibrium state is that with the minimum value of $\Omega/A$. Note that equilibria with the same $\mu$ do not necessarily have the same value of $\bar n$, which is important when considering which equilibria might result from initial conditions via the dynamics. to to The solution corresponding to the uniform density liquid state with $n({{\bm x}})={n_{\text{liq}}}$ can readily be found. In this case we have ${{\cal L}}{{n_{\text{liq}}}}=\sigma(0){n_{\text{liq}}}$, and so we must solve the following algebraic equations for ${n_{\text{liq}}}$: $$\begin{aligned} &\text{DDFT-3,5:}& \log\left(1+{n_{\text{liq}}}\right) - {n_{\text{liq}}}- \sigma(0) {n_{\text{liq}}}&= \mu, \label{eq:DDFTnliq}\\ &\text{{\hbox{PFC-$\gamma$}},$\epsilon$:}& -\tfrac{1}{2}{n_{\text{liq}}}^2 + \tfrac{1}{3}{n_{\text{liq}}}^3 - \sigma(0){n_{\text{liq}}}&= \mu, \label{eq:PFCnliq} \end{aligned}$$ recalling that the value of $\sigma(0)$ depends on whether or not the gradient expansion is carried out (see Fig. \[fig:dispersionGEM4\]). Finding ${n_{\text{liq}}}$ for a given value of $\mu$ is done easily using Newton’s method, and the resulting ${n_{\text{liq}}}$ and specific ${\Omega_{\text{liq}}}$ are shown in Fig. \[fig:liquiddensity\]. In all cases, we see that ${n_{\text{liq}}}$ is an increasing function of $\mu$, while ${\Omega_{\text{liq}}}/A$ is a decreasing function of $\mu$. The figure shows that the specific grand potential for the liquid state predicted by all four models are similar close to $\mu=0$, but the predicted liquid state densities are rather different away from $\mu=0$. This difference originates from the different values of $\sigma(0)$ ($-18.75$ for DDFT-3 and , in contrast to $-4.37$ for DDFT-5 and ). We see from Fig. \[fig:liquiddensity\] that the density of the liquid is erroneously predicted to increase too rapidly as $\mu$ is increased by the gradient expansion theories (DDFT-5 and ). This is because these get the value of the isothermal compressibility $\chi_T$ to be too large [@Jaatinen2009]. This compressibility is related to $\sigma(0)$ via $\chi_T=-\beta/[\sigma(0)\rho_0(1+{n_{\text{liq}}})]$ [@Hansen2013]: see Eq.  and following discussion. Expanding the logarithm makes relatively little difference over this range of densities. Since crystallisation occurs at higher densities, we expect a transition from the liquid to the crystal to occur as $\mu$ increases. At the spinodal the uniform liquid becomes linearly unstable and the patterned state solution branches bifurcate from the liquid at this point. To find these states, we seek a solution of the form $$n({{\bm x}}) = {n_{\text{liq}}}+ {{\delta{\mkern-1mu}n}}({{\bm x}}),$$ where near the bifurcation point ${{\delta{\mkern-1mu}n}}\ll1$, and ${{\delta{\mkern-1mu}n}}$ is of the form $e^{i{{\bm k}}\cdot{{\bm x}}}$, so that ${{\cal L}}{{\delta{\mkern-1mu}n}}=\sigma(k){{\delta{\mkern-1mu}n}}$. Expanding Eqs. (\[eq:betadFdn3\_rewritten\]) and (\[eq:betadFdn5\_rewritten\]–\[eq:betadFdnepsilon\_rewritten\]) in powers of ${{\delta{\mkern-1mu}n}}$ we find that the ${{\cal O}}(1)$ equations to solve are just those for finding the liquid state density, Eqs. (\[eq:DDFTnliq\])–(\[eq:PFCnliq\]). The ${{\cal O}}({{\delta{\mkern-1mu}n}})$ equations are $$\begin{aligned} &\text{DDFT-3,5:}& \left(\frac{1}{1+{n_{\text{liq}}}} - 1 - \sigma(k)\right){{\delta{\mkern-1mu}n}}&= 0, \label{eq:DDFTdeltan}\\ &\text{{\hbox{PFC-$\gamma$}},$\epsilon$:}& \left(-{n_{\text{liq}}}+ {n_{\text{liq}}}^2 - \sigma(k)\right){{\delta{\mkern-1mu}n}}&= 0. \label{eq:PFCdeltan} \end{aligned}$$ The spinodal point for DDFT-3,5 or for ,$\epsilon$ is where there are solutions of the equation with ${{\delta{\mkern-1mu}n}}\neq0$. Since we are looking for a change in stability, we take the extreme value of $\sigma(k)$, i.e., $\sigma(k)=0$ (see Fig. \[fig:dispersionGEM4\]). Then, Eq. (\[eq:DDFTdeltan\]) is solved (with ${{\delta{\mkern-1mu}n}}\neq0$) only for ${n_{\text{liq}}}=0$, which leads to $\mu=0$ from (\[eq:DDFTnliq\]). In contrast, Eq. (\[eq:PFCdeltan\]) with $\sigma(k)=0$ has two solutions, ${n_{\text{liq}}}=0$ and ${n_{\text{liq}}}=1$, leading to $\mu=0$ and $\mu=-\tfrac{1}{6}-\sigma(0)$ from (\[eq:PFCnliq\]). The implication of this is that the PFC has two spinodal points: the liquid loses stability at ${n_{\text{liq}}}=0$ as $\mu$ increases through $0$, but it regains stability at ${n_{\text{liq}}}=1$, which gives $\mu=18.58$ for  and $\mu=4.20$ for . This prediction that the liquid regains stability for higher $\mu$ is a consequence of expanding the logarithm, or equivalently of Taylor expanding the $1/(1+{n_{\text{liq}}})$ term in (\[eq:DDFTdeltan\]) and is confirmed by direct computation of the crystal solutions below. Of course, this prediction is erroneous, since the simulation results for the GEM-4 system [@Mladek2006; @Prestipino2014] show no sign of a second spinodal point or the associated stable second liquid in the equilibrium system phase diagram. In Fig. \[fig:DDFT3solutions\] we display examples of the three different types of periodic solutions that can be found for DDFT-3. These are (i) the crystal solution, which we refer to as ‘up hexagons’, which exhibits a triangular array of isolated density maxima surrounded by hexagonal regions where the density is close to zero. There are also (ii) ‘down hexagons’ which are the opposite, with isolated density minima and hexagonal density maxima. Finally, there is (iii) the stripe state. Depending on the state point these solutions are not necessarily linearly stable. Our naming convention to distinguish the two different hexagonal solutions originates in the convection literature [@Bodenschatz2000]. These solutions were initiated at $\mu\approx0$ and then continued numerically (see Appendix \[app:continuation\]) up to $\mu=10$. For DDFT-3 it is possible to go a bit higher in $\mu$, but with increasing $\mu$ (i.e., increasing average density) the peaks in the density profile get narrower and higher and so more and more grid points are required to resolve these peaks correctly. However, as we show below for some of the other models and for different reasons, it is not possible to continue the solutions this far in $\mu$. The domains on which the profiles are calculated have periodic boundary conditions, with 4 wavelengths in each direction (for stripes), or $8\times\tfrac{8}{\sqrt{3}}$ wavelengths (for hexagons). The wavelength is initially equal to $2\pi$ for $\mu=0$ and is then adjusted by up to about 2% in order to minimise the specific grand potential as $\mu$ is varied; i.e., we minimise $\Omega/A$ with respect to variations in the size of the crystal unit cell or, for the stripe phase, we minimise with respect to variations in the spacing between the stripes – see Appendix B for details. to to to to In Fig. \[fig:DDFTPFCsummary\] we display a series of plots showing the maximum, minimum and average values of the density profiles $1+n$ for the stripe and hexagonal structures as a function of $\mu$. We also plot the specific grand potential $\Omega/A$ for the different structures. Recall that for a given $\mu$ the thermodynamic equilibrium phase corresponds to the global minimum of $\Omega/A$. The results for DDFT-3 are shown in Fig. \[fig:DDFTPFCsummary\](a–c). The (a) stripes originate in a supercritical pitchfork bifurcation at $\mu=0$, and (b) hexagons originate in a transcritical bifurcation at the same value of $\mu$. The density of the up hexagons ranges from about $2\times10^{-5}$ up to about 50, for $\mu=10$. All of these branches can be continued to larger values of $\mu$. DDFT-5, in Fig. \[fig:DDFTPFCsummary\](d–f), initially behaves in the same way, but all three branches have their minimum density heading to zero before $\mu$ gets to 10: this happens at $\mu\approx3.37$ for (d) stripes, and for $\mu\approx0.28$ and $\mu\approx2.73$ for (e) the up and down branches of hexagons, respectively. The numerical method cannot continue the branches beyond these points. We argue in Sec. \[sec:4\] that this is not an artefact of the numerical method, rather it is a genuine feature of solutions of Eq. (\[eq:betadFdn5\_rewritten\]) that the density $1+n$ can go to zero. In this limit, $\log(1+n)\rightarrow-\infty$, but this is balanced by a lack of smoothness in $n({{\bm x}})$: the fourth derivative in ${{{\cal L}_{\text{grad}}}}n$ can go to $+\infty$ and so balance the singularity in $\log(1+n)$. Therefore, $\mu\approx0.28$ is the limit of validity of the DDFT-5 model. The two PFC examples are similar to each other, and it is easier to discuss , in Fig. \[fig:DDFTPFCsummary\](j–l), first. Here, (j) stripes and (k) hexagons bifurcate from the liquid at $\mu=0$, but they rejoin the liquid at $\mu=4.20$ as explained in the discussion following Eqs. (\[eq:DDFTdeltan\]–\[eq:PFCdeltan\]). The maximum and minimum densities for the up and down hexagon cross between the two bifurcations. The behaviour of , in Fig. \[fig:DDFTPFCsummary\](g–i), is similar, though the second bifurcation is at $\mu=18.58$, off the scale of the figure. to to Figure \[fig:DDFTPFCsummary\](c,f,i,l) shows that the curves of the specific grand potential $\Omega/A$ as functions of $\mu$ are very close, so in Fig. \[fig:DDFTPFCrelativeOmega\], we plot instead $(\Omega-{\Omega_{\text{liq}}})/A$ versus $\mu$, where ${\Omega_{\text{liq}}}$ is the specific grand potential for the liquid at the same value of $\mu$. In (a) the DDFT-3 case, the up hexagons clearly have the lowest grand potential for $\mu\geq-2.8$ with the uniform liquid being the global minimum for $\mu<-2.8$, and at no point do stripes come anywhere near, as one should expect from the particle simulation results [@Prestipino2014]. For (b) DDFT-5, the two hexagon branches stop before the stripe branch when their minimum densities go to zero (the limit of validity), but otherwise the relative values for the hexagon and stripe grand potentials is qualitatively similar to DDFT-3. For (c,d) the PFC examples, once again it is easiest to discuss first. In Fig. \[fig:DDFTPFCrelativeOmega\](d), the hexagon and stripe branches bifurcate from the liquid at $\mu=0$ and rejoin the liquid at $\mu=4.20$, with stripes having the lowest grand potential for intermediate values of $\mu$, and up or down hexagons being the lowest grand potential state for smaller or larger values of $\mu$. The behaviour of (c)  is similar, but stretched to larger values of $\mu$ (off scale). The insets in the four panels of Fig. \[fig:DDFTPFCrelativeOmega\] display magnifications that show that the behaviour near the spinodal point at $\mu=0$ is qualitatively similar in all four cases: the up hexagons start with $\Omega>{\Omega_{\text{liq}}}$ for negative $\mu$, but the branch changes direction, forming a cusp, close to which is the thermodynamic coexistence point (Maxwell point), where $\Omega={\Omega_{\text{liq}}}$. The down hexagons start with $\Omega<{\Omega_{\text{liq}}}$ for positive $\mu$, and the stripes, also with $\Omega<{\Omega_{\text{liq}}}$ for positive $\mu$, have a value of the grand potential intermediate between the up and down hexagons. We note that the range of $\mu$ over which this behaviour occurs is about a factor of ten smaller in the DDFT-5 and  cases as compared to DDFT-3, also with a roughly ten-fold drop in the overall range of values of $(\Omega-{\Omega_{\text{liq}}})/A$. to to The observation that the bulk phase behaviour of the system depends only on $\mu$ and the value of $\rho_0\beta\epsilon$ if the pair potential can be written as $u(x_{12})=\epsilon\psi(x_{12})$ – see the discussion around Eq.  – is true for the GEM-4 system. As a consequence, having calculated the coexisting densities for a particular value of $\rho_0\beta\epsilon$, the linear stability threshold, these results can be scaled to give the phase diagram in the full average density $1+\bar{n}$ versus dimensionless temperature $k_BT/\epsilon$ plane, which is one of the usual ways the GEM-4 phase diagram is displayed [@Mladek2006; @Mladek2007; @Prestipino2014; @Archer2014; @Archer2016]. The phase diagrams obtained from doing this are in Fig. \[fig:phase\_diagrams\]. In Fig. \[fig:phase\_diagrams\](a) we display the phase diagram obtained from DDFT-3, which is identical (to within the resolution of the calculations) to that previously calculated in Refs. [@Archer2014; @Archer2016]. For example, when $\beta\epsilon=1$, the average densities $\rho_0(1+\bar{n})R^2$ of the coexisting liquid and the crystal are $5.41$ and $5.68$, respectively. As a result of the scaling behaviour, the coexisting densities (binodals) are two straight lines going from the origin and passing through these two points. In Fig. \[fig:phase\_diagrams\](b) we display the phase diagram obtained for the DDFT-5. The binodals are a little closer to the linear stability threshold line than for DDFT-3, but other than that, it looks similar overall. Note however that the up hexagon branch cannot be continued beyond $\mu\approx0.28$ (where the minimum density goes to zero): this line is indicated as the ‘limit of validity’. Beyond this line, in the bottom right region of the phase diagram, there is no up hexagon solution to the equations, for the reasons discussed in Sec. \[sec:4\]. In Fig. \[fig:phase\_diagrams\](c) we display the phase diagram for  and in (d) for . The binodals almost overlie each other, so the predicted difference between the average densities of the liquid and the crystal at coexistence are much smaller than that predicted by DDFT-3 and DDFT-5. Furthermore, on moving to higher average densities or to lower temperatures $k_BT/\epsilon$ one encounters the stripe phase, followed by the down hexagon phase and then finally the uniform liquid becomes stable again. The prediction of the occurrence of these later phases is of course wrong, signifying a breakdown in the accuracy of the PFC theory at even the qualitative level. Before finishing this section, we note that it is possible to extend the gradient expansion in (\[eq:cLgradGEM4\]) by including higher powers of the Laplacian. For example, Ref. [@Jaatinen2009] proposed an eighth-order fitting (EOF), which in our notation is $${{{\cal L}_{\text{grad-8}}}}n({{\bm x}}) = - \gamma(1+\nabla^2)^2 n({{\bm x}}) - E_B (1+\nabla^2)^4 n({{\bm x}}), \label{eq:cLgradJaatinen}$$ where $\gamma$ fits the curvature of the dispersion relation as before, and $E_B$ allows the eigenvalue $\sigma(0)$ of ${{\cal L}}$ to be matched as well, i.e., allows the model to match correctly the isothermal compressibility $\chi_T$. An example of the dispersion relation for this operator is shown as a dotted line in Fig. \[fig:dispersionGEM4\]. This EOF version of the theory, with ${{{\cal L}_{\text{grad-8}}}}$ in (\[eq:cLgradJaatinen\]), improves over the standard version, with (\[eq:cLgradGEM4\]), since $\sigma(0)$ for ${{\cal L}}$ and ${{{\cal L}_{\text{grad-8}}}}$ are the same. Therefore, the liquid properties of DDFT-5 match those of DDFT-3, and the liquid properties of  match those of , once ${{{\cal L}_{\text{grad}}}}$ is replaced by ${{{\cal L}_{\text{grad-8}}}}$. However, the drawbacks of the gradient expansion are still present. With ${{{\cal L}_{\text{grad-8}}}}$, the values of $\mu$ at which the DDFT-5 stripe and hexagon densities go to zero are larger, but this undesirable feature is only deferred, not eliminated. The reason is that the singularity in the logarithm is now balanced against an eighth-order derivative. Note too that introducing even higher derivatives does not cure this problem, it just pushes the singularity to higher order. In addition, with ${{{\cal L}_{\text{grad-8}}}}$ the second liquid spinodal in the model is still present, it is just pushed to higher values of $\mu$ (similar to the value for ) and since this second spinodal is present, there is still a range of values of $\mu$ for which stripes have the lowest specific $\Omega$. Effect of the approximations {#sec:4} ============================ The qualitative change in going from a DDFT to a PFC model (dropping the $\nabla\cdot\left[n\nabla{{\cal L}}{n}\right]$ term, assuming constant mobility and expanding the logarithm) is apparent in the phase diagrams shown in Fig. \[fig:phase\_diagrams\], comparing (a,b) to (c,d). Here we discuss in detail additional effects of the approximations. Expanding the logarithm ----------------------- The effect of expanding the logarithm as in Eq. (\[eq:expandlogarithm\]) is very significant. In Sec. \[sec:3\] we demonstrated that this expansion leads to the liquid having a second spinodal point, illustrated in the phase diagrams in Fig. \[fig:phase\_diagrams\](c,d), with the crystal re-melting as the density is increased. The reason for this is that Eq. (\[eq:PFCdeltan\]) can be solved with $n=0$ and $n=1$, while Eq. (\[eq:DDFTdeltan\]) is only solved by $n=0$, with $\sigma(k)=0$ in both cases. Intimately connected to the existence of this second spinodal is the transition from stable up hexagons at the $\mu=0$ spinodal to stable down hexagons at the higher density spinodal. These connections to the spinodals mean that the free energy of the up hexagons increases again compared to the liquid state free energy as the chemical potential is increased, in order to reconnect to the liquid state at the upper spinodal. An intermediate region of stable stripes is not inevitable, but is evident in both PFC examples in Fig. \[fig:DDFTPFCrelativeOmega\](c,d). Taking more terms in the expansion in Eq. (\[eq:expandlogarithm\]) does not help. The highest power should be even (otherwise the free energy is not bounded below), and the improved versions of Eq. (\[eq:PFCdeltan\]), which involves the second derivative of Eq. (\[eq:expandlogarithm\]) with respect to $n$, also have $n=0$ and $n=1$ as (the only) real roots, regardless of how many terms are kept in the expansion of the logarithm. [The exception is if only terms up to $n^2$ are kept in (\[eq:expandlogarithm\]); in this case, $c^{(3)}$ and $c^{(4)}$ (if they are non-zero) provide the stabilizing nonlinearities and may also lead to a spurious spinodal.]{} to to Gradient expansion of ${{\cal L}}$ ---------------------------------- As discussed in Sec. \[sec:3\] above, the results in Fig. \[fig:DDFTPFCsummary\](d,e) suggest that for certain values of $\mu$, the DDFT-5 equation (\[eq:betadFdn5\_rewritten\]) has solutions for the density $1+n$ which go to zero at certain places. In contrast, the density in the PFC models appears to stay away from zero (although there is no reason for it to do so and there would be no singularity if it did), and in DDFT-3, the density minimum gets smaller and smaller as $\mu$ increases, but remains positive, without the sharp cutoff seen in DDFT-5. In this section we argue that the density reaching zero is not an artefact of numerical difficulties, rather it is a feature of the DDFT-5 equation (\[eq:betadFdn5\_rewritten\]). [Here, we focus on singularities in the solution, not on stability. Our discussion in this section is mainly framed in terms of the stripe solution, which is unstable, but the stable up-hexagon branch has similar issues, as is illustrated in Fig. \[fig:DDFTPFCsummary\](e).]{} Figure \[figApp:DDFT5singularity\](a) shows that even close to the end of the branch of DDFT-5 stripes, the density profile $1+n$ remains smooth, but since its minimum at $x=x_{\rm min}$ is very close to zero ($1+n(x_{\rm min})\approx5\times10^{-7}$), therefore the logarithm $\log(1+n)$ is sharply spiked towards large negative values at $x_{\rm min}$. Writing out the terms in (\[eq:betadFdn5\_rewritten\]) for a density profile only varying in the $x$-direction, we have: $$\log\left(1+n\right) + (\gamma-1) n + \gamma n_{xx} + \gamma n_{xxxx} - \mu = 0, \label{eq:betadFdn5_rewrittenagain}$$ which suggests that the only way to balance a large negative contribution from $\log(1+n)$ is to have a large positive $\gamma{n_{xxxx}}$. Figure \[figApp:DDFT5singularity\](b) shows that these two terms (solid lines with markers at the grid points) do indeed balance each other. The figure also shows that the other terms in (\[eq:betadFdn5\_rewrittenagain\]) are well behaved and that the equation is satisfied at each grid point. Therefore this singularity is not a numerical artefact, but rather a genuine feature of DDFT-5 stripes: the minimum density goes to zero at a certain finite value of $\mu$. In Fig. \[figApp:DDFT5singularity\](c,d) we show for comparison results from DDFT-3: the stripes have sharply peaked density maxima and density minima just as small as in DDFT-5, but all terms in Eq. (\[eq:betadFdn3\_rewritten\]) (Fig. \[figApp:DDFT5singularity\]d) are well behaved. As $\mu$ is further increased, the minimum of the density in DDFT-5 gets closer to zero, so the logarithm of the density goes further towards $-\infty$ and correspondingly $\gamma{n_{xxxx}}$ goes towards $+\infty$. Fig. \[fig:DDFTPFCsummary\](d) shows that the density minimum gets to zero at a finite value of $\mu$. We have not been able to develop a consistent asymptotic approximation for this limit in the DDFT-5 equation. However, to illustrate that apparently smooth solutions with logarithmic singularities in their fourth derivatives can easily be found, consider for example taking $\gamma=1$ in (\[eq:betadFdn5\_rewrittenagain\]) and taking a density profile that has a quadratic minimum at $x=x_{\rm min}=0$: $$1 + n(x) = Ax^2 + Bx^4 + Cx^4\log(x^2), \label{eqApp:densityexpansion}$$ where $A$, $B$ and $C$ are constants. For small $x$, the largest of these three terms is $Ax^2$, so $\log(1+n)\approx\log(Ax^2)$, which goes to $-\infty$ as $x\rightarrow0$. The other terms are $n_{xx}\approx2A+{{\cal O}}(x^2,x^2\log(x^2))$ and $n_{xxxx}\approx24C\log(x^2) + 24B + 100C$. Adding these three together requires $1+24C=0$ to cancel the logarithmic singularity at $x=0$, and the remaining terms are constants or go to zero as $x\rightarrow0$. As can be seen from Fig. \[figApp:DDFT5singularity\], having an adequate resolution for our numerical calculations was a challenge but for different reasons for the different models. In DDFT-3, the density maxima can be sharply peaked while the logarithm of the density is smooth, so inadequate resolution in the density field prompts an increase in the number of grid points (we implement automatic regridding, as discussed in Appendix \[app:continuation\]). In contrast, in DDFT-5, the density field can be smooth but with minima very close to zero, so its logarithm has very sharp negative peaks. In this case, inadequate resolution in the logarithm of the density prompts regridding. The difference is that in DDFT-5, the equation involves derivatives so any problem is magnified, while in DDFT-3, the equation involves convolutions that smooth out any problems. These arguments indicate that a singular solution to the DDFT-5 equation (\[eq:betadFdn5\_rewritten\]) of the type seen in Fig. \[figApp:DDFT5singularity\] is possible, with the density going to zero, and that this is not a problem of inadequate numerical resolution, but rather a consequence of replacing the convolution in DDFT-3 with derivatives. A full asymptotic theory should result in a prediction for the value of $\mu$ at which the branch terminates. The stripe solutions of the DDFT-3 equation (\[eq:betadFdn3\_rewritten\]) can also have small density but without any singularity in the solution. One mode approximation for DDFT {#sec:5} =============================== The data displayed in Fig. \[figApp:DDFT5singularity\](a,c) lead to an interesting observation: in (a) DDFT-5, the logarithm of the density is sharply negatively peaked, while the density is smooth (at least up to its second derivative), slowly varying and resembles a cosine. In contrast, in (c) DDFT-3, the density is sharply peaked, while the logarithm of the density is slowly varying and resembles a cosine. One of the attractions of PFC theory is that it has slowly varying solutions that are well represented by a few Fourier modes [@Elder2002; @Elder2004; @Emmerich2012; @Wu2010], and this carries over to some extent to DDFT-5. Such Fourier representations of the density profiles in DDFT-3 are unsatisfactory, apart from for the unstable solutions very close to the spinodal point, since any solution of reasonable amplitude is sharply peaked. to However, the data in Fig. \[figApp:DDFT5singularity\] suggest that representing the *logarithm* of the density with a few Fourier modes should work well in DDFT-3. In this section, we elaborate how such a theory can be developed. The key is to write Eq. (\[eq:betadFdn3\_rewritten\]) in terms of $\phi({{\bm x}})\equiv\log(1+n({{\bm x}}))$, and approximate $\phi$ by a few Fourier modes. As a first step, we write out ${{\cal L}}$ using Eq. (\[eq:defnL\_GEM4\]) and re-write (\[eq:betadFdn3\_rewritten\]) as: $$\log(1+n({{\bm x}})) + \rho_0\beta\epsilon \!\!\int\! \psi(|{{\bm x}}-{{\bm x}}_2|) n({{\bm x}}_2){{{\rm d}}{{\bm x}}}_2 - \mu = 0, \label{eq:betadFdn3_rewrittenagain}$$ where $\psi(|{{\bm x}}-{{\bm x}}_2|)=e^{-|{{\bm x}}-{{\bm x}}_2|^4/R^4}$. The convolution term (including the $\rho_0\beta\epsilon$ prefactor) in this equation is $-n({{\bm x}})-{{\cal L}}n({{\bm x}})$. We know the eigenvalues of ${{\cal L}}$: ${{\cal L}}\exp(ikx)=\sigma(k)\exp(ikx)$, which means that the convolution term acting on a Fourier mode $\exp(ikx)$ has eigenvalue $-(1+\sigma(k))$. We also know that for high wavenumbers, the convolution averages to zero, and indeed $\sigma(k)\rightarrow-1$, as can be seen in Fig. \[fig:dispersionGEM4\]. We focus first on stripes, which have Fourier components only at integer wavenumbers, and notice that $1+\sigma(2)$ is already very small (less than $0.01$), because $\hat{\psi}(2)$ is small. This implies that the Fourier components of the convolution term at $k=2$ and higher will be much smaller than the Fourier components at $k=0$ and $k=1$ – regardless of the spectrum of $n$ itself. The other two terms in Eq. (\[eq:betadFdn3\_rewrittenagain\]) are $\mu$, which is constant ($k=0$ only), and $\log(1+n)$, so $\log(1+n)$ can only have significant Fourier components at $k=0$ and $k=1$: there is nothing to balance modes with $|k|\geq2$. This explains why $\log(1+n)$ in the lower left panel of Fig. \[figApp:DDFT5singularity\] is smooth, and it is also why approximating the logarithm of the density by a few Fourier modes must work regardless of the amplitude of the modulations in the density, or how sharply they are spiked, or of the value of $\mu$. For the stripe phase we write $$\log(1+n(x)) = \phi(x) = \phi_0 + \phi_1 e^{ix} + {\bar \phi}_1 e^{-ix}, \label{eq:logdensityFourier}$$ where $\phi_0$ and $\phi_1$ are constants (real and complex, respectively) that we need to find. This can easily be generalised for hexagons and other periodic phases by adding more modes in . The $k=0$ and $k=1$ components of $\exp\left(\phi_1e^{ix}+\text{c.c.}\right)$, where $\text{c.c.}$ denotes the complex conjugate, can be expressed in terms of integrals, defining two functions $f_0(\phi_1)$ and $f_1(\phi_1)$ given by $$\begin{split} f_0(\phi_1) &= \frac{1}{2\pi}\int_0^{2\pi} \exp\left(\phi_1e^{ix}+\text{c.c.}\right){{\rm d}}{}x, \\ f_1(\phi_1) &= \frac{1}{2\pi}\int_0^{2\pi} e^{-ix}\exp\left(\phi_1e^{ix}+\text{c.c.}\right){{\rm d}}{}x, \label{eq:f0f1defn} \end{split}$$ i.e., $f_0$ is modified Bessel function of the first kind of order zero and $f_1$ is a Fourier transform generalisation of $f_0$. Using these functions, $n(x)=e^{\phi(x)}-1$ can be written in terms of its Fourier components as $$\begin{split} e^{\phi(x)}-1 &= e^{\phi_0}f_0(\phi_1) - 1\\ &\quad{} + \left(e^{\phi_0}f_1(\phi_1)e^{ix} + \text{c.c.}\right)\\ &\quad{} + \text{modes with $|k|\geq2$.} \label{eq:densityFourier} \end{split}$$ The modes with $|k|\geq2$ in (\[eq:densityFourier\]) are large in amplitude, but they are reduced in significance in Eq. (\[eq:betadFdn3\_rewrittenagain\]) by the convolution, as explained above. The action of the convolution on modes with $|k|<2$ can be written in terms of $\sigma(k)$. Retaining only these terms, we are left with the $k=0$ and $k=1$ components of (\[eq:betadFdn3\_rewrittenagain\]): $$\begin{split} \phi_0 + (1+\sigma(0))\left(1-e^{\phi_0}f_0(\phi_1)\right) - \mu &= 0,\\ \phi_1 - e^{\phi_0}f_1(\phi_1)(1+\sigma(1)) &= 0. \label{eq:DDFT3onewaveeqns} \end{split}$$ Notice that the only information remaining from the GEM-4 potential is the values of $\sigma(0)$ and $\sigma(1)$, i.e., the values of $\hat{u}(0)$ and $\hat{u}(1)$. Recall also that if the reference density $\rho_0$ is chosen to be the value at the spinodal, then we have $\sigma(1)=0$. These equations can also be written in terms of the pair potential as $$\begin{split} \phi_0 - \rho_0\beta\hat{u}(0)\left(1-e^{\phi_0}f_0(\phi_1)\right) - \mu &= 0,\\ \phi_1 + e^{\phi_0}f_1(\phi_1)\rho_0\beta\hat{u}(1) &= 0. \label{eq:DDFT3onewaveeqns_u} \end{split}$$ The two equations in (\[eq:DDFT3onewaveeqns\]) can easily be solved for $\phi_0$ and $\mu$ in terms of $\phi_1$, from which the density can be reconstructed. The agreement between this and the full solutions of DDFT-3 is astonishing. Figure. \[figApp:DDFT3onemodeapprox\](a) shows a 1D example at $\mu=28.9$, with the full solution as a black line and the approximate solution as a dashed line. Even though the density varies by two orders of magnitude, the two are almost indistinguishable. There is similar excellent agreement with the branch of stripe solutions (Fig. \[figApp:DDFT3onemodeapprox\]b) in 2D DDFT-3 (recall that the GEM-4 potential has different values of $\sigma(0)$ in 1D and 2D). For 2D hexagons, the approach is similar, with the two $e^{\pm{ix}}$ terms in Eq. (\[eq:f0f1defn\]) replaced by six similar terms, with wavenumbers ${{\bm k}}$ that are uniformly spaced around a circle of radius 1 in ${{\bm k}}$-space (c.f. Eq.  in Appendix B). The agreement in this case (Fig. \[figApp:DDFT3onemodeapprox\]c) is also very good. If one adds a further six modes with $|{{\bm k}}|=\sqrt{3}$, then the agreement is as good as that for stripes. These approximate solutions can easily be continued up to $\mu=100$ without difficulties, where we observe very sharply peaked density maxima and extremely small but non-zero values for the density minimum. Discussion and conclusions {#sec:6} ========================== In this paper, starting from DDFT, we have presented a step-by-step derivation of PFC theory, at each stage explaining the consequences of the approximations. The approximations can be listed under three main groupings: (i) making a truncated functional Taylor expansion approximation for the excess Helmholtz free energy, and then making the RPA/RY approximation. This leads to DDFT-3 in our classification. (ii) Neglecting the $\nabla\cdot[n\nabla{{\cal L}}n]$ term, which effectively also forces making a Taylor expansion of the logarithmic ideal gas term and assuming constant mobility. (iii) Replacing the nonlocal convolution in ${{\cal L}}$ with a local gradient expansion. The consequence of (ii) is to introduce a second spinodal into the phase diagram and to significantly alter the relative stabilities of the different periodic states, in particular making striped states to become an equilibrium phase for some state points, which is contrary to the physics. The consequence of making (iii) without first making (ii), is to generate a theory (DDFT-5) that has a no-solution region in the phase diagram, such as that displayed for the GEM-4 model in Fig. \[fig:phase\_diagrams\](b). All these consequences have been illustrated for the GEM-4 system, chosen because DDFT-3 is fairly accurate for this model for temperatures $k_BT/\epsilon>0.1$, allowing us to see the influence of the subsequent approximations. [Throughout, there is good quantitative agreement between DDFT-3,  and the EOF versions of DDFT-5 and  (data not shown) *only* for unstable small amplitude solutions close to the spinodal point. The region of quantitative agreement agreement between DDFT-3 and  is circled in red in Fig. \[fig:DDFTPFCsummary\](b,h). Beyond this region, the agreement between the four theories is at best qualitative.]{} Given all these problematic consequences for PFC theory, especially the issues related to Taylor expanding the logarithmic ideal gas term, it raises the question of why then is PFC theory so successful? In our view, there are several reasons for this. The first reason is that PFC theory is qualitatively correct near to the spinodal. Therefore, it can satisfactorily describe the coexistence between the liquid and the crystal phase, which is often an important aspect in applications of the theory. Second, despite the approximations, PFC theory still incorporates some very important physics: (i) the free energy satisfies the correct symmetries, (ii) the dynamical equation gives a time evolution that decreases the free energy monotonically over time and (iii) the current is proportional to the gradient of the chemical potential. These are all important features for describing many phenomena. Also, many of the features that PFC theory is used to describe are generic, and the model parameters can be scaled to fit (for example) iron [@Jaatinen2009] and graphene [@Fan2017], but could equally well be scaled to match other materials, with similar good agreement. This universality (having the correct symmetries etc.) underlines the importance of PFC theory as a powerful model of generic features of crystalization, but it means that PFC theory will in general not be able to predict any unusual (non-generic) behaviour. Our results in Sec. \[sec:5\] for the GEM-4 system showing that one can derive a very accurate one-mode amplitude equation approximation for the field $\phi({{\bm x}})=\log(\rho({{\bm x}})/\rho_0)$, rather than the density itself, gives a tantalising hint as to how PFC-type theories may more properly be derived and what the order parameter field in PFC theory really represents: Should we consider the PFC order parameter to be a scaled logarithm of the density distribution or some other similar function of the density, rather than being proportional to the density profile itself? On the basis of the work presented here, the answer to this question is ‘probably yes’, but clearly more work is required to fully address this. Returning to theories for the density profile, in our view it is preferable to retain the logarithmic ideal gas term in the approximation used for the Helmholtz free energy functional, since this is required to have physical (i.e., positive) density profiles, and also because with this the DDFT dynamics leads to the (correct physics) linear diffusion equation term – see Eq. . The difficult consequence of retaining this in the free energy is that one then has to deal with terms in the dynamical equation of the form $\nabla\cdot[n\nabla{{\cal L}}n]$, which makes solving numerically far more difficult. However, this term also contributes to making the crystalline phase more stable than the stripe phase, which also makes it important. The crucial contribution of this term can especially be seen in DDFT-3, since in this version of the theory it is clearly required for stabilizing the crystal structure. [In general, we would advocate using one of DDFT-1, DDFT-2 or DDFT-3 (depending on how ${{\cal F_{\text{ex}}}}$ is treated) over all existing PFC theories, for studying the properties of real materials.]{} It is worth noting that in all the approximations made here, we only consider those that retain the form of a dynamics that decreases the free energy monotonically over time – see Eq. . This is a feature of both DDFT and PFC theory. In our view this structure is important and should not be broken by any approximations made, i.e., any that might be made in future in attempting to avoid any of the above mentioned issues. It is also worth noting that, while we have not discussed the consequences of the LDA in going from DDFT-1 to DDFT-2 (see Eq. (\[eq:definecs\])), the polynomial terms in the chemical potential (\[eq:betadFdn2\]) can potentially lead to the same problem of having a second spinodal, even while retaining the logarithm term. This applies to DDFT-4 as well. In this paper we have largely focussed on making our arguments in two dimensions, in order to keep the presentation as simple and comprehensible as possible. However, we should emphasis that all of our arguments apply for three dimensional (3D) systems. For example, at the higher temperatures we have focussed on here, the 3D GEM-4 model exhibits at equilibrium a single fluid phase and two crystalline phases: the body-centered cubic phase at lower densities and the face-centered cubic structure at higher densities. These are all accurately predicted by the 3D version of DDFT-3 [@Mladek2006]. There are no columnar or lamellar phases, which are the 3D equivalents of the stripe phase. On the other hand, the 3D versions of the PFC theories presented here all predict a lamellar phase at some state points [@Thiele2013], the 3D generalisations of the down hexagons and a second spinodal with the uniform liquid becoming the equilibrium phase at higher densities. [It would be interesting to explore how the dynamics of defects, the elasticity and the plasticity of crystals differs between DDFT and ]{}, and our results are also relevant to binary systems. In the derivation in Ref. [@Huang2010a] of a PFC theory for binary mixtures, the generalisation of the $\nabla\cdot[n\nabla{{\cal L}}n]$ term is retained until the last moment in the derivation, but then dropped for the same reasons that is is neglected for one-component systems. Given the importance of this term for one-component systems, it is surely also important for stabilizing crystal structures in binary systems. [Note also that when determining mechanical properties such as elastic constants, the terms in the free energy that are linear in $n$ can be important [@Wang2018b]. These have been neglected here throughout since such terms do not contribute to determining density profiles.]{} The singularity observed for DDFT-5 as the chemical potential $\mu$ (or equivalently, the average density $1+\bar{n}$) is increased was found by continuing equilibrium solutions determined at lower values of $\mu$. One aspect that needs further investigation relates to determining the influence of this when DDFT-5 is solved for state points where the final equilibrium crystal (or stripe) solution for the density profile does not exist. For example, a situation we have in mind is that studied by van Teeffelen  [@Teeffelen2009] consisting of a solidification front propagating into the unstable liquid. These authors compared results for this situation between (in our terminology) DDFT-3, DDFT-5 and . Their results are for two-dimensional dipolar colloidal particles. From the DDFT-5 results (PFC1 in their terminology) displayed in Figs. 4 and 5 of Ref. [@Teeffelen2009], it can be seen that they did not consider values of the coupling parameter large enough to encounter any of the singularities; the density profiles stay well away from zero. It would be interesting to quench deeper into the crystal phase to study the evolution of the density distribution towards the singular state. However, the numerics to resolve this accurately would surely be difficult. One aspect of PFC theory that the derivation from DDFT highlights is that in general one is not free to independently vary the parameters $r$ and $\bar{n}$ in Eqs.  and . For example, for the GEM-4 model there are certain values of $r$ that are not generated by the mapping from DDFT. Of course, by changing the pair potential, different combinations of the PFC model parameters can become accessible. We should also recall that although we have illustrated many of our conclusions by considering the soft-core GEM-4 model, PFC theory can be derived for systems of particles with hard potentials since it is the pair direct correlation function $c^{(2)}({{\bm x}}_1,{{\bm x}}_2)$ that enters the theory; this quantity is finite for all values of ${{\bm x}}_1$ and ${{\bm x}}_2$. As a final point, we mention that our results will also be of interest to the pure mathematics community. DDFT-3 is also referred to as the McKean–Vlasov equation and in this context there are a number of recent interesting rigorous results [@Carrillo2018; @Gomes2019]. Our results for DDFT-5, showing that for a finite value of $\mu$ there is a singularity with the density profile going to zero, may well be of interest to those who study the mathematics of solutions to partial differential equations with compact support – see for example Ref. [@Bernis1992]. For values of $\mu$ beyond the singular point where $1+n({{\bm x}})\to0$ it may be that the solutions become complex. If one were interested to find these solutions, we believe it might require treating $\mu$ as a complex variable. Of course, all of this is out of the realm where the model represents a theory for matter. This work was supported in part by a L’Or[é]{}al UK and Ireland Fellowship for Women in Science (PS), by the EPSRC under grants EP/P015689/1 (AJA, DR) and , and by the Leverhulme Trust (RF-2018-449/9, AMR). We are grateful for conversations with Tapio Ala-Nissilä, Daniele Avitabile, Ken Elder, Zhi-Feng Huang, Kai Jiang, Edgar Knobloch, Ron Lifshitz, Chris Malcotte, Daniel Read, Uwe Thiele, Steve Tobias, Gyula Tóth, Laurette Tuckerman and Joanna Tumelty. We are grateful also for constructive comments from two anonymous referees. Linear theory for GEM-4 {#app:LinearGEM4} ======================= In this appendix, we discuss how we compute the linear theory for the GEM-4 potential in (\[eq:GEM4Potential\]). To be specific, in a two-dimensional periodic domain, the eigenvalue $\sigma(k)$ is defined by ${{\cal L}}e^{i{{\bm k}}\cdot{{\bm x}}} = \sigma(k) e^{i{{\bm k}}\cdot{{\bm x}}}$ and (\[eq:defnL\_GEM4\]), which can be written as $$\sigma(k) = - 1 - \rho_0\beta\epsilon \!\!\int\! e^{-|{{\bm x}}-{{\bm x}}_2|^4/R^4} e^{i{{\bm k}}\cdot({{\bm x}}_2-{{\bm x}})}{{{\rm d}}{{\bm x}}}_2, \label{eqApp:defn_sigmak_GEM4}$$ where the integral is taken over the periodic domain (the GEM-4 potential is replaced by its periodic extension). We set ${{\bm k}}=(k,0)$, we integrate from $[-N\pi/k,N\pi/k]$ in each dimension, and we choose the integer $N$ large enough that the GEM-4 exponential is effectively zero at the boundaries; $N=4$ suffices. We then scale ${{\bm x}}$ by a factor of $k$, replacing ${{\bm x}}_2-{{\bm x}}$ by ${{\bm x}}/k$, so that the integral becomes: $$\sigma(k) = - 1 - \rho_0\beta\epsilon \!\!\iint_{-N\pi}^{N\pi}\! e^{-|{{\bm x}}|^4/(k^4R^4)} e^{ix}\frac{{{{\rm d}}{{\bm x}}}}{k^2}, \label{eqApp:defn_sigmak_GEM4_scaled}$$ where $x$ is the first component of ${{\bm x}}$. With this scaling, the limits of the integral do not depend on $k$. -------------- -------- ----------------------- ---------- ------------- -------------------- Dimension $R$ $\rho_0\beta\epsilon$ $\gamma$ $\sigma(0)$ $E_B$ \[0.5ex\] \[-1.5ex\] 1 4.5918 1.1629 3.2051 $-10.680$ $\phantom{0}7.475$ 2 5.0962 0.2455 4.3692 $-18.752$ $14.383$ 3 5.5719 0.0455 5.6889 $-31.305$ $25.616$ -------------- -------- ----------------------- ---------- ------------- -------------------- : Linear theory for the GEM-4 potential in one, two and three dimensions. Solving $\frac{{{\rm d}}\sigma}{{{\rm d}}{k}}(1)=0$ and $\sigma(1)=0$ gives $R$ and $\rho_0\beta\epsilon$, while $\gamma$ and $\sigma(0)$ are computed from $\frac{{{\rm d}}^2\sigma}{{{\rm d}}{k^2}}(1)$ and (\[eqApp:defn\_sigmak\_GEM4\]) respectively. We also give the values of $E_B$ for the EOF in Eq. .[]{data-label="tabApp:GEM4parameters"} We choose $R$ and $\rho_0\beta\epsilon$ so that $\sigma(1)=0$ and $\frac{{{\rm d}}\sigma}{{{\rm d}}{k}}(1)=0$. The derivative of $\sigma$ with respect to $k$ is $$\frac{{{\rm d}}\sigma}{{{\rm d}}{k}} = - \rho_0\beta\epsilon \!\!\iint_{-N\pi}^{N\pi}\! \left(\frac{4|{{\bm x}}|^4 - 2k^4R^4}{k^7R^4}\right) e^{-|{{\bm x}}|^4/(k^4R^4)} e^{ix}{{{\rm d}}{{\bm x}}}. \label{eqApp:defn_sigmak_GEM4_derivative}$$ Evaluating this at $k=1$ and removing the constant factor outside the integral gives a function $F(R)$: $$F(R) = \iint_{-N\pi}^{N\pi}\! \left(\frac{4|{{\bm x}}|^4 - 2R^4}{R^4}\right) e^{-|{{\bm x}}|^4/R^4} e^{ix}{{{\rm d}}{{\bm x}}}. \label{eqApp:GEM4_FRs}$$ We solve the equation $F(R)=0$ using Newton’s method to give $R$. We then calculate $\rho_0\beta\epsilon$ by requiring that $\sigma(1)=0$ in (\[eqApp:defn\_sigmak\_GEM4\_scaled\]). Values for $R$ and $\rho_0\beta\epsilon$ in one, two and three dimensions are given in table \[tabApp:GEM4parameters\]. We compute the GEM-4 dispersion relation in Fig. \[fig:dispersionGEM4\] in a similar way, and use the second derivative of (\[eqApp:defn\_sigmak\_GEM4\_scaled\]) with respect to $k$, at $k=1$, to find the $\gamma$ parameter (also given in table \[tabApp:GEM4parameters\]) in ${{{\cal L}_{\text{grad}}}}$. The table also gives $\sigma(0)$, since this is useful for computing properties of the liquid, as well as $E_B$, the coefficient in the eighth-order model of [@Jaatinen2009], given in Eq. (\[eq:cLgradJaatinen\]). Numerical method: continuation {#app:continuation} ============================== We use numerical continuation to solve the four equations (\[eq:betadFdn3\_rewritten\]) and (\[eq:betadFdn5\_rewritten\])–(\[eq:betadFdnepsilon\_rewritten\]) for $n({{\bm x}})$ as the parameters vary. Our approach is based on [@Doedel1991] for the pseudo-arclength continuation method, and we use the approach advocated by [@Kelley2003] to solve the large linear systems at each Newton step. The main parameters are the chemical potential $\mu$, the parameters in the linear operators ${{\cal L}}$ and ${{{\cal L}_{\text{grad}}}}$, and the domain size. In this discussion, we focus on $\mu$ as the parameter that is varied. Pseudo-arclength continuation ----------------------------- The main idea behind pseudo-arclength continuation is to suppose that we are looking to calculate a branch of solutions $n({{\bm x}})$ depending on the parameter $\mu$. The branch may have folds (as in Fig. \[fig:DDFTPFCsummary\]), and the method should be able to go around these. The method defines a parameter (the arclength $s$) that increases or decreases monotonically along the branch (including its folds) such that both $n({{\bm x}})$ and $\mu$ can be regarded as single-valued functions of $s$. Then the equation to be solved is $${{\cal G}}(n(s),\mu(s)) = 0, \label{eqApp:equationtosolve}$$ where ${{\cal G}}$ represents the equation we are solving for $n$. Instead of thinking of $n$ as being a function of position ${{\bm x}}$, we represent $n$ as a series of values $n_i$ on $N$ grid points ${{\bm x}}_i$ ($i=1,\dots,N$), so $n$ is now a vector in $\mathbb{R}^N$, and ${{\cal G}}$ is a function from $\mathbb{R}^{N+1}\rightarrow\mathbb{R}^{N}$. Equation (\[eqApp:equationtosolve\]) represents $N$ equations for $N+1$ unknowns, and so it is supplemented by an orthogonality condition, that the next point on a branch should lie in a plane orthogonal to a line connecting the two previous points. It is this that allows the branch following technique to go around folds. If we have two points on the branch $(n(s),\mu(s))$ at $s_0$ and $s_1$, then we take the derivatives of $n$ and $\mu$ with respect to the arclength to be approximately $$\frac{{{\rm d}}{n}}{{{\rm d}}{s}} = S \frac{n(s_1)-n(s_0)}{s_1-s_0}, \quad \frac{{{\rm d}}\mu}{{{\rm d}}{s}} = S \frac{\mu(s_1)-\mu(s_0)}{s_1-s_0}, \label{eqApp:pseudoarclengthstep}$$ with the scaling factor $S$ chosen so as to satisfy $$\frac{1}{N}\sum_{i=1}^N \left(\frac{{{\rm d}}{n_i}}{{{\rm d}}{s}}\right)^2 + \left(\frac{{{\rm d}}\mu}{{{\rm d}}{s}}\right)^2 = 1. \label{eqApp:arclengthnormalization}$$ The $\frac{1}{N}$ prefactor means that the parameterization of the branch by the arclength is essentially independent of the number of grid points. The method then proceeds in a predictor–corrector fashion. The predictor step, with a target stepsize $\Delta{s}$ provides $(n_2,\mu_2)$: $$n_2 = n(s_1) + \Delta{s}\frac{{{\rm d}}{n}}{{{\rm d}}{s}}, \quad \mu_2 = \mu(s_1) + \Delta{s}\frac{{{\rm d}}\mu}{{{\rm d}}{s}}. \label{eqApp:pseudoarclengthpredictor}$$ Then, $(n_2,\mu_2)$ is used as an initial iterate for a Newton solver for equation (\[eqApp:equationtosolve\]), supplemented by the condition that the Newton iterates lie in a plane orthogonal to the line given in (\[eqApp:pseudoarclengthpredictor\]), parameterised by $\Delta{s}$. This means that we are solving ${{\cal H}}(n,\mu)=0$, where ${{\cal H}}$ is $\mathbb{R}^{N+1}\rightarrow\mathbb{R}^{N+1}$, with the first $N$ equations in ${{\cal H}}$ being the same as ${{\cal G}}$, and the last equation being $$(n - n_2) \cdot \frac{{{\rm d}}{n}}{{{\rm d}}{s}} + (\mu - \mu_2) \frac{{{\rm d}}\mu}{{{\rm d}}{s}} = 0. \label{eqApp:pseudoarcorthogonality}$$ To be precise, we take $${{\cal H}}(n,\mu)=\begin{pmatrix} \frac{1}{\sqrt{N}} P\cdot {{\cal G}}(n,\mu)\\ \textrm{Eq.~(\ref{eqApp:pseudoarcorthogonality})} \end{pmatrix}, \label{eqApp:fullsystem}$$ where $P$ is a linear preconditioner for ${{\cal G}}$ (see below). The $\sqrt{N}$ scaling means that the norm $\lVert{{\cal H}}(n,\mu)\rVert$ (the square root of the sum of the squares of its components) is independent of the number of grid points $N$, and it also means that the equations in ${{\cal G}}$ and the orthogonality condition (\[eqApp:pseudoarcorthogonality\]) are given a similar weighting by the Newton method. Solving ${{\cal H}}(n,\mu)=0$ results in a new point on the branch of solutions, $(n(s_2),\mu(s_2))$, where $s_2$ is given by $$s_2 = s_1 + \frac{1}{N}(n(s_2)-n(s_1))\cdot\frac{{{\rm d}}{n}}{{{\rm d}}{s}} + (\mu(s_2)-\mu(s_1))\frac{{{\rm d}}\mu}{{{\rm d}}{s}}, \label{eqApp:newpseudoarc}$$ with $\frac{{{\rm d}}{n}}{{{\rm d}}{s}}$ and $\frac{{{\rm d}}\mu}{{{\rm d}}{s}}$ given by (\[eqApp:pseudoarclengthstep\]). This last equation comes from replacing $n_2$ by $n(s_2)$ and $\mu_2$ by $\mu(s_2)$ in (\[eqApp:pseudoarclengthpredictor\]) and finding a $\Delta{s}=s_2-s_1$ from $1/N$ times the first equation plus the second equation. This is not quite the actual change to the arclength that was achieved in the step, and the approximation is the reason that the method is called the pseudo-arclength method. Newton’s method --------------- For Newton’s method, we define $X=(n,\mu)$ and solve ${{\cal H}}(X)=0$. We start with $X_0$ given by the predictor step above in (\[eqApp:pseudoarclengthpredictor\]), and follow [@Kelley2003], at each step solving the linear equation $$\frac{\partial{{\cal H}}}{\partial{X}} \cdot {{\delta{\mkern-1mu}X}}_n = - {{\cal H}}(X_n), \label{eqApp:linearNewtonEqns}$$ where $\frac{\partial{{\cal H}}}{\partial{X}}$ is the $(N+1)\times(N+1)$ matrix of derivatives of ${{\cal H}}$ with respect to $X$, and then improving our estimate of the root by using ${{\delta{\mkern-1mu}X}}$. The Newton method proceeds until convergence, defined by $$\lVert{{\cal H}}(X_n)\rVert < N_\text{abs} + N_\text{rel}\lVert{{\cal H}}(X_0)\rVert,$$ where $N_\text{abs}$ and $N_\text{rel}$ are the Newton absolute and relative convergence tolerances respectively, typically $10^{-10}$ and $10^{-8}$. We also monitored the maximum of $|{{\cal H}}(n({{\bm x}}),\mu)|$ across the domain, and this was typically no larger than ten times $N_\text{abs}$, so the equations are well satisfied at each point in space as well as in norm. The linear equations in (\[eqApp:linearNewtonEqns\]) are solved to find ${{\delta{\mkern-1mu}X}}_n$ using <span style="font-variant:small-caps;">Matlab</span>’s biconjugate gradient stabilized (l) (`bicgstabl`) method. This allows the matrix–vector multiplications to be evaluated using a function (rather than by explicitly computing a large matrix). The method is iterative, and proceeds until $$\left\lVert\frac{\partial{{\cal H}}}{\partial{X}} \cdot {{\delta{\mkern-1mu}X}}_n + {{\cal H}}(X_n)\right\rVert < L_\text{rel}\lVert{{\cal H}}(X_n)\rVert,$$ where the relative tolerance $L_\text{rel}$ of the linear solver is chosen so as to balance the number of Newton steps against the number of `bicgstabl` iterations. Based on [@Kelley2003], we choose $$L_\text{rel} = 0.1\sqrt{\lVert{{\cal H}}(X_n)\rVert} + \frac{N_\text{abs}}{\lVert{{\cal H}}(X_n)\rVert},$$ subject to the constraint that $L_\text{rel}$ should be no larger than $0.1$. The effect of this is that in the initial first or second of the Newton iterations, when $\lVert{{\cal H}}(X_n)\rVert$ is at its largest, the linear solver is not asked to work too hard to solve (\[eqApp:linearNewtonEqns\]), since any reasonably good approximate solution is likely to improve the estimate of the root, and an absolutely perfect solution isn’t going to do much better. In the middle stages of the Newton iterations, when $\lVert{{\cal H}}(X_n)\rVert$ is about $10^{-6}$, the tolerance $L_\text{rel}$ is about $2\times10^{-4}$, which is not good enough for quadratic convergence of Newton’s method, but is good enough to provide two or three orders of magnitude improvement to the quality of the solution at a considerably lower cost. In the final stages of the Newton iterations, $L_\text{rel}=0.1$, good enough for polishing the solution to the tolerance $N_\text{abs}$ while not attempting to solve the linear problem down to round-off error. Also based on [@Kelley2003], we implement the Armijo rule, which ensures that each Newton step results in an improvement to the solution. The idea is that the solution of the linear equation (\[eqApp:linearNewtonEqns\]) should give the correct direction for improving the solution of ${{\cal H}}(X)=0$, but taking a full step may not actually result in an improvement, so instead we set $$X_{n+1} = X_n + 2^{-j} {{\delta{\mkern-1mu}X}}_n, \label{eqApp:armijo}$$ where $j=0,1,2,\dots$ is chosen to be the smallest such that $$\lVert{{\cal H}}(X_{n+1})\rVert < \lVert{{\cal H}}(X_{n})\rVert. \label{eqApp:armijocondition}$$ In most cases, the first ($j=0$) Armijo step satisfies (\[eqApp:armijocondition\]), equivalent to the normal Newton method, but when the density is close to zero in the DDFT calculations, and small changes in density lead to large changes in its logarithm, the Armijo rule is helpful. We do not use a preconditioner in the GEM-4 calculations (so $P$ in (\[eqApp:fullsystem\]) is the identity), but in the gradient expansion calculations, a preconditioner is helpful. In Fourier space, ${{{\cal L}_{\text{grad}}}}$ can easily be inverted, so the preconditioner is ${{{\cal L}_{\text{grad}}^{-1}}}$ when the absolute value of the eigenvalue $\sigma(k)$ is greater than $1$, otherwise the preconditioner is the identity. This has the effect of reducing the number of iterations needed to solve (\[eqApp:linearNewtonEqns\]) by a factor of 10 or even 100. A sample <span style="font-variant:small-caps;">Matlab</span> code to solve Eq. (\[eq:betadFdn3\_rewritten\]) for DDFT-3 by Newton’s method (without the continuation aspect) is given in the supplementary material. Additional considerations ------------------------- We start the computation of each branch close to $n=0$ and $\mu=0$ using an approximate solution derived from weakly nonlinear theory. For example, for hexagons in DDFT-3, we take $$n({{\bm x}}) = \frac{\mu}{\sigma(0)}\left(-1 + e^{i{{\bm k}}_1\cdot{{\bm x}}} + e^{i{{\bm k}}_2\cdot{{\bm x}}} + e^{i{{\bm k}}_3\cdot{{\bm x}}} + \text{c.c.}\right), \label{eqApp:weaklynonlinearhexagons}$$ where the initial value of $\mu$ is small, $\sigma(0)$ comes from Table \[tabApp:GEM4parameters\], ${{\bm k}}_1=(1,0)$, ${{\bm k}}_2=(-\frac{1}{2},\frac{\sqrt{3}}{2})$, ${{\bm k}}_3=(-\frac{1}{2},-\frac{\sqrt{3}}{2})$, and $\text{c.c.}$ stands for complex conjugate. The equations are posed on periodic domains and we use $N_x\times{N_y}$ grid points, depending on the number of wavelengths in the domain and the nature of the solution. The GEM-4 hexagonal calculations require $8\times\frac{8}{\sqrt{3}}$ wavelength domains, with resolutions starting at $80\times48$ Fourier modes close to onset. At larger amplitude, $512\times320$ Fourier modes (or even more) are needed, especially if the density maxima are sharply peaked (in DDFT-3) or if the density minima are very close to zero (in DDFT-5). In order to accommodate the changing needs for resolution along a branch, we monitor whether the solutions are well resolved and implemented automatic regridding, so as to maintain enough grid points to resolve the solution well, regardless of what features emerge as $\mu$ is varied. Typically we require that the amplitudes of the highest-wavenumber Fourier modes be no higher than $10^{-10}$ times the largest Fourier amplitude. We also implement automatic pseudo-arclength stepsize control: $\Delta{s}$ is increased by a factor of 1.1 (up to a maximum of 0.1) if the Newton method converges quickly (in fewer than 5 iterations), or is decreased by a factor of 2 (down to a minimum of $10^{-6}$) if it converges slowly (more than 8 iterations) or fails. Finally, we adjust the domain size continuously along each branch so as to minimise the specific grand potential $\Omega/A$. This is done by adding an extra parameter (the domain stretch factor $K$), so that the real domain size is $KL_x\times{K}L_y$ instead of an unstretched $L_x\times{L_y}$. The number of grid points is not altered. Then the real GEM-4 potential is proportional to $\psi(|{{\bm x}}|)=\exp\left(-K^4|{{\bm x}}|^4/R^4\right)$, where ${{\bm x}}$ is the unstretched coordinate on the unaltered grid. Quantities like the mean value of $n$ are the sum of the values of $n$ at each grid point divided by the number of grid points, and so are not affected by the stretch factor. The only parts of $\Omega/A$ that are affected are the convolutions (for ${{\cal L}}$ and the GEM-4 potential) and the spatial derivatives (for ${{{\cal L}_{\text{grad}}}}$). In the case of the GEM-4 potential, when considering the specific grand potential $\Omega_3/A$ arising from (\[eq:DDFTF3\]) for example, and ${{\cal L}}$ given by (\[eq:defnL\_GEM4\]), the integrals in $\Omega_3/A$ are proportional to $K^2$, with additional $K$ dependence coming from the GEM-4 potential itself. Therefore, $$\frac{{{\rm d}}(\Omega_3/A)}{{{\rm d}}{K}} \propto \int n({{\bm x}}) \left(2 K \psi \otimes n + K^2 \frac{\partial\psi}{\partial K} \otimes n\right)\,{{{\rm d}}{{\bm x}}},$$ where $\otimes$ represents the convolution integral evaluated on the unstretched grid, and the ${{\bm x}}$ integral is also on the unstretched grid. In the case of ${{{\cal L}_{\text{grad}}}}$, Laplacians on the real grid are a factor of $K^{-2}$ times Laplacians on the unstretched grid, so ${{\rm d}}(\Omega/A)/{{\rm d}}{K}$ is evaluated accordingly. In both cases, an extra equation (${{\rm d}}(\Omega/A)/{{\rm d}}{K}=0$) is added to ${{\cal H}}$ in (\[eqApp:fullsystem\]), and this is solved alongside all the other equations. The Jacobian also needs to be evaluated. In practice this made little difference in the cases considered here, and the domain stretch factor largely stayed between $0.98$ and $1.02$. 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Sometimes you find the next thing you’re meant to fall in love with completely by accident. In late 2011, a venue that seemed to be the only one in the area to draw any of the bands that my husband, Steve, and I actually want to pay money to see shut down due to a classic dispute between owners. The Southgate House split in two; one of the owners reinvented the historic Newport home Southgate House as the Thompson House, and the other owner cleverly transformed a beautiful, old church building in to the Southgate House Revival. So far the two venues have truly been like two halves of a split personality, with Southgate House Revival continuing to draw many of the alternative folk, punk, and bluegrass groups that we once sought out at the original Southgate House, and Thompson drawing bands that can range from questionable (Trapt) to fantastic (Bad Books featuring Kevin Devine). Steve and I were scrolling through the list of upcoming shows at Southgate House Revival when I saw the name Shovels & Rope. I had a nagging feeling I’d once told myself to look them up after hearing a song I liked on local radio station WNKU, so I switched over to YouTube and typed their name in the search bar. Steve and I are both pretty set in our musical tastes and band preferences – open to new bands but rarely finding one to get excited about. We’re both the kind of people that seek out new or reissued releases from groups we already like versus searching for the fresh or unknown. I couldn’t even write a “best of 2012″ list this past new year, because the only things I bought were releases from First Aid Kit, Tegan and Sara, and Tilly & the Wall, and being the only albums I bought certainly doesn’t make those the best releases of 2012. However, we’re both music lovers, and the hope is always alive that there are new, wonderful songs to discover. Thus, when we began watching the first Shovels & Rope video that a YouTube search produced, we were pleasantly surprised. Extremely, pleasantly, surprised. They. Were. Fantastic. We watched the entire video for “Gasoline” in a stunned silence, a buzz of excitement palpable in the space between us as we watched the two members, Michael Trent and Cary Ann Hearst, flawlessly play a raucous tune under a tree. I clicked on another video, and another, and another. We listened to Shovels & Rope’s toe tapping, rebel-with-a-heart country the rest of the night. We smiled watching the obviously in love (and married!) duo croon to each other in the melancholy ballad “Lay Low.” We laughed watching the two shout at each other in the rowdy call and response song “Tell the Truth,” and I danced around in the kitchen while a video of their album’s title track “O’ Be Joyful” played. The night ended in exclamations of future plans to start a husband and wife band of our own. Are you in love yet? Hearing Shovels & Rope is delightful in its own right, but watching the duo is even better. There’s just something utterly charming about them, and that’s why I think this post will be lost in an amorous sea of thousands after their performance on the Late Show with David Letterman airs on Wednesday, Jan. 30th. I’m not bitter, though, about their surely forthcoming notoriety. If a rude businesswoman asks me if I’ve ever heard of Shovels & Rope, I won’t even let it get to me (read: like that time I got all uptight about Mumford & Sons). The Letterman performance will kick off a two week sweep of the Southeastern US for Shovels & Rope, including a stop at Newport’s Southgate House Revival on Feb. 5th. You can keep up with Cary and Michael – which I highly suggest you do! – at www.shovelsandrope.com. My reasoning, of course, flawed, through perhaps not in the way you might think: I actually like the Decemberists quite a bit. In fact, it’s my enjoyment of their last album, The Hazards of Love, which had me rooting against their latest work, The King Is Dead, available now from Capitol (and, according to sales figures, appears to be doing quite well). Hearing the buzz anticipating King as well as its lead off single “Down By The Water,” I was turned off by the glee at declaring this a “triumphant return” after the “failure” of Hazards. I had enjoyed Hazards proggy arrangements, its twisting narrative. It wasn’t perfect, there were moments of fey self-indulgence, but I admired the chutzpa and the vision and besides, I don’t feel they’ve written a better song than “The Wanting Comes In Waves / Repaid.” Therefore, when King was being hailed as more of a “straightforward rock record,” I bristled. I didn’t really believe the band could function as “straightforward rock,” their virtue being found in their gleeful anachronism and goofy flights of fancy. My first listen confirmed a few things: yes, this is a much more straightforward pop/folk/rock record. Yes, it sounds like the Decemberists. No, Colin Meloy still uses words like “anon” and writes about being a galley man on pirate ships. Still, the question remained: Did I like it? Honestly, I wasn’t quite sure. I wasn’t disabused of my concerns. It’s distinctly different from Hazards. Another objection arose: perhaps it’s part-and-parcel with writing a “straightforward” record, but there’s a real lack of originality to the songs. Even in spite of that unmistakable voice at the fore, there’s not much going on here that’s new. “Don’t Carry It All” is a take on Tom Petty; “Down By The Water” lives in the shadow of “Losing my Religion;” and “Rise to Me” is a bit of a soulless rewrite of “Wild Horses” by the Stones. Furthermore, the inclusion of R.E.M.’s Peter Buck and Gillian Welch seems to me to confirm the deliberateness of these nods, a stab at silencing the critics who calling them wank-rock. There’s an aspect of pandering to it that, to quote Mr. Meloy on “Down By The Water,” “rubs me wrong.” However, even from that first uncomfortable listen, something hooked me. Even in spite of the lack of originality, there’s a tunefulness to the tracks that is undeniable. My head screams to not give them the satisfaction of knowing that a few lazy rips of classic rock is all it takes to woo me; but my heart knows no such logic. The songs are well constructed, the melodies undeniable, the arrangements showing a band in rare form. The boys (and the random girl or two) in the Decemberists know what they’re doing and, frankly, there’s real heart in most of the songs. This is the most interesting aspect of the record; it’s divisive to the long time fans of the D. The soul they were missing when they were singing shanties about fairies kidnapping children and mariner’s revenge manifests here. Perhaps it’s the diminished expectations, the well-worn territory allowing for more directness. The songs grow on you, their melodies catching you first, then the richness of feeling behind it. It might not be original, but it feels authentic. I still haven’t been convinced that the entire effort isn’t some crass stab at a degree of critical and popular respect (or at least to get people off their back for Hazards). It seems like a wasted opportunity: These bands that the Decemberists draw “inspiration” from have made careers defining themselves as musical iconoclasts, which the Decemberists have arguably achieved on prior records. It seems a misstep to transition away from the sound that distinguished them from their esteemed competition into something more familiar. Still, there’s something to be said for subverting expectation and crafting a tight, tuneful record along well-traveled lines after one of their most lofty, challenging ones. I can’t say I recommend it on principle, but still an enjoyable work. Sweater weather is one thing you can count on to come around year-round. It’d be unheard of to wear a down puffy jacket in the depths of summer, and donning a halter top in the below-zero temperatures of winter might get you tossed in a straightjacket. But no matter what the season, there will always be a time for sweaters. That’s why Light in August’s Sweater Weather is so aptly titled. From start to finish, the nine-track album is breezy enough for spring, warm enough for summer, tranquil enough for autumn and stark enough for winter. Any time of year, the music fits your ear. Lead vocalist, guitarist and songwriter Alex Wand put his work in good hands with Jim Roll, a veritable Midwest superstar when it comes to producing, mixing, mastering and playing music. Roll has worked on such fine albums as Frontier Ruckus’ The Orion Songbook and Deadmalls & Nightfalls, as well as Chris Bathgate’s Cork Tale Wake and Grey Buried by Drunken Barn Dance. He is also working with Gun Lake on their debut album, now scheduled for release in February. Gun Lake singer/guitarist Mark Fain provides backing vocals on two songs from Sweater Weather. “Seraphim” is a bright serenade to a long-distance love, and “Winter Clothes” a somewhat more melancholy declaration of the same, insisting “I would fit you into my life, but it’s best that you stay gone.” Something tells me Wand carries a torch for some girl who’s gone East. Just a guess. Whatever the reality, Wand’s got one generous muse to which he alludes on the opening and closing tracks, “Muse (Part I)” and “Muse (Part II),” resembling a fusion of Indian and Chinese folk styles with sitar, flute and a timpani drum sound. An Andrew Bird sensibility enters on “The First Days of May,” and sticks around throughout the album. “Water” flows in with the pitter-pat of drums and a brief tinkle of keys, just like a soothing rainfall, and “Weather Reports” is an amicable dueling of the flutes, blowing lightly along with cheery guitar, drum thumps and Wand’s floral voice. Like a gray day when the greenery is in full bloom, Sweater Weather placates while it elevates, a nice companion to springtime gloom. Any day you think you may need to wear a sweater, you may think to pop in Sweater Weather, because it’ll complement the mood just right. It’s diffcult to review an album that has already been reviewed perfectly by a friend and colleague only a few weeks earlier. You see, if it weren’t for Abby Holmes, I wouldn’t be saying this. If it weren’t for Abby Holmes being such a talented wordsmith and solid reviewer, I would never have begged her to write for my site, Radio Free Chicago. I would have never come around to the multi-layered pop wonders of John Vanderslice or taken a shine to The Moondoggies. Sure, I introduced Abby to some local favorites of mine like Lightning Love and Chris Bathgate but it’s Abby that undoubtedly wins this round of recommending great music with Ben Weaver. If it weren’t for Abby, I’d be completely unaware of the subdued beauty Weaver has to offer, blissfully ignorant to the lilting melodies and softly sung romanticism Weaver brings forth on Mirepoix And Smoke and for that, I would be that much less content. From the opening notes of “Grass Doe”, it’s hard not be taken by Ben Weaver’s music. He sounds at times like a more darkly upbeat version of Iron & Wine, a more accessible Bonnie “Prince” Billy, maybe even a hushed, less obviously county take on Justin Townes Earle, with whom Weaver shares a label in Chicago’s Bloodshot Records. Weaver displays his artful ability to weave an unforgettable story immediately in “Grass Doe”, telling the tale of love gone by the wayside in such masterfully poetic lines as “Their legs were twisted up in each other as the rain came down like watermelon seeds” and, later on in the tune, “There’s never gonna be another one like her and now you see her everywhere you go, like a tag under an overpass”. Near everyone’s loved and the vast majority of those people have lost as well. I know I certainly have. And it’s that fact that Weaver capitalizes upon, taking his own heartache, stated so poetically again and again and set to the simple backdrop of a fingerpicked guitar, a slight riff plucked on a banjo. By the time Mirepoix And Smoke closes, on the gentle notes of “The Rooster’s Wife”, you feel as if, to quote Abby Holmes, you’ve just listened to a “button-up flannel set to music”. Despite the fact that I’d never heard Weaver before listening to Mirepoix And Smoke, something about him made me feel immediately at ease, as though I were listening to the recordings of an old friend who’s reappearance in my life filled a very obvious void. I remember when Bon Iver’s For Emma, Forever Ago came out and I remember the subsequent nights I spent with that record. Never was I more at peace with my life than when I was driving alone, sometime around three or four a.m., with nothing but Justin Vernon’s impeccable falsetto harmonies at my side. Musically, Weaver’s Mirepoix And Smoke only has the rudimentary in common with Bon Iver’s release but to me, the records share a vast amount in common in reference to how they make me feel. Mirepoix And Smoke affects me in a way that only comes across a handful of times every few years. It’s the kind of record that lulls you into a false sense of security with it’s lack of obvious hooks but in the restraint that Weaver repeatedly exhibits, there is a seductive element, reminiscent of the first time you catch a glimpse of a beautiful boy from across a crowded room or a dingy bar, knowing with a foresight you probably don’t possess that this is the boy who will one day come to be your greatest triumph in love before he deftly destroys your heart with a grace that prevents you from harboring feelings of bitterness. There’s beauty in that moment, the discovery of great beauty and even greater potential, and even though you know it might break your heart (because doesn’t it always?), you know that it’s worth the risk, if only for the experience. Weaver sings songs that makes girls want to be the kind of woman he writes about and makes men want to find the kind of lady Weaver tells you of in such deftly written tracks like “City Girl” and “Grass Doe”. Nearly every track on Mirepoix And Smoke is an ode to a long lost dream girl, be her fiction or reality, and half of the beauty of the album lies in that amorous fact. “East Jefferson” features lyrics so strong that you’ll be hard pressed to not envision Weaver’s heroine sitting on the stoop alone, smoking her last cigarette as Weaver narrates the situation: “Cold wind blew through the swings in the park; by dinner time, it was already dark; the rain had turned to snow, everything whiter than a hundred ghosts at the end of the night.” That cinematic quality is one that Weaver exhibits over and over again, and by the time album stand out “Split Ends” rolls around, chances are you’ll want Weaver’s record narrating your life’s most poignant moments of heartache. Building on the folk of yesteryear, occasionally bordering on the subdued country of like-minded fellas like Jacob Jones (only more beautifully subdued) and Jonny Corndawg (only much less raunchy), Weaver takes cues from the gentler moments of Bob Dylan and the more callous moments of Simon & Garfunkle, piecing together a veritable quilt of lullabies and longing with nothing but Erica Froman, his female harmonizer, and an acoustic guitar at his side. Weaver makes music for people who have been put through the ringer by love and, despite the fact that they’re left emotionally raw and slightly bruised, they’re still willing to put themselves on the line for the potential of more and there’s beauty in that willfulness. It’s like Weaver sings on “Drag The Hills”, “I’d rather have scars from the life I lived than have none from the one I missed.” I’ve always loved a good spook story. Blame it on my mom taking me to see The Nightmare Before Christmas one too many times as a child or blame it on the fact that death touched my life one too many times for comfort and, to cope, I’ve taken an intense interest in the macabre. So intense, in fact, that before my foray into music journalism, I was dead set (pun intended) on becoming a mortician. So, naturally, any album that deals with murder, loss, and eloquent tombstone testimonials is right up my alley but never had I heard a pop album on the subject until Jeremy Messersmith‘s The Reluctant Graveyard. As the story goes, Messersmith and his wife moved into a Minneapolis house that neighbored a cemetery, spurring an idea in Messersmith’s hook-laden brain to write a concept album of sorts in which the narrarator of every track experiences death. Naturally, that would have you think that the songwriter had abandoned the infectious melodies of his first two albums for a more somber route. That assumption, however, is blown out of the water immediately. The Reluctant Graveyard opens with a guitar hook that makes Messersmith’s love of ’60’s pioneers The Zombies and Beach Boys apparent and as “Lazy Bones” progresses, the album’s dark concepts dissipate in a pop chorus sunnier than an August day. The trend continues through “Dillinger Eyes”, a song that culminates with Messersmith being shot, his good intentions bleeding out around him, all because he “was born with John Dillinger’s eyes”. The album even verges on Neil Diamond territory with “Violet!” which, to me, is slightly reminiscent of Diamond’s “Sweet Caroline” but hey, maybe that’s just me. Of course, my darker tendencies force me to gravitating to the album’s darkest tracks. Messersmith’s concept becomes clear on “Organ Donor,” a seductively eerie standout of a track that has Messersmith confessing that, after all he’s been through, “I don’t know if I’ll ever be whole again” against an understated background of strings and gentle guitar. Messersmith is never one, however, to overwhelm and it seems that every overtly grim track is sandwiched in between a folky tune that deceives you into thinking Messersmith’s verging on optimism until you take a closer listen. When Messersmith sings about “thinking of the friends (he’s) left behind”, it eventually becomes obvious that he’s not leaving on vacation but instead departing to the great beyond that Messersmith influence Elliott Smith was so fond of musing about. While the concept is clear on The Reluctant Graveyard, it never overpowers the album’s pure catchiness. One of the album’s darkest tracks, “John The Determinist”, is wrought with dark desperation, a man insisting “Oh, you silly things, I’ve got you figured out” although it’s more than apparent that the narrator’s constant need to understand his own life and the lives of others will never leave him with any answers. Even on an album fraught with standouts, nothing can compare to “A Girl, A Boy And A Graveyard”. You see, readers, every year for the past five, I’ve made a “Mental Health Mix”. This mix consists of me choosing one song per month that perfectly encapsulates my life. It doesn’t necessarily have to be a new song, it just has to be a song that describes what I was going through during that month. This year, however, twelve songs aren’t necessary. I mean, I’m still totally going to make “Mental Health Mix 2010″ come December but I could sum up my life simply using “A Girl, A Boy And A Graveyard”. It would take quite literally a dissertation to discuss why every single line of that song affects my life so deeply but as Messersmith sings about his Lucy, whose “body’s cold” and “guts are twisted steel”, I can’t help but feel he, without knowing me, perfectly encapsulated every aspect of my personality, from Lucy’s cryptic optimism (“Life’s a game we’re meant to lose (but) stick by me and I will stick by you.”) to her complete fear of vulnerability. Before “A Girl, A Boy And A Graveyard”, I was quite fond of this Messersmith character. Afterward, however, I was completely and utterly in love. Any other musician, I feel, would romanticize his heroine in such a way where she was helpless, feeding upon most every guy’s superhero complex where he gets to swoop in like Superman himself and rescue his lovely Lois Lane. Messersmith, however, compares Lucy to “some kind of Frankenstein, waiting for a shock to bring (her) back to life. But (she doesn’t) want to spend (her) time waiting for lightning to strike.” It goes back to my love of the macabre, of the ghosts of my past that have left me afraid of becoming attached to any one person, afraid of my own “lightning to strike”, for all the reasons Messersmith explores on The Reluctant Graveyard: Despair, tragedy, and the type of catastrophic loss that Messersmith sings about in “Repo Man”, another track that finds a man reflecting on the defeat that has marred his life. Even so, Messermith manages to end his album on two incredibly positive notes. “Deathbed Salesman” finds Messersmith playing the role of just that, a very literal deathbed salesman, and despite the fact that Messersmith is offering his audience their final resting places, he never stops reminding you, while dying is inevitable, “all your friends are there and waiting” and “once you’ll gone, you’ll never want to live again.” The swirling, Beatles-esque chorus, wherein Messermith repeats “This is how it has to end, so love somebody while you can”, recalls Smith once again, this time evoking one of my all time favorite songs, Figure 8‘s “Happiness”, a song who’s lyrics I just so happen to have tattooed on my arm. Messersmith, who released two stellar folk pop albums before The Reluctant Graveyard, finds the perfect balance of tragedy and beauty, of modern and vintage, on his third LP and despite his obvious infatuation with the literal end, Messersmith has managed to produce a remarkably optimistic album. Lyrically, this is the same guy that sang “Even the good times could be so much better…. Even the great times wouldn’t let me down” only now, he knows how to perfectly juxtapose the depression with the good times. I was exposed to rock and roll at a young age. I knew the words to Tom Petty and The Heartbreaker’s “Refugee” when I was well under the age of five and one of the saddest days of my youth was when my dad told me that, during moving from Missouri to Michigan, we’d somehow lost our copy of Buddy Holly’s Greatest Hits. The first cd I ever bought was Abbey Road at eight. I saved up my three-dollars-per-week allowance until I had enough scratch to get my own copy of The Beatles’ legendary disc and, sure, my parents had multiple copies (Vinyl, tape, probably 8-track as well) but how cool was it to have my own? Since then, I’ve expanded my horizons and all but left rock behind in the process. Yeah, I’ve still got the Traveling Wilburies in my iTunes library but I’m sad to say that these days, Jenny Lewis’s cover of “Handle With Care” gets more play than it’s original predecessor. Perhaps that’s why I’ve taken such a shine to Chicago foursome Archie Powell & the Exports. I came across the band shortly after both myself and Powell had moved to the city and was, at the time, going through a huge David Bazan phase. The band, it ends up, had just released a free five song EP on their bandcamp and who am I to refuse free music? Immediately, I was reminded of the bands that got me into music in the first place (Petty, Westerberg) but with a twist of the modern bands that Powell and I both cut our teeth on in high school (The Strokes, Weezer) and that mix kept the band’s sound from being a stale rehash of The Replacements’ Let It Be. Even after the hearty reception the band’s EP received in my household of one, I must admit that I was slightly apprehensive to hear the band’s debut. Why? Well, because while every song on the band’s Loose Change EP was catchy bits of audible bliss with lyrics that, to a girl who had just moved to the exact city Powell talked about in “Moving To The City”, struck a major chord, I couldn’t help but feel that ten plus songs like that had overwhelming odds of growing stale. This, however, was not the first time I’ve been wrong. Skip Work is Archie Powell & the Exports to the extreme. These kids are ready to make a splash and they are not playing around. This isn’t kid’s stuff. This is a band that’s all in, diversifying their sound on more than a few tracks and releasing an album that includes not only the catchy bits of Loose Change, but also some songs that are shockingly different for the Exports. The albums starts out with “Milkman Blues”, a minute and a half long tune that gives you the impression you’ve just popped in a much folkier CD than you actually have. When the song abruptly stops, only to punch your eardrums with the opening notes of lead single “Enough About Me”, it’s as evident to fans of Powell from his debut release as it is new recruits that this band is worth your time, more now than ever. Stand out track “Fightning Words” is a perfect example of this. Admittedly, being used to Powell and company’s straight up rock sound from the previous year’s EP, I hated “Fightning Words” at first. The verses are a spitfire assault of megaphone shouted lyrics while the chorus is signature Powell, megaphone tossed aside. The abrupt juxtaposition of the two versions of Powell present felt jarring, to say the very least but within days, the song had not only grown on me, but quickly became one of my favorite tracks on the album, a song that when I tell people “You need to hear this band!”, ends up being the track I tell them to “wait for” because “this is worth hearing.” Of course, this isn’t to say it’s all about Powell. Sure, he’s a hell of a front man but without the Exports, his talent wouldn’t shine half as much as it does. You see, Archie Powell & the Exports are sort of like the Mighty Morphin’ Power Rangers. Alone, they have spandex and mad ninja skills but together, they can transform into Megazord and that’s when you know some ass is about to get kicked. This, of course, is the proverbial ass of your eardrums, not the ass of Rita Repulsa, but regardless, ass? The Exports are kickin’ it. I never understood when people described a tune as a “bassist’s song” until I heart Okkervil River’s “Lost Coastlines” and goodness, am I ever glad that Okkervil River’s Patrick Pestorius came into my life because without him, I don’t think I’d truly understand the value of Adam Export (Yes, that’s totally his real last name) in Archie Powell & the Exports. Adam’s passion for his craft is showcased song after song and when the Exports bust out their rock numbers, Adam is just as prevalent of an asset to the band as is Powell himself. I feel as if one of the biggest reasons the Exports shine so thoroughly, however, is keyboardist Ryan Export (Where the heck did Powell find all these kids with the last name Export?!), who adds an alt-country flair to Skip Work‘s best tune, swoon-worthy album closer “The Darndest Things”. Elsewhere, Ryan adds an appealing spice where a lesser band would have just put a lackluster guitar solo. To say the Exports function as a musically cohesive unit is completely accurate and even less memorable tracks like “All Tuckered Out” and “Follow Through” are incredibly solid songs. Lyrically, Powell specializes in words that are equal parts sincere and snotty, singing earnestly about the pains of shouldering day to day responsibilities at a factory job where he “doesn’t want to have to fake it anymore” (“Piggy Bank Blues”, “All Tuckered Out”) before launching into a rockin’ tongue-in-cheek number about the shortcomings of his old friend Mattson who, yes, is a real guy that apparently skips out on chill sessions with Powell to watch reruns of the O.C. Powell, as well as all the Exports (which, in addition to Adam and Ryan, include RJ, a recent acquisition that did not appear on Skip Work.) don’t only command your attention but they deserve it. They wear their love of rock on their sleeve and their type of rock is the rock that just doesn’t relay exist anymore, having been replaced in “hipster” culture with synth beats and music laden with irony and kitsch. Sure, Powell might be a snot on occasion but he’s never insincere and if there’s any justice in the world of rock and roll, Archie Powell & the Exports will be big time in no time. You’d be hard-pressed to find a trio more adorable than Lightning Love, and that’s majorly due to vocalist/keyboardist Leah Diehl’s cartoony voice and indie-cute fashion sense (not to mention that darling smile that probably helps her “Friends” forgive her after an embarrassing night of drinking). Anyone who’s ever been 22 can probably relate to Lightning Love’s lyrics, which include the aforementioned chain of alcohol-induced events from “Friends”: “Well, they all had a laugh when I climbed up the shaft/ and I pissed in the elevator in that old parking garage/ but I really wish they hadn’t watched/ And they thought it was cute ’til I kicked off my shoes/ and I started to puke/ and my friends, well, they all walked away/ I thought real friends would have stayed.” Take note: The college/post-college experience that lacks a similar anecdote is truly only a partial experience. Guitarist Ben Collins and drummer Aaron Diehl round out the lineup of sandy-haired youngsters from Michigan. The group’s debut LP, November Birthday, went on sale last year. A dozen electro-pop tunes splay drops of sunshine across these dreary months, like a musical scarf to warm up your wintertime. Diehl’s self-deprecating lyrics are present throughout the record, lamenting professional obligation, a life without ambition, relationship disasters, cold weather, dealing with grown-ups, and becoming a grown-up. But she’s good about reminding herself that it’s all a part of life, so even while it’s getting her down, she doesn’t let it bury her. “I fail at everything/ And every day’s the same/ I’m human, that’s what happens/ and there’s no one else to blame,” Diehl sings on “Girls Are Always Wrong.” The way the child-like musicality conflicts with the mature subject matter in the lyrics is almost in itself the starring point of November Birthday. It’s like Diehl’s way of negotiating a life she’s not ready to leave behind with a life she has to grow into, standing her ground somewhere at the cusp of those two phases. Old enough to drink and smoke and sleep around, but still too young to be expected to know better, consequences be damned. On “Good Time,” the keyboard melody begins forebodingly to illustrate the shame she doesn’t feel about transgressions she doesn’t regret only to cheerily climax with the declaration, “I can’t help having a good time.” And why should she? If there’s any time to be making mistakes, it’s in your early 20s, when you can still sort of get away with it. Diehl puts her heart on her sleeve for “Wait, Wait,” revealing a moment of vulnerability that forces her to evaluate her priorities. Perhaps after the passage of a few more birthdays, Lightning Love will have a new sound to match its growth — but one can secretly hope the trio continues to sound just like it does already, because what Lightning Love has now is pure perfection. Everyone likes a little Ok Go. Now add some Prince and some gritty electronic noise and you have their latest offering from earlier this year, Of the Blue Colour of the Sky. It’s a strange sound, many of you will like it and many of you will hate it. But really, you should at least check it out because it’s sweet! It’s not the pop sound we learned to love from the album Ok Go, but it is the next installment from a band that we know writes good music. The track “White Knuckles” has been getting radio play, at least in my neck of the woods. It harkens back to the indie pop we are used to more so than the rest of the tracks. There are some good hooks, and some beats, but the overall feel of spaciousness takes over and leaves an air of … well, airiness by the end of the album. I have a strange and wonderful relationship with Cursive front man and The Good Life troubadour Tim Kasher. I got into him very early on, via a mixtape (yes, a tape. Yes, I am getting old.) from my middle school’s only other indie kid. The fact that we both wore Chuck Taylors and exchanged mixtapes resulted in some lesbian rumors which I guess makes sense as our relationship was one of self-discovery. Only, instead of discovering each others prepubescent chests, we discovered something far better: Stephen Malkmus. I gave her Elliott Smith and Eels. She gave me The Dismemberment Plan and Cursive. To be more specific, she gave me Cursive’s “Cielings Crack” from their debut release, Such Blinding Eyes For Starving Eyes. And I was smitten. At the time, I was heavy in like with a boy who didn’t see me as anything more than “one of the guys” (probably because of those lesbian rumors) so the sentiments of Kasher’s screams echoed long and hit heavy. “I know I’m just a peon to you, but I deserve more,” he shouted and inside, I said “Eff yeah. I do deserve more. This guy gets it. He gets it!” Some years later, the same friend asked me to recommend her some happy music. I told her I’d been listening to The Ugly Organ a lot. When she informed me that Cursive was just about the furthest thing from “happy music”, I told her that she had done this to me and really, she had. Without her, who knows when I would have discovered Cursive, or The Good Life for that matter, and what affect they would have had on my life. Just like everyone who’s ever been bummed about heartbreak, Album of the Year holds a special place in my heart and “Sierra” gets me to shed a tear every time it comes up on my iTunes library. Black Out has more than a few songs that mirror my life and seeing Cursive play a cover of The Cure’s “Lovecats” to less than 200 die hard Kasher fans in DeKalb, Illinois ranks as one of the best moments of my musical life thus far. I heard rumors of a Tim Kasher solo show at Schubas in Chicago brewing before I heard any announcement of his solo debut, The Game of Monogamy, so when I heard that the man himself was releasing a solo debut, I had two thoughts: 1) Complete and utter fan girl excitement (Oh my god, did I mention the time Tim complimented my shoes you guys?! That totally happened! In real life!) and 2) Wait… Why? You see, Kasher started The Good Life as a way to release all the songs that didn’t fit with his career with Cursive. Since that time, The Good Life has progressed from a electro-indie band to confessional acoustics and Cursive? Well, they just do whatever they damn well please these days. So really, a solo album from Kasher just doesn’t make sense. Surely, one of his outfits could be well suited enough for The Game of Monogamy. Apparently not. The theatrics Kasher displayed with Cursive on The Ugly Organ? Imagine that ten fold and you have the first ten minutes of The Game Of Monogamy. With his solo debut, Kasher has written a mainstream musical, complete with horn sections, pop arrangements, abrupt tempo changes, and storytelling lyrics. The only problem is… well, frankly put, all of these songs would sound better bare bones, Album of the Year style. Another problem? Well, aside from the unabashed mainstream overtones of the album, Kasher has, at one time or another, done everything here before. That isn’t to say the man is losing his talent, as he once expressed fear over in “No News Is Bad News”. Kasher’s songwriting is just as wonderfully, brutally, honest and self-loathing as ever but what Kasher has done on The Game of Monogamy was just better the first time around. Those horn sections that litter “I Think I’m Gonna Die Here”? They just fit better on Happy Hallow. The strings on “There Must Be Something I’ve Lost”? It was cooler when it was on “Driftwood: A Fairytale”. That isn’t to say that The Game Of Monogamy does not have it’s appeal. “No Fireworks”, in particular, is another damn near perfect Kasher composition, the type of song he does so well that manages to do nothing less than encapsulate my personal life with lyrics like “I thought love was supposed to spill from our hearts…. But I can’t feel anything at all.” Much like “What Have I Done” and “Staying Alive” before it, “No Fireworks” is a song for the ages of Amber Valentine, a lament of the sad state of my personal life, a dirge for my once a-flutter heart. If love has to suck so bad, at least I have Kasher to commiserate with me. By the time the album wraps up, with “Monogamy”, if your ears have adjusted to the shock of Kasher’s polished sound, the closing tracks beauty will not be lost on you. Heck, even if your ears haven’t adjusted (as mine seem unable to do), you’ll be able to take note of “Monogamy”. The track sums up the entirety of the album, closing it on the somber note that makes it clear that monogamy was just a “charade”, a “game”, as the album title points out. Once again, just as with all of the endeavors Kasher puts to music, he’s failed. Is it shocking? No. “Ten years of wedded bliss” is not for people like Kasher, over grown teenagers who still lust after high school girlfriends, who want to sleep with every girl they see, just to see what it’s like. For all it’s musical downfalls, The Game Of Monogamy is not a bad album by any means but the fact of the matter is that, honestly, I just can’t help but expect more from Kasher. My exceptions for the man, after all, are quite high. But at the end of the day, “Just Don’t Get Caught”, the country tinged b-side to Kasher’s Cold Love single, is just more appealing to me than anything that made the actual album’s cut. While I am an indie kid through and through, spending many nights in the audible company of Okkervil River and The National, I’ve always had a weakness for the music of my people, the Fench. I like being French for innumerable reasons, including but not limited to my love of the country’s seductive yet relâché take on fashion and the French’s laissez faire attitude towards relationships, but the thing I love most about my heritage is chanson. From Jane Birkin and Francoise Hardy to modern day French multitaskers like Charlotte Gainsbourg and Keren Ann, I’ve always been taken with my proverbial motherland and I’ve always wondered why it is exactly that more French music isn’t widely available and beloved in the states. Part of that, of course, is the fact that some of the loveliest French releases never even make it to America. One of these such releases is the debut full length album by chanteuse Madjo, a classically trained French-Senegalese violinist. If Trapdoor, her September released album, were brought to the U.S., I feel like Madjo could be the next big quirky indie female… if only people could hear her! Madjo is a spritely scamp who’s adorable nature brings to mind, vocally, a mix of Fiona Apple and Imogen Heap and, personality wise, a more three dimensional version of Charlotte Gainsbourg’s enigmatic pal Zoe in The Science of Sleep. Trapdooris split almost evenly between perfectly enunciated English indie pop and edgy, quirky tracks in Madjo’s native toungue. Album opener “Leaving My Heart” spins a jazz tinged web that’s not completely out of step with Fiona Apple, back when she was ripe with Jon Brion’s lovely oddities. As with the rest of the album, “Leaving My Heart” is heavy on layered vocals, making Madjo’s husky and melodic voice even more alluring. The dance beat of “Another Day” is ripe for play at hipster bars while “Le Coeur Hibou” is audible sex appeal, with enticing vocals that only a French woman could provide and a backdrop of multi-layered, echoing instruments. The album’s title track is a clap-along number that spares no expense when it comes to charm and it just begs to be in a Focus Features off-beat romance, during the inevitable “Why don’t we just fall in love?” moment between the two quirky and neurotic main characters. While I am French by birth, I cannot hold my own in a French conversation to save my life. Nevertheless, one of Trapdoor’s most captivating tracks is “Le Nid Des 100 Soucis.” What’s Madjo saying? I haven’t a clue! But the fact that “Le Nid Des 100 Soucis” a straight up, infectious jam is a testament to Madjo’s talent. Nothing to do with the appeal of the song, or Madjo herself, is lost in translation. Amber Valentine is the editor in chief of Radio Free Chicago and you can read her review of Madjo’s first release, a self titled EP, here.

Q: How to determine number of roots and what type for quartic equations? I want to figure out the number of roots and their types in the quartic equation $$x^4-34x^2-x+272=0$$ without actually solving it. Is there such a way to do so? A: With this particular equation, you can let $f(x) = x^4 - 34x^2 - x + 272$ (this is just for convenience), and then calculate: \begin{align*} f(-5) &= 52 \\ f(-4) &= -12 \\ f(0) &= 27 \\ f(4) &= -20 \\ f(5) &= 42 \end{align*} Since this function should look "continuous" when you draw it, and it changes sign between each of -5, -4, 0, 4, 5, this implies that there is a root in between any adjacent pair of each of these numbers, so we have at least two negative real roots and two positive real roots. I suspect that's what your sign changing business is supposed to mean.

NOTE: This disposition is nonprecedential. United States Court of Appeals for the Federal Circuit ______________________ HOWARD GREENBERG, DENISE GREENBERG, PARENTS OF J.G., A MINOR, Petitioners-Appellants v. SECRETARY OF HEALTH AND HUMAN SERVICES, Respondent-Appellee ______________________ 2016-1187 ______________________ Appeal from the United States Court of Federal Claims in No. 1:08-vv-00024-TCW, Judge Thomas C. Wheeler. ______________________ Decided: May 6, 2016 ______________________ HOWARD GREENBERG, DENISE GREENBERG, J.G., Kihei, HI, pro se. HEATHER LYNN PEARLMAN, Torts Branch, Civil Divi- sion, United States Department of Justice, Washington, DC, for respondent-appellee. Also represented by BENJAMIN C. MIZER, RUPA BHATTACHARYYA, VINCENT J. MATANOSKI. 2 GREENBERG v. SECRETARY OF HEALTH ______________________ Before TARANTO, SCHALL, and HUGHES, Circuit Judges. PER CURIAM. When he was one-year old, J.G., child of Howard and Denise Greenberg, received a measles, mumps, and rubella vaccination. One year later, he was diagnosed with a form of autism. A few years after J.G.’s diagnosis, the Greenbergs filed a petition with the United States Court of Federal Claims seeking compensation under the National Vaccine Injury Compensation Program of the National Childhood Vaccine Injury Act of 1986, codified as amended at 42 U.S.C. § 300aa-1 et seq. A special master dismissed their petition as untimely and for failure to show that J.G. suffered a relevant post-vaccination injury, and the Court of Federal Claims entered final judgment. The Greenbergs did not appeal from that judgment, but they sought post-judgment relief by filing a motion for reconsideration. The special master denied their motion, and the Court of Federal Claims affirmed. Because the special master’s refusal to reconsider his decision showed no abuse of discretion, we affirm. BACKGROUND J.G. was born on April 10, 2003. He passed all devel- opmental milestones at several “well-child” doctor visits during his first year. On April 13, 2004, J.G. received a vaccination for measles, mumps, and rubella (MMR). Between that visit and his 15-month well-child visit, the Greenbergs called J.G.’s doctors at least three times, concerned about J.G.’s swollen gums and fussiness (the medical notes refer to molars coming in), bumps on his limbs and torso, and an allergic reaction to peanuts. At his 15- and 18-month well-child visits in July and October 2004, J.G.’s medical records show him continuing to meet all developmental goals. And the medical notes record GREENBERG v. SECRETARY OF HEALTH 3 “none” next to “shot reaction” through J.G.’s first 18 months. At his two-year well-child visit, J.G.’s parents raised concerns about his tantrums, screeching, and limited speech. Half a year later, in January 2006, a pediatrician determined that J.G. had a significant speech delay, unusual behavior patterns, and impaired social interac- tions. In the pediatrician’s opinion, J.G.’s behavior was consistent with Pervasive Developmental Disorder, a variant of autism. On January 14, 2008, the Greenbergs filed, in the Court of Federal Claims, a petition alleging that J.G.’s MMR vaccine caused his autism and that the National Vaccine Injury Compensation Program thus required compensation. To show entitlement to compensation, the Greenbergs needed to show by a preponderance of the evidence either (a) that J.G. had received a vaccine listed on the Vaccine Injury Table and suffered an injury listed on the Table as corresponding to that vaccine (a “table injury”), without additional proof of causation, or (b) that administration of a Table-listed vaccine had actually caused or significantly aggravated some injury not listed on the Table for that vaccine. 42 U.S.C. §§ 300aa-13(a)(1), 300aa-11(c)(1); Cedillo v. Sec’y of Health & Human Servs., 617 F.3d 1328, 1335 (Fed. Cir. 2010). Autism was (and is) not a table injury for the MMR vaccine. 42 U.S.C. § 300aa-14; 42 C.F.R. § 100.3. The court assigned the Greenbergs’ petition to a spe- cial master. 42 U.S.C. §§ 300aa-11(a)(1), 300aa-12(d). Initially, the Greenbergs’ petition was considered during a multi-case proceeding about autism—the Omnibus Au- tism Proceeding. See Cedillo, 617 F.3d at 1334; Haz- lehurst v. Sec’y of Health & Human Servs., 604 F.3d 1343, 1345 (Fed. Cir. 2010). When that proceeding ended, the Greenbergs filed an amended petition, seeking compensa- tion only for a table injury based on the allegation that 4 GREENBERG v. SECRETARY OF HEALTH J.G. had suffered an encephalopathy within 15 days of receiving the April 2004 MMR vaccine. 42 U.S.C. § 300aa-14(a)(II)(B). On December 8, 2014, the special master dismissed the Greenbergs’ petition. Greenberg v. Sec’y of Health & Human Servs., No. 08-24V, 2014 WL 7496604, at *1 (Fed. Cl. Office of Special Masters Dec. 8, 2014). He first con- cluded that their petition was time-barred. J.G. received his MMR vaccine on April 13, 2004, and if his symptoms began within 15 days (as alleged), the petition for com- pensation had to be filed within 36 months of April 28, 2004, 42 U.S.C. § 300aa-16(a)(2), i.e., April 28, 2007. But the Greenbergs filed their petition in January 2008, beyond the due date. Greenberg, 2014 WL 7496604, at *8–9. The special master also found that equitable tolling did not excuse the lateness of the petition, rejecting the argument that the government’s endorsement of certain vaccine studies was fraudulent and prevented a timely filing. Id. at *9–10. The special master alternatively determined that the Greenbergs had failed to demonstrate by a preponderance of the evidence that J.G. had suffered, within 15 days of receiving his MMR vaccine, an “acute encephalopathy,” followed by at least six months of a “chronic encephalopa- thy.” Id. at *13–15 (citing 42 C.F.R. § 100.3(b)(2)). The special master addressed two pieces of evidence concern- ing the onset of the alleged acute encephalopathy. One was a December 2012 letter, in which Mrs. Greenberg stated that the Greenbergs “first noticed that [J.G.] was sick when he had a fever and seemed very sensitive to his surroundings like to light and sound” and “just seemed weak and out of it and very irritable”; the other was an undated letter from Dr. Kevin Passer confirming the consistency of the descriptions in Mrs. Greenberg’s letter with an acute encephalopathy. Id. at *14. Because, however, those letters did not state when J.G. experienced the described symptoms, the special master found them to GREENBERG v. SECRETARY OF HEALTH 5 be insufficient proof of the onset of an acute encephalopa- thy within 15 days of J.G.’s MMR vaccination. Id. at *14. Moreover, the special master found that J.G.’s irritability and sensitivity to his surroundings did not indicate “a significantly decreased level of consciousness,” a defining symptom of an acute encephalopathy. Id. (citing 42 C.F.R. § 100.3(b)(2)(i)). The special master also concluded that another letter by Dr. John Green showed no more than that J.G. suffered a metabolic encephalopathy, a type of encephalopathy not covered by the Vaccine Injury Table. Id. at *15 n.17 (citing 42 C.F.R. § 100.3(b)(2)(iii)). Likewise, none of J.G.’s medical records between his one- year well-child visit (when he received the MMR vaccine) and his two-year visit indicated that J.G. had suffered symptoms of an acute or chronic encephalopathy. Id. at *14. The Greenbergs did not timely file a motion seeking review of the special master’s decision by the Court of Federal Claims. Accordingly, the special master’s deci- sion became a final judgment on January 8, 2015. 42 U.S.C. § 300aa-12(e); U.S. Ct. Fed. Claims, App’x B, Vaccine R. 23 (Vaccine Rule 23). On February 3, 2015, the Greenbergs moved for re- consideration of the special master’s decision. The special master, to whom the motion was assigned, denied the motion on March 20, 2015. He considered the motion under Vaccine Rule 36(a)(2), which allows a petitioner, after entry of judgment, to move “for reconsideration pursuant to [U.S. Ct. Fed. Claims R. (RCFC)] 59 or oth- erwise seek[ ] relief from a judgment or order pursuant to RCFC 60.” Insofar as the motion would be read to seek reconsid- eration of the December 2014 special master’s decision, the special master deemed it untimely and also outside RCFC 59 because the Greenbergs had not sought judicial review of the December 2014 decision. Insofar as the 6 GREENBERG v. SECRETARY OF HEALTH motion would be read to challenge the January 2015 judgment, the special master concluded that RCFC 60(a) was unavailable because the Greenbergs did not allege any “clerical mistakes,” “oversights,” or “omissions” in that judgment. The special master also rejected reconsid- eration under RCFC 60(b). Because Dr. Passer’s and Dr. Green’s letters did not support the Greenbergs’ table encephalopathy claim, the special master determined that oral testimony from the Greenbergs could not change the result, and he ultimately concluded that no other reason justified reconsideration of his no-encephalopathy finding. And the special master again rejected the Greenbergs’ equitable-tolling argument, while adding that, even if their petition had been timely filed, the Greenbergs’ failure to prove a table encephalopathy independently prevented them from receiving compensation. On March 12, 2015, before the special master ruled on the reconsideration motion, the Greenbergs filed a “Notice of Review” in this court. On June 10, 2015, we concluded that we lacked jurisdiction, because our jurisdiction does not encompass direct review of special masters’ decisions. J.A. 42; see Grimes v. Sec’y of Dep’t of Health & Human Servs., 988 F.2d 1196, 1198 (Fed. Cir. 1993). We trans- ferred the Greenbergs’ notice of review to the Court of Federal Claims in part—not for review by that court of the January 2015 judgment (time had run out on obtain- ing any such review), but for possible review of the special master’s March 2015 order denying reconsideration. The Court of Federal Claims, acting “in the interest of justice,” reviewed and affirmed the special master’s order refusing reconsideration as not “arbitrary, capricious, an abuse of discretion, or otherwise not in accordance with law.” Greenberg v. Sec’y of Health & Human Servs., No. 08-24V, 2015 WL 6684703, at *2–3 (Fed. Cl. Nov. 2, 2015). The Greenbergs now appeal from the Court of Federal Claims’ decision. 42 U.S.C. § 300aa-12(f). We have jurisdiction under 28 U.S.C. § 1295(a)(3). GREENBERG v. SECRETARY OF HEALTH 7 DISCUSSION We review de novo the Court of Federal Claims’ affir- mance of the special master’s decision denying reconsid- eration. See Hines v. Sec’y of Dep’t of Health & Human Servs., 940 F.2d 1518, 1523–24 (Fed. Cir. 1991). In effect, we review the special master’s underlying decision, set- ting it aside only if arbitrary, capricious, an abuse of discretion, or otherwise not in accordance with law. See id. at 1524; Vaccine Rule 36(b)(7). The special master determined, and the Greenbergs do not dispute, that the motion for reconsideration should be evaluated only under Vaccine Rule 36(a)(2), and even then only as seeking post- judgment relief under RCFC 60(b). The special master did not abuse his discretion in finding no Rule 60(b) ground justifying reconsideration of his determination that the Greenbergs had failed to show that J.G. suffered a table encephalopathy. In his Decem- ber 2014 decision, the special master correctly described the statutes and regulations governing entitlement to compensation for a table encephalopathy, and he dis- cussed at length the application of those laws to the Greenbergs’ medical records, affidavits, and letters. Regarding J.G.’s alleged table encephalopathy, the Greenbergs have pointed us to no evidence or arguments undermining the adverse finding on that issue, let alone under the demanding standard for Rule 60(b) relief. The special master likewise acted within his discre- tion in rejecting the Greenbergs’ argument that reconsid- eration was warranted because they had been denied an evidentiary hearing before the December 2014 decision. A special master may, but is not required to, conduct an evidentiary hearing. 42 U.S.C. § 300aa-12(d)(3)(B)(v); Vaccine Rule 8(d). The record does not show that the Greenbergs requested an evidentiary hearing, although they could have. See Vaccine Rule 6(b). In any event, the Greenbergs filed at least 140 exhibits and participated in 8 GREENBERG v. SECRETARY OF HEALTH several status conferences with the special masters as- signed to their petition, and they have not identified what additional evidence they would have submitted or argu- ments they would have made at an evidentiary hearing or how such evidence and arguments could have changed the outcome of their case. In these circumstances, we see no abuse of discretion in rejecting the lack-of-evidentiary- hearing basis for reconsideration. Last, the special master did not abuse his discretion in refusing reconsideration even if equitable tolling might have excused the Greenbergs’ untimely petition. The equitable-tolling ruling made no difference to the outcome here, because the special master independently rejected the Greenbergs’ claim on the merits, finding that the Greenbergs had failed to show that J.G. suffered a table encephalopathy. Having already concluded that the special master need not have reconsidered his table- encephalopathy determination, we do not disturb the resolution of the Greenbergs’ equitable-tolling argument. CONCLUSION For the foregoing reasons, the judgment of the Court of Federal Claims is affirmed. AFFIRMED 

FILED NOT FOR PUBLICATION JAN 03 2013 MOLLY C. DWYER, CLERK UNITED STATES COURT OF APPEALS U .S. C O U R T OF APPE ALS FOR THE NINTH CIRCUIT TYRONE WALLACE, No. 11-55872 Plaintiff - Appellant, D.C. No. 2:09-cv-05075-VAP- AGR v. RICHARD TULL, Correctional Officer, MEMORANDUM * Defendant - Appellee. Appeal from the United States District Court for the Central District of California Virginia A. Phillips, District Judge, Presiding Submitted December 19, 2012 ** Before: GOODWIN, WALLACE, and FISHER, Circuit Judges. California state prisoner Tyrone Wallace appeals pro se from the district court’s summary judgment in his 42 U.S.C. § 1983 action alleging excessive force under the Eighth Amendment. We have jurisdiction under 28 U.S.C. §1291. * This disposition is not appropriate for publication and is not precedent except as provided by 9th Cir. R. 36-3. ** The panel unanimously concludes this case is suitable for decision without oral argument. See Fed. R. App. P. 34(a)(2). We review de novo. Jones v. Blanas, 393 F.3d 918, 926 (9th Cir. 2004). We affirm. The district court properly granted summary judgment because Wallace failed to raise a genuine dispute of material fact as to whether the force was not applied in a good faith effort to restore prison discipline. See Whitley v. Albers, 475 U.S. 312, 319 (1986); see also Karam v. City of Burbank, 352 F.3d 1188, 1194 (9th Cir. 2003) (speculation as to defendant’s improper motive does not rise to the level of evidence sufficient to state a triable issue of fact). AFFIRMED. 2 11-55872 

One of the most iconic denim jackets ever made is the Levi’s Jeans jacket Lot Number 506XX. The 506XX jacket was originally introduced in 1905. Actually they didn’t call it a jacket in those times, but a blouse. It was made of 9oz. unsanforized AmoskeagManufacturing Company double heavy extra strong quality, that explains the XX. It was around 1917 that the term Number One (Type 1) was used for this Levi’s506XX Jacket. The switch from the word blouse into jacket came in 1938 when the word jacket was used in the western wear catalog, Dude Ranch Duds. Old advertising with the Levi’s 506XX jacket. Notice the price of $2,-. Old Levi’s Jeans cowboys advertising. Note from the advertising: ‘There are lots of blue jeans, but there’s only one Levi’s, the original cowboys pants’. Details from the Levi’s 506XX Jeans jacket The Levi’s Jeans 506XX jacket has a simple, but timeless classic design. The fit of the jacket is short and boxy with pleats on the front. The horizontal seams holding down the pleats which could be removed to give the wearer extra room. The first edtions of the 506XX models has only one front pocket, a silver buckle cinch in the back and no flap covering the front pocket. The later models of the 506XX jacket has a flap over the pocket, rivets in the corners for strength and a bronze buckle back. The silver editions became more valuable than the bronze ones. The famous Red Tab – Big E – appeared on the 506XX models in 1936, the year Levi’s introduced the famous Red Tab. The word Levi’s was only written on one side of the Red Tab. This changed by the birth of the follow-up jacket in 1953, the 507XX model (also known as Type 2). During WorldWar II Levi’s simplified the 506XX jacket to save materials (metal, fabric and thread) for the War. They skipped the flap above the pocket, and they used 4 buttons on the jacket instead of 5. The buttons on the ‘War’ edition had donut buttons featuring laurel leaves, this as a symbol of peace. Laurel leaf button on a pair of LVC 501 fit 1944. Pic by End Clothing. Levi’s Jeans made in total 6 different versions of the 506XX jacket in the past; Version #1: 1905 – Release of the original first version. Version #2: 1928 – The pocket flap was introduced. Version #3: 1936 – The Red Tab was introduced for the first time, but without ‘R’ and ‘LEVIS’ (Big E) only on one side of the tab. Version #4: 1941 – During WWII Levi’s Jeans removed the pocket flap and added donut buttons to simplify the jacket. Version #5: 1944 – The cinch back slider was introduced for the first time. Version #6: 1947 – The pocket flap was reintroduced. The Levi’s Jeans 506XX jacket was a companion to work trousers. They became very popular amongst farmers and labor workers and later for cowboys too. From the ’40’s, and later, famous people as actors and singers were spotted wearing the Levi’s 506XX jacket. It was a style for the true rebels. Singer and actor Bing Crosby Actor Robert Mitchum Young Mick Jagger wearing the 506XX jacket Original vintage Levi’s 506XX Jeans jacket from the ’40’s This original vintage Levi’s 506XX (Type 1)Jeans jacket is part of my private denim collection. The jacket was first in hands from a Levi’s Jeans collector from Texas. He bought the jacket from a farmer family when they cleaned the abandoned house from a family farmer member after he died. The farmer wore this jacket while working on his farm. The jacket has an extremely light natural sun bleach look from working for years on his land. The jacket shows a lot of rusty spots as the old abandoned farm was very soggy. Almost all the buttons are broken and missing, only a few are left. The jacket has a bronze buckle back, with slider instead of pin teeth, and has still the ‘one side’ LEVIS– Big E – written Red Tab. The worn-out frayed cuffs are reinforced with rivets for extra strength. The red line in the selvage on the inside is completely washed-out. The fabric is in pretty good condition as you realise that its used for hard labor on the land during the ’40’s. This original vintage Levi’s 506XX Jeans jacket is a perfect example of the American workers lifestyle.

Constraints on speciation suggested by comparing lake-stream stickleback divergence across two continents. Adaptation to ecologically distinct environments can coincide with the emergence of reproductive barriers. The outcome of this process is highly variable and can range along a continuum from weak population differentiation all the way to complete, genome-wide divergence. The factors determining how far diverging taxa will move along this continuum remain poorly understood but are most profitably investigated in taxa under replicate divergence. Here, we explore determinants of progress towards speciation by comparing phenotypic and molecular divergence within young (<150 years) lake-stream stickleback pairs from Central Europe to divergence in older (thousands of years) archetypal lake-stream pairs from Vancouver Island, Canada. We generally find relatively weak divergence in most aspects of foraging morphology (gill raker number, body shape) in the European pairs, although substantial adaptive divergence is seen in gill raker length. Combined with striking overall phenotypic differences between the continents, this argues for genetic and time constraints on adaptive divergence in the European pairs. The European lake-stream pairs also do not display the strong habitat-related differentiation in neutral (microsatellite) markers seen in the Canadian watersheds. This indicates either the lack of strong reproductive barriers owing to weak adaptive divergence, or alternatively that neutral markers are poorly suited for detecting reproductive barriers if these emerge rapidly. Overall, our comparative approach suggests constraints on speciation due to genetic architecture and limited time for divergence. The relative importance of these factors remains to be quantified by future investigation.

The cigarette package of this type comprises an inner pack and a parallelepiped outer box enclosing the inner pack. The inner pack includes a bundle of rod-shaped smoking articles, such as filter cigarettes, and an inner wrapper covering the bundle. The outer box includes a box body open at the upper end thereof, and a lid joined to the box body at a rear edge of the open end of the box body, which functions as a hinge. The outer box is formed by folding a blank around the inner pack. The lid of the outer box can be a hinged lid in the shape of a box, or a tongue lid having a tongue. While the hinged lid is fitted on top of the open end of the box body, the tongue lid has an upper wall for covering the open end of the box body and a tongue extending from the upper wall designed such that when the upper wall closes the open end of the box body, the tongue overlies the front wall of the box body. [Patent Document 1] Japanese Unexamined Patent Publication No. Hei 11-49134 It is desirable that at the time the above-described cigarette package is made, the lid of the outer box should be joined to the box body by a tearable separation line. The provision of such separation line is effective in deterring people from tampering with the cigarette package. More specifically, when the lid is first opened, the lid is torn from the box body along the separation line, and the torn separation line leaves break marks to the box body as well as the lid, which marks indicate that the lid has already been opened. Generally, the separation line is provided as a perforated line formed in the blank for the outer box in advance. The perforated line, i.e., the separation line has a lot of joins connecting the adjacent perforations. Thus, when the lid is first opened, first, a join at one end of the separation line suffers a break, and the adjacent joins suffer such break one after another, so that the separation line completely breaks. In other words, the separation line breaks in the manner that a crack spreads. However, the crack does not infallibly spread along the separation line, but can spread deviating from the separation line. Such deviating crack can give an undesired break to the tongue lid and/or the box body, and therefore lead to a damaged appearance of the cigarette package opened. Such trouble can be avoided by making the joins of the separation line shorter so that the joins can be broken easily. This can, however, cause a problem that in the process of making a cigarette package, specifically in folding the blank, the blank splits along the separation line so that the cigarette package fails to be made.

Q: Access function pointers outside of the instance I have a class as outlined below: class InputReader { public: typedef void (*handler)(std::string, int); static void errorHandler(std::string error, int severity); //Supplies a default error handler static void warningHandler(std::string error, int severity); //Supplies a default warning handler handler errorH; handler warningH; InputReader(std::string pwd = "", handler eHandler = errorHandler, handler wHandler = warningHandler); bool readFile(std::string filename); std::vector<first> mesh; //other irrelevant objects that need to be read into }; first is a struct: struct first { std::string filename; double scale; }; With Mooing Duck's help I have: std::istream& operator>>(std::istream& file, first& obj) { std::string symbol; while(file >> symbol) { if (symbol[0] == '#') { std::getline(file, symbol); } else if (symbol == FIRSTTAGEND) { break; } else if (symbol == FILEPATH) { if (!(file >> '=' >> obj.filename)) std::cerr << symbol << " is incorrectly formatted"; //This needs to use errorH } else if (symbol == SCALE) { if (! (file >> '=' >> obj.scale) ) std::cerr << symbol << " is incorrectly formatted"; //This needs to use errorH } else { //not a member: failure std::cerr << symbol << " is not a member of first"; file.setstate(file.rdstate() | std::ios::badbit); break; } } return file; } std::istream& operator>>(std::istream& file, InputReader& obj) { std::string symbol; while(file >> symbol) { if (symbol[0] == '#') { std::getline(file, symbol); } else if (symbol == FIRSTTAGBEG) { first t; if (file >> t) obj.mesh.push_back(t); } else { obj.errorH(symbol + " is not a member of the input reader.", 1); file.setstate(file.rdstate() | std::ios::badbit); } } return file; } bool InputReader::readFile(std::string filename) { std::ifstream infile; infile.open(filename.c_str()); infile >> *this; return true; } errorH gets set upon construction of the InputReader object. It can be provided by the user of the class, otherwise, it uses the default ones I provide. The only problem is that I can't access errorH when first is being read into. How can I solve this? Problem constraints: External libraries are not allowed. C++11/C++OX is not allowed. A: You naturally have few options: Do not use >> operator and create a function that does the same but accepts three arguments - an input stream, a first object and an input reader object. Define operator >> that accepts a tuple of first and InputReader as a second argument (i.e. a pair of pointers). For example: std::istream& operator>>(std::istream& file, std::pair<first *, InputReader *> & obj). You can extend this list indefinitely as long as you have a good imagination. Hope it helps. UPDATE: Here goes some simple example: #include <string> #include <vector> #include <fstream> #include <iostream> struct first { std::string filename; double scale; }; class InputReader { public: typedef void (*handler)(const std::string &, int); InputReader(const std::string & pwd = std::string(), handler eHandler = errorHandler, handler wHandler = warningHandler); bool readFile(const std::string & filename); static void errorHandler(const std::string & error, int severity); static void warningHandler(const std::string & error, int severity); handler errorH; handler warningH; first firstobj; std::vector<first> mesh; }; std::istream & operator >> (std::istream & file, std::pair<first, InputReader *> & obj) { std::string symbol; while (file >> symbol) { if (symbol[0] == '#') { std::getline(file, symbol); } else if (symbol == "FIRSTTAGEND") { break; } else if (symbol == "FILEPATH") { if (!(file >> obj.first.filename)) obj.second->errorHandler(symbol + " is incorrectly formatted", 1); } else if (symbol == "SCALE") { if (!(file >> obj.first.scale)) obj.second->errorHandler(symbol + " is incorrectly formatted", 1); } else { //not a member: failure std::cerr << symbol << " is not a member of first"; file.setstate(file.rdstate() | std::ios::badbit); break; } } return file; } std::istream & operator>>(std::istream & file, InputReader & obj) { std::string symbol; while (file >> symbol) { if (symbol[0] == '#') { std::getline(file, symbol); } else if (symbol == "FIRSTTAGBEG") { std::pair<first, InputReader *> t(first(), &obj); if (file >> t) obj.mesh.push_back(t.first); } else { obj.errorH(symbol + " is not a member of the input reader.", 1); file.setstate(file.rdstate() | std::ios::badbit); } } return file; } bool InputReader::readFile(const std::string & filename) { std::ifstream infile; infile.open(filename.c_str()); infile >> *this; return true; }