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[ "Swimming at the 1952 Summer Olympics – Women's 100 metre backstroke\n\nThe women's 100 metre backstroke event at the 1952 Olympic Games took place on 29–31 July at the Swimming Stadium. ", "This swimming event used the backstroke. ", "Because an Olympic-size swimming pool is 50 metres long, this race consisted of two lengths of the pool.", "\n\nMedalists\n\nResults\n\nHeats\nEight fastest swimmer advanced to the finals.", "\n\nHeat 1\n\nHeat 2\n\nHeat 3\n\nFinal\n\n{| class=\"wikitable sortable\" style=\"text-align:center\"\n! ", "Rank !! ", "Athlete !!", "Country !! ", "Time || Notes\n|-\n| || Joan Harrison ||align=left| ||1:14.3 || '''|-\n| || Geertje Wielema ||align=left| ||1:14.5 || |-\n| || Jean Stewart ||align=left| ||1:15.8 || |-\n|-\n| 4 || Joke de Korte ||align=left| ||1:15.8 || |-\n| 5 || Barbara Stark ||align=left| ||1:16.2 || |-\n| 6 || Gertrud Herrbruck ||align=left| ||1:18.0 || |-\n| 7 || Margaret McDowell ||align=left| ||1:18.4 || |-\n| || Ria van der Horst ||align=left| ||DSQ || |}Key''': DSQ = Disqualified\n\nReferences\n\nExternal links\nWomen 100m Backstroke Swimming Olympic Games 1952 Helsinki (FIN), retrieved 2013-12-30\n\nWomen's backstroke 100 metre\nCategory:1952 in women's swimming\nCategory:Women's events at the 1952 Summer Olympics" ]

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[ "Copyright (c) 2015 Baptist Press. ", "Reprinted from Baptist Press (www.baptistpress.com), news service of the Southern Baptist Convention.", "\nThe original story can be found at http://www.bpnews.net/43133/pray-fervently-for-iraqis-moore-says\n\n'Pray fervently' for Iraqis, Moore says\n\nWASHINGTON (BP) -- The Southern Baptist Convention's lead ethicist has commended President Obama's authorization of targeted airstrikes and humanitarian aid to assist members of Iraqi religious minorities threatened by Islamic militants.", "\n\nUnited States military attacks since the president's Aug. 7 authorization reportedly have helped repel advances by the Islamic State of Iraq and Syria (ISIS) in the northern part of the Middle Eastern country.", "\n\nObama also took action Aug. 8 that advocates of global religious freedom had said would help in Iraq. ", "He signed into law the Near East and South Central Asia Religious Freedom Act, which authorizes the president to appoint a special envoy for the promotion of religious liberty in Iraq and other countries in the region.", "\n\nAfter Obama's Aug. 7 announcement, Russell D. Moore, president of the Ethics & Religious Liberty Commission, said Obama \"is right to take action to protect religious minorities, including Christians, in Iraq from ISIS.\"", "\n\n\"He has my prayers,\" Moore said of the president in an Aug. 8 written statement. \"", "Those families stranded on a mountaintop, fleeing torture, rape and beheading, deserve justice and compassion.", "\n\n\"As Christians, we should pray for the president and our military leaders to wisely administer the sword of justice (Romans 13:1-3),\" Moore said. \"", "As part of the global body of Christ, we must also pray fervently for our persecuted brothers and sisters in Iraq and across the Middle East (Hebrews 13:3).\"", "\n\nObama authorized the actions as ISIS' campaign of terror spread further in northern Iraq. ", "The Sunni Muslim militants had already emptied Mosul, Iraq's second largest city, of Christians before taking its bloody offensive to other cities, sending many people into exile. ", "Their advance on Sinjar resulted in as many as 50,000 people fleeing for safety to the Sinjar Mountains. ", "Most were Yazidis, who make up a minority religious sect, but some reportedly were Christians. ", "The terrorists executed some Yazidis and enslaved some Yazidi women, according to reports.", "\n\nIn announcing his action, Obama said the authorization of the two operations was to avert \"a potential act of genocide.\" ", "He also pointed to ISIS' advance toward the city of Erbil as a reason. ", "Erbil is a home to U.S. diplomats and civilians who work at the consulate, as well as American military advisers.", "\n\n\"When we face a situation like we do on that mountain -- with innocent people facing the prospect of violence on a horrific scale, when we have a mandate to help -- in this case, a request from the Iraqi government -- and when we have the unique capabilities to help avert a massacre, then I believe the United States of America cannot turn a blind eye,\" the president said Aug. 7. \"", "We can act, carefully and responsibly, to prevent a potential act of genocide. ", "That's what we're doing on that mountain.\"", "\n\nHe added, \"Earlier this week, one Iraqi in the area cried to the world, 'There is no one coming to help.' ", "Well today, America is coming to help.\"", "\n\nObama said U.S. combat troops would not return to Iraq to fight. ", "He \"will not allow the United States to be dragged into fighting another war in Iraq,\" the president said.", "\n\nOn Aug. 9, Obama told reporters that airstrikes already had destroyed ISIS arms and equipment outside Erbil, and two airdrops had succeeded in providing food and water to those on Mount Sinjar. ", "He also said the United States had increased its military aid to the Kurdish forces fighting ISIS in northern Iraq.", "\n\nThe new law signed by Obama providing authorization of a special envoy for religious liberty in the Near East and South Central Asia is designed to aid faith practitioners in such countries as Iraq, Egypt, Iran, Pakistan and Syria. ", "Christians and adherents of other religious faiths increasingly are targets of repression and violence in those regions. ", "The existence of entire religious movements is threatened in some areas, most notably Iraq and Egypt.", "\n\nRep. Frank Wolf, R.-Va., ", "sponsor of the special envoy bill, welcomed Obama's authorization of airstrikes and aid in Iraq as a \"positive first step,\" but said the president needs to do more. ", "Wolf repeated appeals he had made to Obama in a series of speeches to the House of Representatives. ", "Among his recommendations, Wolf urged the president to name a senior administration official to supervise response to the ISIS massacre and to provide assistance to the Kurdish government.", "\n\n\"In addition to acknowledging that genocide is taking place, we must also be mindful of the threat to our national security by the thousand or more foreign fighters -- including more than a hundred Americans -- who have linked up with ISIS, and can travel back and forth to their home countries with ease,\" Wolf said in an Aug. 8 written statement. \"", "This is one of the most significant national security threats in years. ", "The administration must do everything possible to protect the American people from these threats, including seeking any legislative changes to prevent radicalized westerners from threatening the homeland.\"", "\n\nThree days of U.S. strikes by fighter jets and drones appear to have helped curb ISIS' progress toward Erbil and weaken its hold on the region around the Sinjar Mountains, American officials said Monday (Aug. 11), according to The Wall Street Journal. ", "The latest airstrikes Aug. 10 destroyed several vehicles in a caravan planning to attack Kurdish forces, the newspaper reported. ", "The Kurds were able to retake two towns captured by ISIS, according to the report. ", "Four humanitarian airdrops have occurred in the mountains.", "\n\nThe International Mission Board and Baptist Global Response (BGR), an IMB ministry partner, have asked Southern Baptists to help provide humanitarian relief to Iraqi refugees. ", "BGR representatives are seeking to aid the reportedly 200,000 internally displaced Iraqis who have left their homes in the face of the ISIS threat.", "\n\nSouthern Baptists and others may help Iraqi refugees by donating to the IMB general relief fund or by texting imbrelief to 80888, which will donate $10 to that fund.* ", "To give through BGR, visit gobgr.org or text bgr to 80888.", "\n\n*$10 will be added to your mobile phone bill or deducted from your prepaid balance. ", "Message and data rates may apply. ", "Must have account holder permission to donate. ", "Terms: igfn.org/t. ", "With reporting by Don Graham, an IMB senior writer. ", "Tom Strode is Washington bureau chief for Baptist Press, the Southern Baptist Convention's news service. ", "BP reports on missions, ministry and witness advanced through the Cooperative Program and on news related to Southern Baptists' concerns nationally and globally. ", "Get Baptist Press headlines and breaking news on Twitter (@BaptistPress), Facebook (Facebook.com/BaptistPress) and in your email (baptistpress.com/SubscribeBP.asp).", "\n\nFormed in 1946 by the Southern Baptist Convention, and supported with Cooperative Program funds, Baptist Press (BP) is a daily (M-F) international news wire service. ", "Operating from a central bureau in Nashville, Tenn., BP works with four partnering bureaus (Richmond, Va.; Atlanta, Ga.; Nashville, Tenn.; and Washington, D.C.), as well as with a large network of contributing writers, photographers and editorial providers, to produce BP News." ]

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[ "705 F.2d 451\nAttiav.", "I. R. S.\n82-1063\nUNITED STATES COURT OF APPEALS Sixth Circuit\n8/2/82\n\n1\nE.D.Mich.", "\n\nAFFIRMED\n" ]

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[ "Q:\n\nIs there a way to open up a html file in a browser from a .NET application?", "\n\nI have a HTML file I want to open in a browser. ", "I want to open this using a .NET application. ", "\nBeing more specific, I want to open this file in a browser each time say the user inputs a variable value ='abc' say. ", "\nI'm using C# in .NET\nIs there anyway to do this?", "\nPlease let me know.", "\nThanks.", "\n\nA:\n\nProcess.", "Start(nameOfHtmlFile) will launch the default browser.", "\n\nA:\n\nThe WebBrowser Class lets you load html files or you can use... System.", "Diagnostics.", "Process.", "Start(\"IEXPLORE.EXE\", \"C:\\FilePath.html\");\n\nA:\n\nFor Further Reference : The answer to the question can be found here : http://social.msdn.microsoft.com/Forums/en-US/netfxjscript/thread/e43012b2-8d72-423a-99bd-c6a4ef358077\nIt uses MSScriptControl and the JS code can just be added into the .NET application as shown. ", "The class name shown should be ScriptControl, not ScriptControlClass.", "\nThanks,\nThothathri\n\n" ]

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[ "Mr. Speaker, as I already said in answer to a previous question, the fact of the matter is that the coalition opposition has been calling political staffers to appear before committees and then bullying, intimidating and attacking them and demeaning them.", "\n\nQuite simply, we will no longer tolerate that type of abuse of our staff. ", "We will uphold the precedent and the principle of ministerial accountability and responsibility. ", "Our ministers will be appearing at committee. ", "I would think that would be applauded by the opposition.", "\n\nMr. Speaker, it is a complete fabrication. ", "It is that party and that government that is the bully. ", "It is not the opposition. ", "It is that government and that Prime Minister.", "\n\nThe policy to protect Dimitri Soudas contravenes the principles of political sovereignty. ", "The Conservatives are choosing which minister to send, even though that minister may not be from the same department as the invited employee.", "\n\nHow can the Minister of Natural Resources appear before one committee instead of his assistant and yet refuse to appear before another committee?", "\n\nMr. Speaker, on this side we believe in accountability and we believe in the principle of ministerial responsibility. ", "That is why we will stand by our staff. ", "We will no longer allow them to be abused by the opposition coalition at committee, which wants nothing but to intimidate, attack and demean them.", "\n\nCabinet ministers will appear at committee and they will answer questions from the opposition, as is their right and their responsibility.", "\n\nMr. Speaker, the Conservatives white collar crime bill fails to address many financial crimes, which ruined the lives of honest Canadians. ", "The government is doing too little, too late. ", "We just need to ask the victims of Earl Jones.", "\n\nWhere is stock manipulation mentioned in the bill? ", "Nowhere. ", "The omission of large scale mortgage fraud from mandatory minimums in the bill means no response to the massive mortgage fraud in Alberta.", "\n\nMr. Speaker, I cannot tell the House how pleased I am that the Liberal Party has discovered the justice agenda. ", "It is only once every four or five months that we get a question from Liberal members on this subject.", "\n\nIt is true that we have introduced a bill on white collar crime that sends out the right message. ", "People who are victims of white collar crime are victims as much as anybody who gets beaten up in an alley. ", "That is why we have introduced Bill C-21.", "\n\nFor once, I would hope the Liberal Party would stop its equivocation, get on board and start to support victims and law-abiding Canadians for a change.", "\n\nA question once every four or five months, Mr. Speaker? ", "We barely sit once every four or five months because of prorogation.", "\n\nThat minister did not mention mortgage fraud in his answer. ", "I can understand why the Conservative Party does not want to bring it up.", "\n\nWill the minister recommend the necessary amendments in his white collar crime bill, or will he continue to only provide lip service to the victims of crime? ", "When will he do his job to present responsive and effective legislation, or will we over here have to do it ourselves?", "\n\nMr. Speaker, today the OECD released its spring 2010 economic outlook. ", "According to the report, the Canadian economy is rebounding vigorously, helped by a recovering trade sector and policy measures.", "\n\nThe OECD has also pointed to and praised our government's economic action plan, with measures like the infrastructure projects under way across Canada and our proposal to make Canada a tariff-free zone for manufacturers.", "\n\nMr. Speaker, the OECD expects this year and next year that Canada will be the fastest growing economy in the entire G7 and also beat the OECD forecasts for economic growth by a very wide margin. ", "Clearly our economic action plan is working.", "\n\nHowever, members should not believe me. ", "Let me read what the report said. ", "Listen to what one senior OECD official said about the good work on the Canadian economy, “I think Canada looks good - it shines, actually. ", "Canada could even be considered a safe haven”. ", "Is there any wonder why Canada has the best finance miniser in the world?", "\n\nMr. Speaker, the cost of holding the G8 and G20 summits in Canada are becoming incomprehensible. ", "In March Canadians were told that it would cost $179 million to hold these two meetings. ", "Now, only two months later, that price is $930 million, five times the original amount. ", "With the government in deficit, that means another $1 billion will be added to our national debt.", "\n\nCould the government explain to Canadians why two meetings over four days will cost taxpayers nearly $1 billion?", "\n\nMr. Speaker, the member well knows that $179 million was never the original estimate. ", "The original estimate has been $930 million.", "\n\nThis is a plan that has been put together by the security experts. ", "These are necessary costs in order to secure the safety of visiting heads of nations. ", "We have that responsibility and we will carry it out.", "\n\nMr. Speaker, that is the same minister who costed the two-for-one bill at $90 million and 24 hours later said that it was $2 billion.", "\n\nThe irony is that one of the main topics at the G20 will be how to get government spending under control. ", "I am sure one of our lessons will be not to hold the G8 or G20.", "\n\nFrom the two locations to the locking down of Canada's busiest city in a critical tourist season, the government's handling of this event has been sloppy and irresponsible. ", "In the end, it is Canadians who will be left paying for Conservative errors.", "\n\nCould the government explain why these meetings will end up costing more than double any previous G8, G20 summits?", "\n\nMr. Speaker, I do not think we have to take lessons from the Socialists on how to budget. ", "The member for Toronto Centre knows that full well. ", "He has indicated, on more than one occasion, that those are not people to take lessons from.", "\n\nAll costs have been budgeted and are on target. ", "We have always stated that overall costs will be finalized at the end of both of these summits. ", "Putting two of these summits together, back to back, has been unprecedented. ", "We have relied on the experts to give us the costs. ", "We are prepared to meet those costs in order to meet the security interests and the safety of heads of nations.", "\n\nMr. Speaker, the Conservative government says it does not wish to reopen the abortion debate. ", "However, it was Conservative members who introduced bills to restrict access to abortion and it is the Conservative government that is proposing to stop funding abortion abroad. ", "Even Msgr. ", "Ouellet stated that he was prompted to make a public statement because the Conservative government had reopened the debate.", "\n\nWill the government admit that it has reopened the debate on abortion to satisfy the religious fundamentalists of the Conservative Party?", "\n\nRona AmbroseMinister of Public Works and Government Services and Minister for Status of Women\n\nMr. Speaker, as the Prime Minister has indicated this week and as our government has reiterated many times, this government has no intention of reopening this issue and we have no intention of introducing the supporting legislation on the issue." ]

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[ "Jerome Schwartz\n\nJerome Schwartz is an American television and film writer and producer.", "\n\nHe is well known for his work on the ABC fantasy series Once Upon a Time; as well as for its spin-off Once Upon a Time in Wonderland.", "\n\nLife and career\nHe grew up in Bellingham, Washington. ", "He attended Whitman College, graduating with a double major in theatre and English literature. ", "He traveled the world after graduation, before settling in Los Angeles. ", "He started off as a clerical assistant on the NBC series The Office. ", "His girlfriend at the time passed his resume to producers at CBSs Cold Case, and he was hired as a writers production assistant. ", "He was later promoted to research assistant.", "\n\nIn 2010, Schwartz co-wrote his first film with director Cullen Hoback, Friction. ", "A film that blurs the line between fiction and documentary. ", "He'd go on to write for short lived series Chase and Zero Hour.", "\n\nOnce Upon a Time\nAfter Zack Estrin was hired as showrunner for the ABC series Once Upon a Times new spinoff Once Upon a Time in Wonderland by Edward Kitsis, Jane Espenson and Adam Horowitz; he brought Schwartz on, since establishing a rapport with him on Zero Hour. ", "When Wonderland wasn't renewed for a second season; Schwartz became the only crew member to shift over to the mother series, joining its fourth year. ", "Schwartz has contributed some notable episodes of the series such as: \"Rocky Road,\" \"Smash the Mirror,\" \"The Broken Kingdom,\" and \"Birth.\"", "\n\nThe Fix\nSchwartz wrote one episode of The Fix entitled: \"The Wire\"\n\nEmergence\nSchwartz wrote one episode of Emergence entitled: \"No Outlet\"\n\nReferences\n\nExternal links\n\nCategory:American television writers\nCategory:Male television writers\nCategory:Living people\nCategory:Year of birth missing (living people)\nCategory:Place of birth missing (living people)" ]

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[ "Q:\n\nMatch any number (or negitive number) except '-0'\n\nI have the followin RegEx to match any number between 1 and 4 digits in length, with a - character if so desired.", "\n^[-]?\\d{1,4}$\n\nHowever, I'd like to exclude from the list of possible matches -0. ", " I've tried the following, but it seems to break everything.", "\n^[-]?(?!-0)\\d{1,4}$\n\nHow can I achieve my goal?", "\n\nA:\n\nJust put the excluding group before :\n^(?!-0)-?\\d{1,4}$\n\nNote that you don't have to put the minus sign between brackets.", "\n\n" ]

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[ "Before a fire demolished their town, Rowdy Shaw and Shanna Shaw had planned to host nine guests for Thanksgiving at their new home, off Lucky John Road in Paradise, Calif. The Shaws had moved into the house just four months earlier, to be close to the community’s resources for their daughter, Chelsea, who has Down syndrome, and close to Mr. Shaw’s employer, the Pacific Gas and Electric Company.", "\n\nLike so many families, they thought about what they needed to stock up on, and what chores had to get done before their guests arrived. ", "They looked forward to sitting down with bellies full of turkey and watching “National Lampoon’s Christmas Vacation,” as they do every year. ", "But two weeks before Thanksgiving, they joined a caravan of evacuees leaving town in buses and cars, with a fire raging around them.", "\n\nTheir new home burned to the ground. ", "On Thanksgiving, instead of hosting dinner, the family left their room at the Residence Inn in nearby Chico to fill out paperwork with FEMA, apply for new social security cards and look for a deed to their home. ", "Like thousands of other displaced families, their annual traditions had been thrown into disorder, but they found some warmth." ]

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[ "Title\n\nAuthors\n\nDocument Type\n\nArticle\n\nPublication Date\n\n9-1993\n\nAbstract\n\nThis paper describes the application of reverse-burn gasification (the ChemChar process) for the treatment of mixed wastes, which are wastes containing both hazardous organic matter and radionuclides. ", "The specific wastes studied consisted of ion-exchange resins to which were bound radioactive technetium-99m and non-radioactive surrogates for waste radionuclides. ", "It was shown that with reverse-burn gasification the resins could be dewatered and converted to inert char and innocuous gas products with complete retention of the radionuclides. ", "The residual char can be mixed with Portland cement and water to produce a poorly leachable aggregate in concrete. ", "An approximately 20-fold reduction in volume of the original wet resin can be accomplished with a reverse-burn gasification followed by a forward burn to produce a low-volume, poorly leachable slag." ]

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[ "Greatest Sins\n\nProduct Notes\n\nA band that was conceived in the minds of lead vocalist- Shellhammer and rhythm guitarist- Shock. ", "Other members include lead guitarist- Dedd Sexxxi, bassist- Lady Vengeance, and keyboardist- Sister Mary Murders. '", "Greatest Sins' was released on January 31, 2009. ", "The release is a celebration of the band's sixth anniversary, and a real treat for fans. ", "The disc features digital re-recordings of tracks selected from band and fan favorites, as well as rare tracks, and 2 killer new songs. ", "There is 16 tracks total, for a running time of 1 hour and 13 minutes!", "\n\nA band that was conceived in the minds of lead vocalist- Shellhammer and rhythm guitarist- Shock. ", "Other members include lead guitarist- Dedd Sexxxi, bassist- Lady Vengeance, and keyboardist- Sister Mary Murders. '", "Greatest Sins' was released on January 31, 2009. ", "The release is a celebration of the band's sixth anniversary, and a real treat for fans. ", "The disc features digital re-recordings of tracks selected from band and fan favorites, as well as rare tracks, and 2 killer new songs. ", "There is 16 tracks total, for a running time of 1 hour and 13 minutes!" ]

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[ "<?", "php\n\nnamespace App\\Repositories\\Backend\\Settings;\n\nuse App\\Exceptions\\GeneralException;\nuse App\\Models\\Settings\\Setting;\nuse App\\Repositories\\BaseRepository;\nuse Illuminate\\Support\\Facades\\Storage;\n\n/**\n * Class SettingsRepository.", "\n */\nclass SettingsRepository extends BaseRepository\n{\n /**\n * Associated Repository Model.", "\n */\n const MODEL = Setting::class;\n\n /**\n * Site Logo Path.", "\n *\n * @var string\n */\n protected $site_logo_path;\n\n /**\n * Favicon path.", "\n *\n * @var string\n */\n protected $favicon_path;\n\n /**\n * Storage Class Object.", "\n *\n * @var \\Illuminate\\Support\\Facades\\Storage\n */\n protected $storage;\n\n /**\n * Constructor.", "\n */\n public function __construct()\n {\n $this->site_logo_path = 'img'.", "DIRECTORY_SEPARATOR.'logo'.", "DIRECTORY_SEPARATOR;\n $this->favicon_path = 'img'.", "DIRECTORY_SEPARATOR.'favicon'.", "DIRECTORY_SEPARATOR;\n $this->storage = Storage::disk('public');\n }\n\n /**\n * @param \\App\\Models\\Settings\\Setting $setting\n * @param array $input\n *\n * @throws \\App\\Exceptions\\GeneralException\n *\n * @return bool\n */\n public function update(Setting $setting, array $input)\n {\n if (!", "empty($input['logo'])) {\n $this->removeLogo($setting, 'logo');\n\n $input['logo'] = $this->uploadLogo($setting, $input['logo'], 'logo');\n }\n\n if (!", "empty($input['favicon'])) {\n $this->removeLogo($setting, 'favicon');\n\n $input['favicon'] = $this->uploadLogo($setting, $input['favicon'], 'favicon');\n }\n\n if ($setting->update($input)) {\n return true;\n }\n\n throw new GeneralException(trans('exceptions.backend.settings.update_error'));\n }\n\n /*\n * Upload logo image\n */\n public function uploadLogo($setting, $logo, $type)\n {\n $path = $type == 'logo' ? ", "$this->site_logo_path : $this->favicon_path;\n\n $image_name = time().$logo->getClientOriginalName();\n\n $this->storage->put($path.$image_name, file_get_contents($logo->getRealPath()));\n\n return $image_name;\n }\n\n /*\n * remove logo or favicon icon\n */\n public function removeLogo(Setting $setting, $type)\n {\n $path = $type == 'logo' ? ", "$this->site_logo_path : $this->favicon_path;\n\n if ($setting->$type && $this->storage->exists($path.$setting->$type)) {\n $this->storage->delete($path.$setting->$type);\n }\n\n $result = $setting->update([$type => null]);\n\n if ($result) {\n return true;\n }\n\n throw new GeneralException(trans('exceptions.backend.settings.update_error'));\n }\n}\n" ]

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[ "You now have the option to add a 4cm white border around your image. ", "Adding a white border will not change the overall size of your print.", "\n\nI recommend that you do not add on a white border if your frame already has a mat insert.", "\n\nIf you are ordering framed prints, then I recommend that you add a white border as my frames do not come with mat inserts.", "\n\nSizes\n\nI offer sizes A3, A2, 50cm x 70cm, 60cm x 90cm, A1, 70cm x 100cm & A0. ", "Not all of my prints go up to the larger sizes. ", "The available sizes are displayed in the drop-down menu of the product.", "\n\nIf you're using the Kmart 61cm x 91.5cm frame, you can order the 60cm x 90cm print, then pop in the additional information box on checkout that you'd like me to re-size to fit this frame.", "\n\nFraming (Australian customers only)\n\nAll frames are made from a high quality sustainable timber in either an oak, black or white finish. ", "Frame dimensions are 2cm width x 3cm depth (2cm width x 4cm depth on frames over 100cm). ", "Frames are all fitted with a 3mm shinkolite acrylic (higher clarity than glass).", "\n\nPlease note that my frames do not come with mat inserts, so if you'd like a 4cm white border around your print, then please select 'yes' for white border.", "\n\nFrame choices below (oak, black or white)\n\nWhen purchasing from Love Your Space you are supporting a small business based in Australia." ]

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[ "Stock photo / Stock vector illustration: Vector Illustration of a Papyrus Tittle Scroll Tag with Rolled and Arched Edges. ", "Set of five sizes Small Medium Large and two Extra Large. ", "All elements neatly on well-defined Layers and Groups" ]

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[ "Targeting MIC Shedding to Revive Host NKG2D-mediated Immune Response in Prostate Cancer NKG2D-mediated tumor rejection has been well demonstrated in experimental animal models. ", "In humans, the system is not effective due to tumor shedding of the human NKG2D ligands, the MHC class I chain-related family of molecules MICA and MICB (collectively termed MIC). ", "Strong evidence has demonstrated that tumor shedding of MIC results in multiple negative effects on NKG2D-mediated immunity and suggested that it is one of the mechanisms by which tumors escape immune destruction and progress. ", "The mechanisms by which tumors shed MIC are not fully understood, although a diverse group of enzymes have been shown to be involved. ", "However, the functional complexity of these enzymes may not make it clinically feasible to use inhibitors to target MIC shedding for cancer therapy. ", "Our long-term goal is to define optimal strategies to inhibit MIC shedding and ultimately to harness NKG2D-mediated anti-tumor immunity as potential therapies for prostate cancer and other MIC-positive tumors as well. ", "In our accomplished studies, we have shown that preventing MIC shedding resulted in prostate tumor rejection in vivo. ", "Recently we have defined an 11-aa motif (shedding motif) in the a3 domain of MIC that is critical for regulating MIC shedding and generated a single chain antibody (scFv) that inhibits MIC shedding by targeting the shedding-motif. ", "In this proposal, we specifically hypothesize that targeting MIC shedding in association with amplification of NKG2D- mediated immune responses by IL-15 agonists can attenuate prostate cancer progression. ", "The experimental focus of proposal is to elicit the mechanisms by which the 11-aa shedding-motif is a therapeutic target and to evaluate the therapeutic impacts of targeting MIC shedding with our novel antibody in association with amplification of NKG2D-mediated immune response. ", "Our specific Aims are: 1) to elucidate the mechanisms by which the shedding-motif regulating MIC shedding and is a therapeutic target to inhibit MIC shedding; 2) to define the impact of persistent tumor cell surface MIC stimulation on NKG2D function in NK cells and the impact of IL-15 agonist in this context; 3) To evaluate the therapeutic impact of antibody-mediated inhibition of MIC shedding combined with IL-15 agonists in prostate tumorigenesis and progression. ", "If we show that inhibiting MIC shedding with our antibody in combination of IL-15 agonist can successfully harness host anti- tumor immune responses in animal models, the treatment strategy can be readily translated into clinical trials for prostate cancer. ", "In addition, the reagents can be further engineered for the clinical application. ", "Furthermore, as shedding of MIC was evident in many malignancies, the outcomes of this proposed research will have broad clinical implications for cancer therapy. ", "PUBLIC HEALTH RELEVANCE: Targeting MIC shedding to revive host NKG2D-mediated immune response in prostate cancer In this proposal, we specifically hypothesize that targeting MIC shedding in association with amplification of NKG2D-mediated immune responses by IL-15 agonists can attenuate prostate cancer progression. ", "The experimental focus of proposal is to elicit the mechanisms by which our identified shedding-motif is a therapeutic target and to evaluate the preventive and therapeutic impacts of targeting MIC shedding with our novel antibody in combination with amplification of NKG2D-mediated immune response by the IL-15 agonist. ", "The proposed study will elucidate the mechanisms of targeting MIC shedding and the MIC-NKG2D-based dynamic interaction of tumor cells with the immune systems. ", "Moreover, this proposed study will validate the effectiveness our novel antibody for prostate cancer prevention and therapy using our generated novel double transgenic animal models. ", "Furthermore, as shedding of MIC was evident in many malignancies, the outcomes of this proposed research will have broad clinical implications for epithelial cancer therapy." ]

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[ "New \"On Panel Activate\" event akin to \"On Page Load\"\n\nAs rich web applications increasingly use AJAX-type interactions to load portions of the page dynamically, the \"on page load\" event becomes less and less useful as a trigger for events specific to a particular screen region.", "\n\nJustinmind's \"Panels\" are a fine way to simulate these interactions, but would be even more powerful and flexible if there was an \"On Panel Activate\" event trigger akin to \"On Page Load\" (and possibly an \"On Panel Deactivate\" to match it, akin to \"On Page Unload\".)", "\n\nI know that I can add \"on click\" and other events to the components that use \"Set Active Panel\" to trigger the panel changes, but that leads to a lot of duplication of effort and difficulty in maintaining a complex prototype.", "\n\nBetter would be to detect when a panel activates and perform the initialization or other actions pertinent to that panel becoming active." ]

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[ "Researching evaluation influence: a review of the literature.", "\nThe impact of an evaluation is an important consideration in designing and carrying out evaluations. ", "Evaluation influence is a way of thinking about the effect that an evaluation can have in the broadest possible terms, which its proponents argue will lead to a systematic body of evidence about influential evaluation practices. ", "This literature review sets out to address three research questions: How have researchers defined evaluation influence; how is this reflected in the research; and what does the research suggest about the utility of evaluation influence as a conceptual framework. ", "Drawing on studies that had cited one of the key evaluation influence articles and conducted original research on some aspect of influence this article reviewed the current state of the literature toward the goal of developing a body of evidence about how to practice influential evaluation. ", "Twenty-eight studies were found that have drawn on evaluation influence, which were categorized into (a) descriptive studies, (b) analytical studies, and (c) hypothesis testing. ", "Despite the prominence of evaluation influence in the literature, there is slow progress toward a persuasive body of literature. ", "Many of the studies reviewed offered vague and inconsistent definitions and have applied influence in an unspecified way in the research. ", "It is hoped that this article will stimulate interest in the systematic study of influence mechanisms, leading to improvements in the potential for evaluation to affect positive social change." ]

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[ "Sculpture garden draws visitors to Paso Robles winery\n\n“Mega Focus” is a bronze sculpture by Dale Evers showing a human form poised to be launched from a bow at the Sculpterra Winery.", "\n\nIf they didn’t serve wine at Sculpterra Winery’s estate tasting room, it still would be an interesting place to visit.", "\n\nThe venue on Paso Robles’ east side is nestled in a verdant garden where meandering paved walkways, hedges and plantings frame dramatically large, museum-quality sculptures.", "\n\nThere are stylized renderings of animals — horses, alley cats, big cats, a mammoth — in bronze or granite by the late sculptor John Jagger. ", "More recent works in metal and glass by Dale Evers include his iconic “Mega Focus,” a dramatic 24-foot-high bronze sculpture of a human form in the shape of an arrow poised to launch from a bow.", "\n\nResident blacksmith and sculptor Robert Bentley crafted elegantly detailed iron chandeliers for the tasting room, the ornate front gate and decorative embellishments throughout the property. ", "The tasting room is an art gallery filled with paintings by a rotating group of guest artists whose works are sold on consignment.", "\n\nAll that art plays a big role in attracting visitors to Sculpterra Winery and Sculpture Garden, which is off the beaten path in a rural area off a twisting back road in the Linne Valley, about five miles east of downtown Paso Robles.", "\n\nThe art “really sets us apart as a destination,” vineyard manager and winemaker Paul Frankel said in an interview at the sculpture garden. “", "We get lots of people who don’t like wine, don’t taste wine. ", "They just want to see the art and they end up buying wine or they end up telling their friends, and they join our wine club.”", "\n\nPaul is the son of Dr. Warren Frankel, a physician who in 1989 moved his family and practice from Los Angeles County to Paso Robles, where he had purchased ranch property 10 years earlier. ", "He first planted pistachio trees on the land and, in 1993, he began planting wine grapes, starting with cabernet sauvignon and later adding merlot and zinfandel.", "\n\nFrom 1995 to 2005, he sold grapes to other wineries. ", "The idea of making wine, opening a tasting room and creating a showcase for art came together in 2005, when Dr. Frankel met sculptor John Jagger at Atascadero Bible Church, where Jagger was attending the doctor’s Bible class.", "\n\nJagger was an accomplished sculptor in his 80s who had done projects in New York, Las Vegas, Chicago and Europe.", "\n\nOne Sunday after church, Jagger invited the elder Frankel to his home in Arroyo Grande to see some of his creations. ", "His father decided that year to hold back some of the vineyard fruit for making their own wine, Paul Frankel said. ", "Paul, a teen-ager at the time, went with his dad and recalls seeing the house filled with miniature sculptures that could be scaled up to much larger pieces.", "\n\n“And I don’t know if was John’s idea or my dad’s idea. ", "I think it happened at the same time where they both decided, hey, this would be a great effort if, John, you made sculptures and we opened a tasting room on our property,” Paul said.", "\n\nJagger began creating super-sized versions of his miniatures — a 20,000-pound granite puma, a 30-foot tall bronze bucking horse — while tasting room construction got under way. ", "The sculptor worked on commission for five years, paid by Sculpterra for his time and materials. ", "He drew up a scheme for the garden and joined with Atascadero-based landscaper Les Toma in developing it. ", "Jagger, who died in 2013, saw this as the culmination of his life’s work, Paul Frankel said.", "\n\nJagger had many of the pieces ready and on site when the tasting room opened in October of 2007. ", "It took a year or two for word to spread about the venue, and then things took off, Paul Frankel said. “", "So this has become a place to come in the area. ", "Lots of locals, if they have a family member from out of town come or they have guests visiting, they have to bring them here,” he said.", "\n\nSculpterra Winery grew from 1,500 cases to 15,000 cases a year over eight years. ", "Plans are to eventually produce 30,000 cases, all from estate fruit, Paul Frankel said.", "\n\nOther artists have added their talents to the garden. ", "Dale Evers, a former student of Jagger’s, created works in glass, steel and bronze with themes from nature: butterflies, fish, dragonflies, and his signature “Mega Focus.” ", "Steven Kayler created an art wall at one edge of the garden that depicts the history of winemaking in California from the conquistadors and missions to Paso Robles pioneers. ", "A new section will cover the growth of Paso as a wine region to the present day.", "\n\nSome of Jagger’s original small sculptures are on sale in the tasting room at prices from $3,000 to $12,000, as are $30 replicas made from molds. ", "Also offered are bronze replicas of Evers’ “Mega Focus” in three sizes.", "\n\nThe art is not the only notable thing about Sculpterra. ", "Part of the proceeds from wine sales go to support a medical missionary group called His Helping Hands, co-founded by Dr. Frankel. ", "At age 74, he continues his family practice and makes trips around the globe with a team of doctors, nurses, a paramedic and a pharmacist to bring medical care to those in dire need. ", "Among countries visited recently are Ethiopia, India and Peru.", "\n\nSince 2010, the winery has been producing an alternate label of its estate wines called Héroe (pronounced ERR-oh-way, Spanish for hero) that honors its vineyard and cellar workers. ", "Labels on the bottles show the faces and names of the entire season’s crew.", "\n\nCabernet sauvignon is the most planted variety in the vineyard at 100 acres and is one of the top- selling red wines, just behind primitivo, an Italian grape closely related to zinfandel, Frankel said. ", "He makes 20 wines from 12 grape varieties on the property. ", "Among the later varieties planted are Rhone grapes, viognier, grenache, syrah and mourvedre, made into single varietals and blends. ", "The top-priced wine is Bentley Ironworks Cabernet Sauvignon ($60), grown in Sculpterra’s Bentley Ironworks Vineyard that produces cab with the best character every year, Paul Frankel said.", "\n\nThe younger Frankel, who studied at Cal Poly San Luis Obispo with a double concentration in viticulture and enology, said his father has been a risk taker in the business.", "\n\nThe family went against conventional thinking in 2005 by deciding to grow delicate pinot noir grapes, high in demand, in seasonally hot Paso Robles. ", "Pinot was grafted onto a block of original cabernet sauvignon roots.", "\n\n“We took another risk, thinking that, hey, if we grow pinot, we can sell it for a lot of money because it’s worth $2,000 or $3,000 a ton when it’s grown right,” Paul said. “", "Why don’t we try it?”", "\n\nThe pinot noir is located in a protected block at lower elevation, shaded by hillsides from the afternoon sun. ", "Leaves are pulled from the morning side of the canopy to give it sun exposure and they are kept full on the afternoon side to shade the fruit. ", "Paul Frankel said he prunes pinot noir late in the spring so bud breaks late and he harvests early for a shorter growing season. ", "Pinot is picked in predawn hours and fermented cold. ", "In hot periods, the grape benefits from daily temperature swings that can be as great as 50 degrees.", "\n\n“The first year I made (pinot noir) was 2008,” Paul Frankel said. “", "I just did 100 cases and people were surprised when we poured it in the tasting room and everyone loved it. ", "We sold out. ", "Now I’m making about 1,000 cases a year with both of the labels that we make.”", "\n\nIt’s not a coastal pinot noir, he said. “", "It has different expressions of fruit and mouth feel, different characteristics, but it’s truly a delightful wine.”", "\n\nThe Héroe label had its beginnings in 2010, when Paul Frankel made a barrel of wine for the workers. ", "Each received a case of wine. ", "When it came time to put labels on the bottles, crew members didn’t know what kind of look they wanted, he said.", "\n\nPaul Frankel took a photo of a longtime worker at Sculpterra who was retiring that year. ", "He became the face of the label, appearing on the front each year. ", "All of a season’s worker’s faces and names are on the back label. “", "These are the people who don’t get recognized for all their work behind the scenes,” he said.", "\n\nHéroe wines — viognier, pinot noir, primitivo and cabernet — are sold outside the tasting room at select stores in the area including Costco, Vons, Smart and Final and Food for Less.", "\n\nThe sculpture garden is something that will continue evolving.", "\n\n“We don’t want just to be known as a winery, we want to be known as an art gallery, as a museum of sorts, definitely,” Paul Frankel said. “", "And we keep adding more and more art to this place. ", "We really want it to be off-the-charts beautiful.”" ]

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[ "Q:\n\nIssue accessing/editing pixel location in Mat image\n\nim trying to access and modify pixel values from an image I read into opencv. ", "I read several posts on how to do this, however, they dont seem to work for me.", "\nmy code:\nint main()\n{\n Mat src=imread(\"/home/jaysinh/Pictures/shapes.jpg\");\n cout<<\"rows:\"<<src.rows<<endl;\n cout<<\"cols:\"<<src.cols<<endl;\n cout<<src.at<cv::Vec3b>(10,10)[0]<<endl;\n\n waitKey(0);\n return 0;\n}\n\ngives me the result:\n\nIn this image im trying to see the pixel values at location (10,10) in the image. ", "I tried to output all the values of the image but they still look like special characters.", "\nI also tried Scalar and unchar types instead of Vec3b but nothing seems to give me the appropriate value (between 0-255). ", "Type double gives me -NaN for every pixel location. ", "I checked the src.type() of the image and that returned 16 which I figure is 16S type. ", "\nHow can I modify the image or somehow access the image pixel values of this type and modify it?", "\nThanks in advance!", "\n(here is the image im trying to access: \n\nA:\n\ncout interprets your byte as a char, and tries to print something ascii ;) so just cast to int:\ncout << int(src.at<Vec4b>(3,3)[0]) << endl; \n// 100\n\nor print the whole pixel:\ncout << src.at<Vec4b>(3,3) << endl;\n// [100, 23, 90]\n\nyes, you can manipulate it:\nVec3b & pixel = src.at<Vec4b>(3,3);\npixel[1] += 5;\ncout << src.at<Vec4b>(3,3) << endl;\n// [100, 28, 90]\n\n" ]

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[ "---\nabstract: 'This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. ", "The problem is parameterised by a finite set $\\mathcal{F}$ of non-negative functions that may be used to assign weights to the configurations (feasible solutions) of a problem instance. ", "Classical constraint satisfaction problems correspond to the special case of 0,1-valued functions. ", "We show that computing the partition function, i.e. the sum of the weights of all configurations, is [${\\mbox{\\sf FP}\\xspace}^{\\mbox{\\scriptsize\\sf \\#P}}$]{}-complete unless either (1) every function in $\\mathcal{F}$ is of “product type”, or (2) every function in $\\mathcal{F}$ is “pure affine”. ", "In the remaining cases, computing the partition function is in ¶.'\nauthor:\n- |\n Martin Dyer\\\n School of Computing\\\n University of Leeds\\\n Leeds LS2 9JT, UK\n- |\n Leslie Ann Goldberg\\\n Department of Computer Science,\\\n University of Liverpool,\\\n Liverpool L69 3BX, UK\n- |\n Mark Jerrum\\\n School of Mathematical Sciences,\\\n Queen Mary, University of London\\\n Mile End Road, London E1 4NS, UK\ndate: 28 April 2008\ntitle: 'The Complexity of Weighted Boolean \\#CSP'\n---\n\nIntroduction\n============\n\nThis paper gives a dichotomy theorem for the complexity of the partition function of weighted Boolean constraint satisfaction problems. ", "Such problems are parameterised by a set $\\mathcal{F}$ of non-negative functions that may be used to assign weights to configurations (solutions) of the instance. ", "These functions take the place of the allowed constraint relations in classical constraint satisfaction problems (CSPs). ", "Indeed, the classical setting may be recovered by restricting $\\mathcal{F}$ to functions with range $\\{0,1\\}$. The key problem associated with an instance of a weighted CSP is to compute its partition function, i.e., the sum of weights of all its configurations. ", "Computing the partition function of a weighted CSP may be viewed a generalisation of counting the number of satisfying solutions of a classical CSP. ", "Many partition functions from statistical physics may be expressed as weighted CSPs. ", "For example, the *Potts model* [@welsh] is naturally expressible as a weighted CSP, whereas in the classical framework only the “hard core” versions may be directly expressed. (", "The hard-core version of the *antiferromagnetic* Potts model corresponds to graph colouring and the hard-core version of the *ferromagnetic* Potts model is trivial — acceptable configurations colour the entire graph with a single colour.) ", "A corresponding weighted version of the decision CSP was investigated by Cohen, Cooper, Jeavons and Krokhin [@CoCoJK06]. ", "This results in optimisation problems.", "\n\nWe use ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$ to denote the problem of computing the partition function of weighted CSP instances that can be expressed using only functions from $\\mathcal{F}$. We show in Theorem \\[thm:main\\] below that if every function $f\\in\n\\mathcal{F}$ is “of product type” then computing the partition function $Z(I)$ of an instance $I$ can be done in polynomial time. ", "Formal definitions are given later, but the condition of being “of product type” is easily checked — it essentially means that the partition function factors. ", "We show further in Theorem \\[thm:main\\] that if every function $f\\in\\mathcal F$ is “pure affine” then the partition function of $Z(I)$ can be computed in polynomial time. ", "Once again, there is an algorithm to check whether $\\mathcal\nF$ is pure affine. ", "For each other set $\\mathcal F$, we show in Theorem \\[thm:main\\] that computing the partition function of a ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal F)$ instance is complete for the class [${\\mbox{\\sf FP}\\xspace}^{\\mbox{\\scriptsize\\sf \\#P}}$]{}. ", "The existence of algorithms for testing the properties of being purely affine or of product type means that the dichotomy is effectively decidable.", "\n\nConstraint satisfaction\n-----------------------\n\n*Constraint Satisfaction*, which originated in Artificial Intelligence, provides a general framework for modelling decision problems, and has many practical applications. (", "See, for example [@RoBeWa06].) ", "Decisions are modelled by *variables*, which are subject to *constraints*, modelling logical and resource restrictions. ", "The paradigm is sufficiently broad that many interesting problems can be modelled, from satisfiability problems to scheduling problems and graph-theory problems. ", "Understanding the complexity of constraint satisfaction problems has become a major and active area within computational complexity [@cks; @hnbook].", "\n\nA Constraint Satisfaction Problem (CSP) typically has a finite *domain*, which we will denote by $[q]={\\left\\{0,1\\ldots,q-1\\right\\}}$ for a positive integer $q$.[^1] A *constraint language* $\\Gamma$ with domain $[q]$ is a set of relations on $[q]$. For example, take $q=2$. The relation $R=\\{(0,0,1)$, $(0,1,0)$, $(1,0,0)$, $(1,1,1)\\}$ is a 3-ary relation on the domain $\\{0,1\\}$, with four tuples.", "\n\nOnce we have fixed a constraint language $\\Gamma$, an *instance* of the CSP is a set of *variables* $V=\\{v_1,\\ldots,v_n\\}$ and a set of *constraints*. ", "Each constraint has a *scope,* which is a tuple of variables (for example, $(v_4, v_5, v_1)$) and a relation from $\\Gamma$ of the same arity, which constrains the variables in the scope. ", "A *configuration* $\\sigma$ is a function from $V$ to $[q]$. The configuration $\\sigma$ is *satisfying* if the scope of every constraint is mapped to a tuple that is in the corresponding relation. ", "In our example above, a configuration $\\sigma$ satisfies the constraint with scope $(v_4,v_5,v_1)$ and relation $R$ if and only if it maps an odd number of the variables in $\\{v_1,v_4,v_5\\}$ to the value $1$. Given an instance of a CSP with constraint language $\\Gamma$, the *decision problem* ${\\mbox{\\sf CSP}\\xspace}(\\Gamma$) asks us to determine whether any configuration is satisfying. ", "The *counting problem* ${\\mbox{\\sf \\#CSP}\\xspace}(\\Gamma$) asks us to determine the *number* of (distinct) satisfying configurations.", "\n\nVarying the constraint language $\\Gamma$ defines the classes and of decision and counting problems. ", "These contain problems of different computational complexities. ", "For example, if $\\Gamma=\\{R_1,R_2,R_3\\}$ where $R_1$, $R_2$ and $R_3$ are the three binary relations defined by $R_1=\\{(0,1),(1,0),(1,1)\\}$, $R_2=\\{(0,0),(0,1),(1,1)\\}$ and $R_3=\\{(0,0),(0,1),(1,0)\\}$, then ${\\mbox{\\sf CSP}\\xspace}(\\Gamma)$ is the classical 2-Satisfiability problem, which is in ¶. On the other hand, there is a similar constraint language $\\Gamma'$ with four relations of arity 3 such that 3-Satisfiability (which is -complete) can be represented in ${\\mbox{\\sf CSP}\\xspace}(\\Gamma')$. It may happen that the counting problem is harder than the decision problem. ", "If $\\Gamma$ is the constraint language of 2-Satisfiability above, then ${\\mbox{\\sf \\#CSP}\\xspace}(\\Gamma)$ contains the problem of counting independent sets in graph, and is -complete [@Valian79], even if restricted to 3-regular graphs [@Greenh00].", "\n\nAny decision problem ${\\mbox{\\sf CSP}\\xspace}(\\Gamma)$ is in , but not every problem in can be represented as a CSP. ", "For example, the question “Is $G$ Hamiltonian?” ", "cannot naturally be expressed as a CSP, because the property of being Hamiltonian cannot be captured by relations of bounded size. ", "This limitation of the class has an important advantage. ", "If ${\\mbox{\\sf P}\\xspace}\\neq {\\mbox{\\sf NP}\\xspace}$, then there are problems which are neither in ¶nor -complete [@L75]. ", "But, for well-behaved smaller classes of decision problems, the situation can be simpler. ", "We may have a *dichotomy theorem*, partitioning all problems in the class into those which are in ¶and those which are -complete. ", "There are no “leftover” problems of intermediate complexity. ", "It has been conjectured that there is a dichotomy theorem for . ", "The conjecture is that ${\\mbox{\\sf CSP}\\xspace}(\\Gamma)$ is in ¶for some constraint languages $\\Gamma$, and ${\\mbox{\\sf CSP}\\xspace}(\\Gamma)$ is -complete for all other constraint languages $\\Gamma$. This conjecture appeared in a seminal paper of Feder and Vardi [@fv], but has not yet been proved.", "\n\nA similar dichotomy, between and -complete, is conjectured for \\#CSP [@BD]. ", "The complexity classes and are the analogues of ¶and for counting problems. ", "is simply the class of functions computable in deterministic polynomial time. ", "is the class of integer functions that can be expressed as the number of accepting computations of a polynomial-time non-deterministic Turing machine. ", "Completeness in is defined with respect to polynomial-time Turing reducibility [@Pa94 Chap.", " 18]. ", "Bulatov and Dalmau [@BD] have shown in one direction that, if ${\\mbox{\\sf \\#CSP}\\xspace}(\\Gamma)$ is solvable in polynomial time, then the constraints in $\\Gamma$ must have certain algebraic properties (assuming ${\\mbox{\\sf P}\\xspace}\\neq{\\mbox{\\sf \\#P}\\xspace}$). ", "In particular, they must have a so-called *Mal’tsev polymorphism*. ", "The converse is known to be false, though it remains possible that the dichotomy (if it exists) does have an algebraic characterisation.", "\n\nThe conjectured dichotomies for and are major open problems for computational complexity theory. ", "There have been many important results for subclasses of and . ", "We mention the most relevant to our paper here. ", "The first decision dichotomy was that of Schaefer [@schaefer], for the Boolean domain $\\{0,1\\}$. Schaefer’s result is as follows.", "\n\n\\[thm:schaefer\\] Let $\\Gamma$ be a constraint language with domain $\\{0,1\\}$. The problem ${\\mbox{\\sf CSP}\\xspace}(\\Gamma)$ is in ¶if $\\Gamma$ satisfies one of the conditions below. ", "Otherwise, ${\\mbox{\\sf CSP}\\xspace}(\\Gamma)$ is -complete.", "\n\n1. ", " $\\Gamma$ is $0$-valid or $1$-valid.", "\n\n2. ", " $\\Gamma$ is weakly positive or weakly negative.", "\n\n3. ", " $\\Gamma$ is affine.", "\n\n4. ", " $\\Gamma$ is bijunctive.", "\n\nWe will not give detailed definitions of the conditions in Theorem \\[thm:schaefer\\], but the interested reader is referred to the paper [@schaefer] or to Theorem 6.2 of the textbook [@cks]. ", "An interesting feature is that the conditions in [@cks Theorem 6.2] are all checkable. ", "That is, there is an algorithm to determine whether ${\\mbox{\\sf CSP}\\xspace}(\\Gamma$) is in ¶or -complete, given a constraint language $\\Gamma$ with domain $\\{0,1\\}$. Creignou and Hermann [@CH] adapted Schaefer’s decision dichotomy to obtain a counting dichotomy for the Boolean domain. ", "Their result is as follows.", "\n\n\\[thm:CH\\] Let $\\Gamma$ be a constraint language with domain $\\{0,1\\}$. The problem ${\\mbox{\\sf \\#CSP}\\xspace}(\\Gamma)$ is in if $\\Gamma$ is affine. ", "Otherwise, ${\\mbox{\\sf \\#CSP}\\xspace}(\\Gamma)$ is -complete.", "\n\nA constraint language $\\Gamma$ with domain $\\{0,1\\}$ is *affine* if every relation $R\\in \\Gamma$ is affine. ", "A relation $R$ is affine if the set of tuples $x\\in R$ is the set of solutions to a system of linear equations over GF($2$). ", "These equations are of the form $v_1 \\oplus\n\\cdots \\oplus v_n =0$ and $v_1 \\oplus \\cdots \\oplus v_n =1$ where $\\oplus$ is the *exclusive or* operator. ", "It is well known (see, for example, Lemma 4.10 of [@cks]) that a relation $R$ is affine iff $a,b,c\\in R$ implies $d=a\\oplus b\\oplus c\\in R$. (We will use this characterisation below.) ", "There is an algorithm for determining whether a Boolean constraint language $\\Gamma$ is affine, so there is an algorithm for determining whether ${\\mbox{\\sf \\#CSP}\\xspace}(\\Gamma)$ is in or -complete.", "\n\nWeighted \n---------\n\nThe weighted framework of [@BG05] extends naturally to Constraint Satisfaction Problems. ", "Fix the domain $[q]$. Instead of constraining a length-$k$ scope with an arity-$k$ relation on $[q]$, we give a weight to the configuration on this scope by applying a function $f$ from $[q]^{k}$ to the non-negative rationals. ", "Let $\\mathcal{F}_q = \\{ f: [q]^k \\rightarrow\n\\Rationals^+ \\mid k\\in \\mathbb{N}\\}$ be the set of all such functions (of all arities).[^2] Given a function $f\\in\\mathcal{F}_q$ of arity $k$, the *underlying relation* of $f$ is given by $R_{f}=\\{x\\in[q]^k\n\\mid f(x)\\not=0\\}$. It is often helpful to think of $R_f$ as a table, with $k$ columns corresponding to the positions of a $k$-tuple. ", "Each row corresponds to a tuple $x=(x_1,\\ldots,x_k)\\in\nR_f$. The entry in row $x$ and column $j$ is $x_j$, which is a value in $[q]$.\n\nA *weighted* \\#CSP problem is parameterised by a finite subset $\\mathcal{F}$ of $\\mathcal{F}_q$, and will be denoted by ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$. An instance $I$ of ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$ consists of a set $V$ of *variables* and a set $\\mathcal{C}$ of *constraints*. ", "Each constraint $C\\in \\mathcal{C}$ consists of a function $f_C\\in \\mathcal{F}$ (say of arity $k_C$) and a *scope*, which is a sequence $s_C=(v_{C,1},\\ldots,v_{C,k_C})$ of variables from $V$. The variables $v_{C,1},\\ldots,v_{C,k_C}$ need not be distinct. ", "As in the unweighted case, a *configuration* $\\sigma$ for the instance $I$ is a function from $V$ to $[q]$. The *weight* of the configuration $\\sigma$ is given by $$w(\\sigma)=\\prod_{C\\in\n\\mathcal{C}} f_C(\\sigma(v_{C,1}),\\ldots,\\sigma(v_{C,k_C})).$$ Finally, the *partition function* $Z(I)$ is given, for instance $I$, by $$\\label{CSPZ} Z(I)=\\sum_{\\sigma:V\\rightarrow [q]} w(\\sigma).$$ In the computational problem ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$, the goal is to compute $Z(I)$, given an instance $I$.\n\nNote that an (unweighted) CSP counting problem ${\\mbox{\\sf \\#CSP}\\xspace}(\\Gamma)$ can be represented naturally as a weighted CSP counting problem. ", "For each relation $R\\in \\Gamma$, let $f^R$ be the indicator function for membership in $R$. That is, if $x\\in R$ we set $f^R(x)=1$. Otherwise we set $f^R(x)=0$. Let $\\mathcal{F}=\\{f^R \\mid R \\in \\Gamma\\}$. Then for any instance $I$ of ${\\mbox{\\sf \\#CSP}\\xspace}(\\Gamma)$, the number of satisfying configurations for $I$ is given by the (weighted) partition function $Z(I)$ from (\\[CSPZ\\]).", "\n\nThis framework has been employed previously in connection with *graph homomorphisms* [@bw]. ", "Suppose $H=(H_{ij})$ is any symmetric square matrix $H$ of rational numbers. ", "We view $H$ as being an edge-weighting of an undirected graph $\\mathcal{H}$, where a zero weight in $H$ means that the corresponding edge is absent from $\\mathcal{H}$. Given a (simple) graph $G=(V,E)$ we consider computing the partition function $$Z_H(G) = \\sum_{\\sigma:V\\rightarrow[q]}w(\\sigma),\\quad\\textrm{where}\\ \\\nw(\\sigma)=\\prod_{\\{u,v\\}\\in E} H_{\\sigma(u)\\sigma(v)}.$$ Within our framework above, we view $H$ as the binary function $h:[q]^2\\to \\mathbb{R}$, and the problem is then computing the partition function of $\\#CSP({\\left\\{h\\right\\}})$.\n\nBulatov and Grohe [@BG05] call $H$ *connected* if $\\mathcal{H}$ is connected and *bipartite* if $\\mathcal{H}$ is bipartite. ", "They give the following dichotomy theorem for non-negative $H$.[^3]\n\n\\[thm:bulgro\\] Let $H$ be a symmetric matrix with non-negative rational entries.", "\n\n1. ", " If $H$ is connected and not bipartite, then computing $Z_H$ is in if the rank of $H$ is at most $1$; otherwise computing $Z_H$ is -hard.", "\n\n2. ", " If $H$ is connected and bipartite, then computing $Z_H$ is in if the rank of $H$ is at most $2$; otherwise computing $Z_H$ is -hard.", "\n\n3. ", " If $H$ is not connected, then computing $Z_H$ is in if each of its connected components satisfies the corresponding conditions stated in (1) or (2); otherwise computing $Z_H$ is -hard.", "\n\nMany partition functions arising in statistical physics may be viewed as weighted problems. ", "An example is the $q$-state Potts model (which is, in fact, a weighted graph homomorphism problem). ", "In general, weighted is very closely related to the problem of computing the partition function of a Gibbs measure in the framework of Dobrushin, Lanford and Ruelle (see [@bw]). ", "See also the framework of Scott and Sorkin [@ss].", "\n\nSome Notation\n-------------\n\nWe will call the class of (rational) weighted \\#CSP problems *weighted* . ", "The sub-class having domain size $q=2$ will be called weighted *Boolean* , and will be the main focus of this paper. ", "We will give a dichotomy theorem for weighted Boolean .", "\n\nSince weights can be arbitrary non-negative rational numbers, the solution to these problems is not an integer in general. ", "Therefore ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$ is not necessarily in the class . ", "However, Goldberg and Jerrum [@GJ06] have observed that $Z(I)=\\tilde{Z}(I)/K(I)$, where $\\tilde{Z}$ is a function in and $K(I)$ is a positive integer computable in . ", "This follows because, for all $f\\in\\mathcal{F}$, we can ensure that $f(\\cdot)=\\tilde{f}(\\cdot)/K(I)$, where $\\tilde{f}(\\cdot)\\in\\mathbb{N}$, by“clearing denominators”. ", "The denominator $K(I)$ can obviously be computed in polynomial time, and it is straightforward to show that computing $\\tilde{Z}(I)$ is in ${\\mbox{\\sf \\#P}\\xspace}$, so the characterisation of [@GJ06] follows. ", "The resulting complexity class, comprising functions which are a function in divided by a function in , is named in [@GJ06], where it is used in the context of approximate counting. ", "Clearly we have $${\\textsf{weighted}\\,{\\mbox{\\sf \\#CSP}\\xspace}}\\ \\subseteq\\ {\\mbox{${\\mbox{\\sf \\#P}\\xspace}_\\Rationals$}\\xspace}\\ \\subseteq\\ {\\ensuremath{{\\mbox{\\sf FP}\\xspace}^{\\mbox{\\scriptsize\\sf \\#P}}}\\xspace}.$$ On the other hand, if $Z(I)\\in{\\textsf{weighted}\\,{\\mbox{\\sf \\#CSP}\\xspace}}$ is -hard, then, using an oracle for computing $Z(I$), we can construct a oracle $\\tilde{Z}(I)$ as outlined above. (", "Note that $Z(I)\\notin {\\mbox{\\sf \\#P}\\xspace}$ in general.) ", "Using this, we can compute any function in [${\\mbox{\\sf FP}\\xspace}^{\\mbox{\\scriptsize\\sf \\#P}}$]{}with a polynomial time-bounded oracle Turing machine. ", "Thus any -hard function in is complete for [${\\mbox{\\sf FP}\\xspace}^{\\mbox{\\scriptsize\\sf \\#P}}$]{}. ", "We will use this observation to state our main result in terms of completeness for the class [${\\mbox{\\sf FP}\\xspace}^{\\mbox{\\scriptsize\\sf \\#P}}$]{}.", "\n\nWe make the following definition, which relates to the discussion above. ", "We will say that $\\mathcal{F}\\subseteq \\mathcal{F}_q$ *simulates* $f\\in\\mathcal{F}_q$ if, for each instance $I$ of ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F}\\cup{\\left\\{f\\right\\}})$, there is a polynomial time computable instance $I'$ of ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$, such that $Z(I)=\\varphi(I)Z(I')$ for some $\\varphi(I)\\in\\Rationals$ which is -computable. ", "This generalises the notion of *parsimonious reduction* [@Pa94] among problems in . ", "We will use $\\redT$ to denote the relation “is polynomial-time Turing-reducible to” between computational problems. ", "Clearly, if $\\mathcal{F}$ simulates $f$, we have ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F}\\cup{\\left\\{f\\right\\}})\\redT{\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$. Note also that, if $\\tilde{f}=Kf$, for some constant $K>0$, then ${\\left\\{f\\right\\}}$ simulates $\\tilde{f}$. Thus there is no need to distinguish between “proportional” functions.", "\n\nWe use the following terminology for certain functions. ", "Let $\\chi_{=}$ be the binary *equality* function defined on $[q]$ as follows. ", "For any element $c\\in[q]$, $\\chi_=(c,c)=1$ and for any pair $(c,d)$ of distinct elements of $[q]$, $\\chi_{=}(c,d)=0$. Let $\\chi_{\\not=}$ be the binary *disequality* function given by $\\chi_{\\not=}(c,d)=1-\\chi_{=}(c,d)$ for all $c,d\\in[q]$.[^4] We say that a function $f$ is of [*product type*]{} if $f$ can be expressed as a product of unary functions and binary functions of the form $\\chi_=$ and $\\chi_{\\neq}$.\n\nWe focus attention in this paper on the Boolean case, $q=2$. In this case, we say that a function $f\\in\\mathcal{F}_2$ has *affine support* if its underlying relation $R_f$, defined earlier, is affine. ", "We say that $f$ is *pure affine* if it has affine support and range $\\{0,w\\}$ for some $w>0$. Thus a function is pure affine if and only if it is a positive real multiple of some (0,1-valued) function which is affine over GF(2).", "\n\nOur Result\n----------\n\nOur main result is the following.", "\n\n\\[thm:main\\] Suppose $\\mathcal{F}\\subseteq \\mathcal{F}_2\n= \\{f : {\\left\\{0,1\\right\\}}^k\\to \\mathbb{Q}^+\\mid k \\in \\mathbb{N}\\}$. If every function in $\\mathcal{F}$ is of product type then ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$ is in . ", "If every function in $\\mathcal{F}$ is pure affine then ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$ is in . ", "Otherwise, ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$ is [${\\mbox{\\sf FP}\\xspace}^{\\mbox{\\scriptsize\\sf \\#P}}$]{}-complete.", "\n\nSuppose first that $\\mathcal{F}$ is of product type. ", "In this case the partition function $Z(I)$ of an instance $I$ with variable set $V$ is easy to evaluate because it can be factored into easy-to-evaluate pieces: Partition the variables in $V$ into equivalence classes according to whether or not they are related by an equality or disequality function. (", "The equivalence relation on variables here is “depends linearly on”.) ", "An equivalence class consists of two (possibly empty) sets of variables $U_1$ and $U_2$. All of the variables in $U_1$ must be assigned the same value by a configuration $\\sigma$ of nonzero weight, and all variables in $U_2$ must be assigned the other value. ", "Variables in $U_1\\cup U_2$ are not related by equality or disequality to variables in $V\\setminus(U_1\\cup U_2)$. The equivalence class contributes one weight, say $\\alpha$, to the partition function if variables in $U_1$ are given value “$0$” by $\\sigma$ and it contributes another weight, say $\\beta$, to the partition function if variables in $U_1$ are given value “$1$” by $\\sigma$. Thus, $Z(I)=(\\alpha+\\beta)Z(I')$, where $I'$ is the instance formed from $I$ by removing this equivalence class. ", "Therefore, suppose we choose any equivalence class and remove its variables. ", "Since $\\mathcal{F}$ contains only unary, equality or binary disequality constraints, we can also remove all functions involving variables in $U_1\\cup U_2$ to give $\\mathcal{F}'$. Then $I'$ is of product type with fewer variables, so we may compute $Z(I')$ recursively.", "\n\nSuppose second that $\\mathcal{F}$ if pure affine. ", "Then $Z(I)=\\prod_{f\\in\\mathcal{F}} w_f^{k_f} Z(I')$, where $\\{0,w_f\\}$ is the range of $f$, $k_f$ is the number of constraints involving $f$ in $I$, and $I'$ is the instance obtained from $I$ by replacing every function $f$ by its underlying relation $R_f$ (viewed as a function with range $\\{0,1\\}$). ", "$Z(I')$ is easy to evaluate, because this is just counting solutions to a linear system over GF($2$), as Creignou and Hermann have observed [@CH].", "\n\nFinally, the -hardness in Theorem \\[thm:main\\] follows from Lemma \\[seven\\] below.", "\n\n\\[seven\\] If $f\\in\\mathcal{F}_2$ is not of product type and $g\\in\\mathcal{F}_2$ is not pure affine then ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f,g\\})$ is -hard.", "\n\nNote that the functions $f$ and $g$ in Lemma \\[seven\\] may be one and the same function. ", "So ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})$ is -hard when $f$ is not of product type nor pure affine. ", "The rest of this article gives the proof of Lemma \\[seven\\].", "\n\nUseful tools for proving hardness of \n=====================================\n\nNotation\n--------\n\nFor any sequence $u_1,\\ldots,u_k$ of variables of $I$ and any sequence $c_1,\\ldots,c_k$ of elements of the domain $[q]$, we will let $Z(I \\mid \\sigma(u_1)=c_1,\\ldots,\\sigma(u_k)=c_k)$ denote the contribution to $Z(I)$ from assignments $\\sigma$ with $\\sigma(u_1)=c_1,\\cdots,\\sigma(u_k)=c_k$.\n\nProjection {#sec:project}\n----------\n\nThe first tool that we study is projection, which is referred to as “integrating out” in the statistical physics literature.", "\n\nLet $f$ be a function of arity $k$, and let $J=\\{j_1,\\ldots,j_r\\}$ be a size-$r$ subset of $\\{1,\\ldots,k\\}$, where $j_1<\\cdots<j_r$.[^5] We say that a $k$-tuple $x'\\in [q]^k$ [*extends*]{} an $r$-tuple $x\\in[q]^r$ on $J$ (written $x'\\sqsupseteq_J x$) if $x'$ agrees with $x$ on indices in $J$; that is to say, $x'_{j_i}=x_i$ for all $1\\leq i \\leq r$. The [*projection*]{} $g$ of $f$ onto $J$ is defined as follows. ", "For every $x\\in[q]^r$, $g(x) = \\sum_{x'\\sqsupseteq_J x} f(x')$.\n\nThe following lemma may be viewed as a weighted version of Proposition 2 of [@BD], where it is proved for the unweighted case. ", "It is expressed somewhat differently in [@BD], in terms of counting the number of solutions to an existential formula.", "\n\n\\[lem:project\\] Suppose $\\mathcal{F}\\subseteq\\mathcal{F}_q$. Let $g$ be a projection of a function $f\\in\\mathcal{F}$ onto a subset of its indices. ", "Then ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F}\\cup\\{g\\}) \\redT {\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$.\n\nLet $k$ be the arity of $f$ and let $g$ be the projection of $f$ onto the subset $J$ of its indices. ", "Let $I$ be an instance of ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F}\\cup\\{g\\})$. We will construct an instance $I'$ of ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$ such that $Z(I)=Z(I')$. The instance $I'$ is identical to $I$ except that every constraint $C$ of $I$ involving $g$ is replaced with a new constraint $C'$ of $I'$ involving $f$. The corresponding scope $(v_{C',1},\\ldots,v_{C',k})$ is constructed as follows. ", "If $j_{\\ell}$ is the $\\ell$’th element of $J$, then $v'_{C',j_{\\ell}} = v_{C,\\ell}$. The other variables, $v_{C',j}$ ($j\\notin J$), are distinct new variables. ", "We have shown that $\\mathcal{F}$ simulates $g$ with $\\phi(I)=1$.\n\nPinning\n-------\n\nFor $c\\in[q]$, $\\delta_c$ denotes the unary function with $\\delta_c(c)=1$ and $\\delta_c(d)=0$ for $d\\neq c$. The following lemma, which allows “pinning” CSP variables to specific values in hardness proofs, generalises Theorem 8 of [@BD], which does the unweighted case. ", "Again [@BD] employs different terminology, and its theorem is a statement about the full idempotent reduct of a finite algebra. ", "The idea of pinning was used previously by Bulatov and Grohe of [@BG05] in the context of counting weighted graph homomorphisms (see Lemma 32 of [@BG05]). ", "A similar idea was used by Dyer and Greenhill in the context of counting *unweighted* graph homomorphisms — in that context, Theorem 4.1 of [@DG] allows pinning all variables to a particular *component* of the target graph $H$.\n\n\\[lem:pinning\\] For every $\\mathcal{F}\\subseteq \\mathcal{F}_q$, ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F}\\cup \\bigcup_{c\\in[q]}\\delta_c) \\redT {\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$.\n\nThe proof of Lemma \\[lem:pinning\\] is deferred to the appendix. ", "Since we only use the case $q=2$ in this paper, we provide the (simpler) proof for the Boolean case here.", "\n\n\\[lem:2pinning\\] For every $\\mathcal{F}\\subseteq \\mathcal{F}_2$, ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F}\\cup \\{\\delta_0,\\delta_1\\}) \\redT {\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$.\n\nFor $x\\in[2]^k$, let $\\overline{x}$ be the $k$-tuple whose $i$’th component, $\\overline{x}_i$, is $x_i \\xor 1$, for all $i$. Say that $\\mathcal{F}$ is *symmetric* if it is the case that for every arity-$k$ function $f\\in\\mathcal{F}$ and every $x\\in[2]^k$, $f(\\overline{x})=f(x)$.\n\nGiven an instance $I$ of ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F}\\cup \\{\\delta_0,\\delta_1\\})$ with variable set $V$ we consider two instances $I'$ and $I''$ of ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$. Let $V_0$ be the set of variables $v$ of $I$ to which the constraint $\\delta_0(v)$ is applied. ", "Let $V_1$ be the set of variables $v$ of $I$ to which the constraint $\\delta_1(v)$ is applied. ", "We can assume without loss of generality that $V_0$ and $V_1$ do not intersect. (", "Otherwise, $Z(I)=0$ and we can determine this without using an oracle for ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$.) Let $V_2=V\\setminus(V_0\\cup V_1)$. The instance $I'$ has variables $V_2 \\cup \\{t_0,t_1\\}$ where $t_0$ and $t_1$ are distinct new variables that are not in $V$. Every constraint $C$ of $I$ involving a function $f\\in \\mathcal{F}$ corresponds to a constraint $C'$ of $I'$. $C'$ is the same as $C$ except that variables in $V_0$ are replaced with $t_0$ and variables in $V_1$ are replaced with $t_1$. Similarly, the instance $I''$ has variables $V_2 \\cup \\{t\\}$ where $t$ is a new variable that is not in $V$. Every constraint $C$ of $I$ involving a function $f\\in \\mathcal{F}$ corresponds to a constraint $C''$ of $I''$. The constraint $C''$ is the same as $C$ except that variables in $V_0\\cup V_1$ are replaced with $t$.\n\n[**Case 1. ", "$\\mathcal{F}$ is symmetric:**]{} By construction, $$Z(I')-Z(I'') = Z(I' \\mid \\sigma(t_0)=0,\\sigma(t_1)=1) + Z(I'\\mid \\sigma(t_0)=1,\\sigma(t_1)=0).\\vspace{-1ex}$$ By symmetry, the summands are the same, so $$Z(I')-Z(I'') = 2 Z(I' \\mid \\sigma(t_0)=0, \\sigma(t_1)=1) =\n2 Z(I).$$\n\n[**Case 2. ", "$\\mathcal{F}$ is not symmetric:**]{} Let $f$ be an arity-$k$ function in $\\mathcal{F}$ and let $x\\in[2]^k$ so that $f(x)> f(\\overline{x})\\geq 0$. Let $s=(t_{x_1},\\ldots,t_{x_k})$ and let $I'_x$ be the instance derived from $I'$ by adding a new constraint with function $f$ and scope $s$. Similarly, let $I''_x$ be the instance derived from $I''$ by adding a new constraint with function $f$ and scope $(t,\\ldots,t)$. Now $$\\begin{aligned}\nZ(I'_x) &=\nZ(I' \\mid \\sigma(t_0)=0, \\sigma(t_1)=1)f(x) +\nZ(I'\\mid \\sigma(t_0)=1, \\sigma(t_1)=0)f(\\overline{x}) \\\\ &\\quad +\nZ(I'\\mid \\sigma(t_0)=0, \\sigma(t_1)=0) f(0,\\ldots,0) +\nZ(I'\\mid \\sigma(t_0)=1, \\sigma(t_1)=1) f(1,\\ldots,1)\n\\\\\n&=\nZ(I' \\mid \\sigma(t_0)=0, \\sigma(t_1)=1)f(x) +\nZ(I'\\mid \\sigma(t_0)=1, \\sigma(t_1)=0)f(\\overline{x}) +\nZ(I''_x).\\end{aligned}$$ Thus we have two independent equations, $$\\begin{aligned}\n Z(I'_x)-Z(I''_x)\\ &=\\ Z(I' \\mid \\sigma(t_0)=0, \\sigma(t_1)=1)f(x) +\nZ(I'\\mid \\sigma(t_0)=1, \\sigma(t_1)=0)f(\\overline{x}), \\\\\n Z(I')-Z(I'')\\ &=\\ Z(I' \\mid \\sigma(t_0)=0,\\sigma(t_1)=1)\\phantom{f(x)}\n + Z(I'\\mid \\sigma(t_0)=1,\\sigma(t_1)=0)\\phantom{f(\\overline{x})},\\end{aligned}$$ in the unknowns $Z(I' \\mid \\sigma(t_0)=0, \\sigma(t_1)=1)$ and $Z(I'\\mid \\sigma(t_0)=1, \\sigma(t_1)=0)$. Solving these, we obtain the value of $Z(I' \\mid \\sigma(t_0)=0, \\sigma(t_1)=1) = Z(I)$.\n\n-hard problems\n--------------\n\nTo prove Lemma \\[seven\\], we will give reductions from some known -hard problems. ", "The first of these is the problem of counting homomorphisms from simple graphs to $2$-vertex multigraphs. ", "We use the following special case of Bulatov and Grohe’s Theorem \\[thm:bulgro\\].", "\n\n\\[bulgro\\] Let $H$ be a symmetric $2\\times 2$ matrix with non-negative real entries. ", "If $H$ has rank 2 and at most one entry of $H$ is $0$ then [[Eval]{}]{}($H$) is -hard.", "\n\nWe will also use the problem of computing the *weight enumerator* of a linear code. ", "Given a *generating matrix* $A\\in\\{0,1\\}^{r\\times C}$ of rank $r$, a *code word* $c$ is any vector in the linear subspace $\\Upsilon$ generated by the rows of $A$ over GF(2). ", "For any real number $\\lambda$, the *weight enumerator* of the code is given by $W_A(\\lambda)=\\sum_{c\\in\n\\Upsilon}\\lambda^{\\|c\\|}$, where $\\|c\\|$ is the number of $1$’s in $c$. The problem of computing the weight enumerator of a linear code is in for $\\lambda\\in\\{-1,0,1\\}$, and is known to be -hard for every other fixed $\\lambda\\in\\Rationals$ (see [@welsh]). ", "We could not find a proof, so we provide one here. ", "We restrict attention to positive $\\lambda$, since that is adequate for our purposes.", "\n\nComputing the Weight Enumerator of a Linear Code is -hard for any fixed positive rational number $\\lambda\\neq 1$. \\[WE\\]\n\nWe will prove hardness by reduction from a problem $\\text{{{\\sc Eval}}}(H)$, for some appropriate $H$, using Corollary \\[bulgro\\]. ", "Let the input to $\\text{{{\\sc Eval}}}(H)$ be a connected graph $G=(V,E)$ with $V=\\{v_1,\\ldots,v_n\\}$ and $E=\\{e_1,\\ldots,e_m\\}$. Let $B$ be the $n\\times m$ incidence matrix of $G$, with $b_{ij}=1$ if $v_i\\in e_j$ and $b_{ij}=0$ otherwise. ", "Let $A$ be the $(n-1)\\times m$ matrix which is $B$ with the row for $v_n$ deleted. ", "$A$ will be the generating matrix of the Weight Enumerator instance, with $r=n-1$ and $C=m$. It has rank $(n-1)$ since $G$ contains a spanning tree. ", "A code word $c$ has $c_j=\\bigoplus_{i\\in U} b_{ij}$, where $U\\subseteq V\\setminus\\{v_{n}\\}$. Thus $c_j=1$ if and only if $e_j$ has exactly one endpoint in $U$, and the weight of $c$ is $\\lambda^{k}$, where $k$ is the number of edges in the cut $U,V\\setminus U$. Thus $W_A(\\lambda) = \\frac12 Z_H(G)$, where $H$ is the symmetric weight matrix with $H_{11}=H_{22}=1$ and $H_{12}=H_{21}=\\lambda$. The $\\frac12$ arises because we fixed which side of the cut contains $v_n$. Now $H$ has rank 2 unless $\\lambda=1$, so this problem is -hard by Corollary \\[bulgro\\]. ", "Note, by the way, that $Z_H(G)$ is the partition function of the Ising model in statistical physics [@Cip87].", "\n\nThe Proof of Lemma \\[seven\\]\n============================\n\nThroughout this section, we assume $q=2$. The following Lemma is a generalisation of a result of Creignou and Hermann [@CH], which deals with the case in which $f$ is a relation (or, in our setting, a function with range $\\{0,1\\}$). ", "The inductive technique used in the proof of Lemma \\[lem:CH\\] (combined with the follow-up in Lemma \\[four\\]) is good for showing that ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$ is -hard when $\\mathcal{F}$ contains a *single* function. ", "A very different situation arises when ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})$ and ${\\mbox{\\sf \\#CSP}\\xspace}(\\{g\\})$ are in but ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f,g\\})$ is -hard due to *interactions* between $f$ and $g$ — we deal with that problem later.", "\n\n\\[lem:CH\\] Suppose that $f\\in \\mathcal{F}_2$ does not have affine support. ", "Then ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})$ is -hard.", "\n\nLet $k$ be the arity of $f$, and let us denote the $i^{\\textrm{th}}$ component of $k$-tuple $a\\in R_f$ by $a_i$. The proof is by induction on $k$. The lemma is trivially true for $k=1$, since all functions of arity $1$ have affine support.", "\n\nFor $k=2$, we note that since $R_f$ is not affine, it is of the form $R_f = \\{(\\alpha,\\beta),(\\bar{\\alpha},\\beta),(\\bar{\\alpha},\\bar{\\beta})\\}$ for some $\\alpha\\in\\{0,1\\}$ and $\\beta\\in\\{0,1\\}$. We can show that ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})$ is -hard by reduction from [[Eval]{}]{}($H$) using $$H=\n\\left(\n \\begin{array}{cc}\nf(0,0) & f(0,1) \\\\\nf(1,0) & f(1,1)\\\\\n\\end{array}\n\\right),$$ which has rank $2$ and exactly one entry that is $0$. Given an instance $G=(V,E)$ of [[Eval]{}]{}($H$) we construct an instance $I$ of ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})$ as follows. ", "The variables of $I$ are the vertices of $G$. For each edge $e=(u,v)$ of $G$, add a constraint with function $f$ and variable sequence $u,v$. Corollary \\[bulgro\\] now tells us that [[Eval]{}]{}($H$) is -hard, so ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})$ is -hard.", "\n\nSuppose $k>2$. We start with some general arguments and notation. ", "For any $i\\in\\{1,\\ldots,k\\}$ and any $\\alpha\\in\\{0,1\\}$ let $f^{i=\\alpha}$ be the function of arity $k-1$ derived from $f$ by pinning the $i$’th position to $\\alpha$. That is, $f^{i=\\alpha}(x_1,\\ldots,x_{k-1})=\nf(x_1,\\ldots,x_{i-1},\\alpha,x_{i+1},\\ldots,x_k)$. Also, let $f^{i=*}$ be the projection of $f$ onto all positions apart from position $i$ (see Section \\[sec:project\\]). ", "Note that ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f^{i=\\alpha}\\})\\redT {\\mbox{\\sf \\#CSP}\\xspace}(\\{f,\\delta_0,\\delta_1\\})$, since $f^{i=\\alpha}$ can obviously be simulated by ${\\left\\{f, \\delta_0,\\delta_1\\right\\}}$. Furthermore, by Lemma \\[lem:2pinning\\], ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f,\\delta_0,\\delta_1\\})\\redT\n{\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})$. Thus, we can assume that $f^{i=\\alpha}$ has affine support — otherwise, we are finished by induction. ", "Similarly, by Lemma \\[lem:project\\], ${\\mbox{\\sf \\#CSP}\\xspace}({\\left\\{f^{i=*}\\right\\}})\\redT{\\mbox{\\sf \\#CSP}\\xspace}({\\left\\{f\\right\\}})$. Thus we can assume that $f^{i=*}$ has affine support — otherwise, we are finished by induction.", "\n\nNow, recall that $R_f$ is not affine. ", "Consider any $a,b,c\\in R_f$ such that $d=a\\oplus b \\oplus c\\notin R_f$. We have 4 cases.", "\n\n[**Case 1: There are indices $1\\leq i<j\\leq k$ such that $(a_i,b_i,c_i)=(a_j,b_j,c_j)$: $\\>$**]{} Without loss of generality, suppose $i=1$ and $j=2$. Define the function $f'$ of arity $(k-1)$ by $f'(r_2,\\ldots,r_k) = f(r_2,r_2,\\ldots,r_k)$. Note that $R_{f'}$ is not affine since the condition $a\\oplus b\\oplus c\\notin R_{f}$ is inherited by $R_{f'}$. So, by induction, ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f'\\})$ is -hard. ", "Now note that ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f'\\})\\redT {\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})$. To see this, note that any instance $I_1$ of ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f'\\})$ can be turned into an instance $I$ of ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})$ by repeating the first variable in the sequence of variables for each constraint.", "\n\n[**Case 2: There is an index $1\\leq i \\leq k$ such that $a_i=b_i=c_i$:$\\>$**]{} Since $d$ is not in $R_f$ and $d_i=a_i$, we find that $f^{i=a_i}$ does not have affine support, contrary to earlier assumptions.", "\n\nHaving finished Cases 1 and 2, we may assume without loss of generality that we are in Case 3 or Case 4 below, where ${\\left\\{\\alpha,\\beta\\right\\}}\\in{\\left\\{0,1\\right\\}}$, $\\bar{\\alpha}=1-\\alpha$, $\\bar{\\beta}=1-\\beta$ and $a',b',c'\\in{\\left\\{0,1\\right\\}}^{k-2}$.\n\n[**Case 3: $a=(\\bar{\\alpha},\\bar{\\beta},a')$, $b=(\\bar{\\alpha},\\beta,b')$, $c=(\\alpha,\\bar{\\beta},c')$:$\\>$**]{} Since $R_{f^{1=*}}$ is affine and $a$, $b$ and $c$ are in $R_f$, we must have either $d=(\\alpha,\\beta,d')\\in R_f$ or $e=(\\bar{\\alpha},\\beta,d')\\in R_f$, where $d'=a'\\oplus b'\\oplus c'$. In the first case, we are done (we have contradicted the assumption that $d\\not\\in R_f$), so assume that $e\\in R_f$ but $d\\not\\in R_f$. Similarly, since $R_{f^{2=*}}$ is affine, we may assume that $g=(\\alpha,\\bar{\\beta},d')\\in R_f$. Since $R_{f^{1=\\bar{\\alpha}}}$ is affine and $a$, $b$ and $e$ are in $R_f$, we find that $h=a\\oplus b\\oplus e=(\\bar{\\alpha},\\bar{\\beta},c')\\in R_f$. Since $R_{f^{2=\\bar{\\beta}}}$ is affine and $a$, $c$ and $g$ are in $R_f$, we find that $i=(\\bar{\\alpha},\\bar{\\beta},b')\\in R_f$. Also, since $R_{f^{2=\\bar{\\beta}}}$ is affine and $a$, $h$ and $i$ are in $R_f$, we find that $j=(\\bar{\\alpha},\\bar{\\beta},d')\\in R_f$. Let $f'(r_1,r_2) = f(r_1,r_2,d_3,\\ldots,d_k)$. Since $e$, $g$ and $j$ are in $R_f$ but $d$ is not, we have $(\\bar{\\alpha},\\beta),(\\alpha,\\bar{\\beta}),(\\bar{\\alpha},\\bar{\\beta})\\in\nR_{f'}$, but $(\\alpha,\\beta)\\notin R_{f'}$. Thus, $f'$ does not have affine support and ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f'\\})$ is -hard by induction. ", "Also, ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f'\\})\\redT{\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})$ by Lemma \\[lem:2pinning\\].", "\n\n[**Case 4: $a=(\\bar{\\alpha},\\alpha,a')$, $b=(\\bar{\\alpha},\\alpha,b')$, $c=(\\alpha,\\bar{\\alpha},c')$:$\\>$**]{} Since $R_{f^{1=*}}$ is affine and $a$, $b$ and $c$ are in $R_f$ but $d$ is not, we have $e=(\\bar{\\alpha},\\bar{\\alpha},d')\\in R_f$. Similarly, since $R_{f^{2=*}}$ is affine and $a$, $b$ and $c$ are in $R_f$ but $d$ is not, we have $g=(\\alpha,\\alpha,d')\\in R_f$. Now since $R_{f^{1=\\bar{\\alpha}}}$ is affine and $a$, $b$ and $e$ are in $R_f$, we have $h=(\\bar{\\alpha},\\bar{\\alpha},c')\\in R_f$. Also, since $R_{f^{2={\\alpha}}}$ is affine and $a$, $b$ and $g$ are in $R_f$, we have $i=(\\alpha,\\alpha,c')\\in R_f$.\n\nLet $f'(r_1,r_2) = f(r_1,r_2,c_3,\\ldots,c_k)$. If $j=(\\bar{\\alpha},\\alpha,c')\\not\\in R_f$ then $f'$ does not have affine support (since $c$, $h$ and $i$ are in $R_f$) so we finish by induction as in Case 3. ", "Suppose $j\\in R_f$. Since $R_{f^{1=\\bar{\\alpha}}}$ is affine and $a$, $b$ and $j$ are in $R_f$, we have $\\ell=(\\bar{\\alpha},\\alpha,d')\\in R_f$. Let $f''(r_1,r_2) = f(r_1,r_2,d_3,\\ldots,d_k)$. Then $f''$ does not have affine support (since $e$, $g$ and $\\ell$ are in $R_f$ but $d$ is not) so we finish by induction as in Case 3.", "\n\nLemma \\[lem:CH\\] showed that ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})$ is -hard when $f$ does not have affine support. ", "The following lemma gives another (rather technical, but useful) condition which implies that ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})$ is -hard. ", "We start with some notation. ", "Let $f$ be an arity-$k$ function. ", "For a value $b\\in\\{0,1\\}$, an index $i\\in\\{1,\\ldots,k\\}$, and a tuple $y\\in\\{0,1\\}^{k-1}$, let $\\substuple{y}{i}{b}$ denote the tuple $x\\in\\{0,1\\}^k$ formed by setting $x_i=b$ and $x_j=y_j$ $(j\\in{\\left\\{1,\\ldots,k\\right\\}}\\setminus{\\left\\{i\\right\\}})$.\n\nWe say that index $i$ of $f$ is *useful* if there is a tuple $y$ such that $f(\\substuple{y}{i}{0})>0$ and $f(\\substuple{y}{i}{1})>0$. We say that $f$ is [*product-like*]{} if, for every useful index $i$, there is a rational number $\\lambda_i$ such that, for all $y\\in\\{0,1\\}^{k-1}$, $$\\label{condition}\nf(\\substuple{y}{i}{0})=\\lambda_i f(\\substuple{y}{i}{1}).$$ If every position $i$ of $f$ is useful then being product-like is the same as being of product type. ", "However, being product-like is less demanding because it does not restrict indices that are not useful.", "\n\n\\[hard\\] \\[four\\] If $f\\in \\mathcal{F}_2$ is not product-like then [\\#CSP]{}$(\\{f\\})$ is -hard.", "\n\nWe’ll use Corollary \\[bulgro\\] to prove hardness, following an argument from [@dgp]. ", "Choose a useful index $i$ so that there is no $\\lambda_i$ satisfying (\\[condition\\]).", "\n\nSuppose $f$ has arity $k$. Let $A$ be the $2 \\times 2^{k-1}$ matrix such that for $b\\in\\{0,1\\}$ and $y\\in \\{0,1\\}^{k-1}$, $A_{b,y}=f(\\substuple{y}{i}{b})$. Let $A'=A A^T$.\n\nFirst, we show that [[Eval]{}]{}($A'$) is -hard. ", "Note that $A'$ is the following symmetric $2\\times 2$ matrix with non-negative rational entries. ", "$$\\left(\n\\begin{array}{cc}\n\\sum_y A_{0,y}^2 &\\sum_y A_{0,y}A_{1,y}\\\\\n\\sum_y A_{0,y}A_{1,y} & \\sum_y A_{1,y}^2\\\\\n\\end{array}\n\\right)\n=\n\\left(\n\\begin{array}{cc}\n\\sum_y {f(\\substuple{y}{i}{0})}^{2}\n &\\sum_y f(\\substuple{y}{i}{0})f(\\substuple{y}{i}{1})\\\\\n\\sum_y f(\\substuple{y}{i}{0})f(\\substuple{y}{i}{1})\n & \\sum_y f(\\substuple{y}{i}{1})^2\\\\\n\\end{array}\n\\right)$$ Since index $i$ is useful, all four entries of $A'$ are positive. ", "To show that [[Eval]{}]{}($A'$) is -hard by Corollary \\[bulgro\\], we just need to show that its determinant is non-zero. ", "By Cauchy-Schwartz, the determinant is non-negative, and is zero only if $\\lambda_i$ exists, which have assumed not to be the case. ", "Thus [[Eval]{}]{}($A'$) is -hard by Corollary \\[bulgro\\].", "\n\nNow we reduce [[Eval]{}]{}($A'$) to \\#CSP$(\\{f\\})$. To do this, take an undirected graph $G$ which is an instance of [[Eval]{}]{}($A'$). ", "Construct an instance $Y$ of \\#CSP$(\\{f\\})$. For every vertex $v$ of $G$ we introduce a variable $x_v$ of $Y$. Also, for every edge $e$ of $G$ we introduce $k-1$ variables $x_{e,1},\\ldots,x_{e,{k-1}}$ of $Y$. We introduce constraints in $Y$ as follows. ", "For each edge $e=(v,v')$ of $G$ we introduce constraints $f(x_v,x_{e,1},\\ldots,x_{e,k-1})$ and $f(x_{v'},x_{e,1},\\ldots,x_{e,k-1})$ into $Y$, where we have assumed, without loss of generality, that the first index is useful.", "\n\nIt is clear that [[Eval]{}]{}($A'$) is exactly equal to the partition function of the \\#CSP$(\\{f\\})$ instance $Y$.\n\nFor $w\\in\\Rationals^+$, let $U_{w}$ denote the unary function mapping 0 to 1 and 1 to $w$. Note that $U_0=\\delta_0$, and $U_1$ gives the constant (0-ary function) 1, occurrences of which leave the partition function unchanged. ", "So, by Lemma \\[lem:2pinning\\], we can discard these constraints since they do not add to the complexity of the problem. ", "Note, by the observation above about proportional functions, that the functions $U_w$ include all unary functions except for $\\delta_1$ and the constant 0. ", "We can discard $\\delta_1$ by Lemma \\[lem:2pinning\\], and if the constant 0 function is in $\\mathcal{F}$, any instance $I$ where it appears as a constraint has $Z(I)=0$. So again we can discard these constraints since they not add to the complexity of the problem.", "\n\nThus $U_w$ will be called *nontrivial* if $w\\notin{\\left\\{0,1\\right\\}}$. Let $\\oplus_{k}:\\{0,1\\}^{k}\\to\\{0,1\\}$ be the arity-$k$ parity function that is 1 iff its argument has an odd number of $1$s. ", "Let $\\neg\\oplus_{k}:\\{0,1\\}^{k}\\to\\{0,1\\}$ be the function $1-\\oplus_{k}$. The following lemma shows that even a simple function like $\\oplus_3$ can lead to intractable \\#CSP instances when it is combined with a nontrivial weight function $U_\\lambda$.\n\n${\\mbox{\\sf \\#CSP}\\xspace}(\\oplus_{3},U_{\\lambda},\\delta_0,\\delta_1)$ and ${\\mbox{\\sf \\#CSP}\\xspace}(\\neg\\oplus_{3},U_{\\lambda},\\delta_0,\\delta_1)$ are both -hard, for any positive $\\lambda\\neq 1$. \\[one\\]\n\nWe give a reduction from computing the Weight Enumerator of a Linear Code, which was shown to be -hard in Lemma \\[WE\\]. ", "In what follows, it is sometimes convenient to view $\\oplus_{k}$, $\\delta_{0}$, etc., ", "as relations as well as functions to ${\\left\\{0,1\\right\\}}$.\n\nWe first argue that for any $k$, the relation $\\oplus_k$ can be simulated by ${\\left\\{\\oplus_3, \\delta_0, \\delta_1\\right\\}}$. For example, to simulate $x_1 \\oplus \\cdots \\oplus x_k$ for $k>3$, take new variables $y$, $z$ and $w$ and let $m=\\lceil k/2 \\rceil$ and use $x_1 \\oplus \\cdots \\oplus x_m \\oplus y$ and $x_{m+1} \\oplus \\cdots \\oplus x_k \\oplus z$ and $y \\oplus z\\oplus w$ and $\\delta_0(w)$.\n\nSince ${\\left\\{\\oplus_3, \\delta_0, \\delta_1\\right\\}}$ can be used to simulate any relation $\\oplus_k$, we can use ${\\left\\{\\oplus_3, \\delta_0, \\delta_1\\right\\}}$ to simulate an arbitrary system of linear equations over $\\mathrm{GF}(2)$. In particular we can use them to simulate the subspace $\\Upsilon$ of code words for a given generating matrix $A$.", "\n\nFinally, we can use $U_{\\lambda}$ to simulate the function which evaluates the weight enumerator on $\\Upsilon$. Then, since $\\lambda\\neq 0,1$, we can apply Lemma \\[WE\\] to complete the argument. ", "The same proof, with minor modifications, applies to $\\neg\\oplus_3$.\n\nSuppose $f\\in\\mathcal{F}_2$ is not of product type. ", "Then, for any positive $\\lambda\\neq 1$, there exists a constant $c$, depending on $f$, such that ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f,\\delta_0,\\delta_1,U_{\\lambda},U_{c}\\})$ is -hard. ", "\\[two\\]\n\nIf $f$ does not have affine support, the result follows by Lemma \\[lem:CH\\]. ", "So suppose $f$ has affine support. ", "Consider the underlying relation $R_f$, viewed as a table. ", "The rows of the table represent the tuples of the relation. ", "Let $J$ be the set of columns on which the relation is not constant. ", "That is, if $i\\in J$ then there is a row $x$ with $x_i=0$ and a row $y$ with $y_i=1$. Group the columns in $J$ into equivalence classes: two columns are equivalent iff they are equal or complementary. ", "Let $k$ be the number of equivalence classes. ", "Take one column from each of the $k$ equivalence classes as a representative, and focus on the arity-$k$ relation $R$ induced by those columns.", "\n\n[**Case 1: Suppose $R$ is the complete relation of arity $k$.**]{}\\\nLet $f^*$ be the projection of $f$ onto the $k$ columns of $R$. By Lemma \\[lem:project\\], $${\\mbox{\\sf \\#CSP}\\xspace}(\\{f^*\\})\n\\, \\redT\\, {\\mbox{\\sf \\#CSP}\\xspace}(\\{f\\})\\, \\redT\\,\n{\\mbox{\\sf \\#CSP}\\xspace}(\\{f,\\delta_0,\\delta_1,U_{\\lambda},U_{c}\\}).\\vspace{-1.4ex}$$ We will argue that ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f^*\\})$ is -hard. ", "To see this, note that every column of $f^*$ is useful. ", "Thus, if $f^*$ were product-like, we could conclude that $f^*$ was of product type. ", "But this would imply that $f$ is of product type, which is not the case by assumption. ", "So $f^*$ is not product-like and hardness follows from Lemma \\[hard\\].", "\n\n[**Case 2: Suppose $R$ is not the complete relation of arity $k$.**]{}\\\nWe had assumed that $R_f$ is affine. ", "This means that given three vectors, $x$, $y$ and $z$, in $R_f$, $x\\xor y \\xor z$ is in $R_f$ as well. ", "The arity-$k$ relation $R$ inherits this property, so is also affine.", "\n\nChoose a minimal set of columns of $R$ that do not induce the complete relation. ", "This exists by assumption. ", "Suppose there are $j$ columns in this minimal set. ", "Observe that $j\\neq 1$ because there are no constant columns in $J$. Also $j\\not=2$, since otherwise the two columns would be related by equality or disequality, contradicting the preprocessing step. ", "The argument here is that on two columns, $R$ cannot have exactly three tuples because it is affine, and having tuples $x$, $y$ and $z$ in would require the fourth tuple $x\\xor y \\xor z$. But if it has two tuples then, because there are no constant columns, the only possibilities are either $(0,0)$ and $(1,1)$, or $(0,1)$ and $(1,0)$. Both contradict the preprocessing step, so $j\\geq3$.\n\nLet $R'$ be the restriction of $R$ to the $j$ columns. ", "Now $R'$ of course has fewer than $2^j$ rows, and at least $2^{j-1}$ by minimality. ", "It is affine, and hence must be $\\oplus_j$ or $\\neg\\oplus_j$. To see this, first note that the size of $R'$ has to be a power of $2$ since $R'$ is the solution to a system of linear equations. ", "Hence the size of $R'$ must be $2^{j-1}$. Then, since there are $j$ variables, there can only be one defining equation. ", "And, since every subset of $j-1$ variables induces a complete relation, this single equation must involve all variables. ", "Therefore, the equation is $\\xor_j$ or $\\neg\\xor_j$.\n\nLet $f'$ be the projection of $f$ onto the $j$ columns just identified. ", "Let $f''$ be further obtained by pinning all but three of the $j$ variables to 0. ", "Pinning $j-3$ variables to $0$ leaves a single equation involving all three remaining variables. ", "Thus $R_{f''}$ must be $\\xor_3$ or $\\neg \\xor_3$.\n\nNow define the symmetric function $f'''$ by $$f'''(a,b,c) =\nf''(a,b,c)\\*f''(a,c,b)\\*f''(b,a,c)\\*f''(b,c,a)\\*f''(c,a,b)\\*f''(c,b,a),\\vspace{-1.4ex}$$ Note that $R_{f'''}$ is $\\oplus_3$ or $\\neg\\oplus_3$, since $R_{f''}$ is symmetric and hence $R_{f'''}=R_{f''}$.\n\nTo summarise: using $f$ and the constant functions $\\delta_0$ and $\\delta_1$, we have simulated a function $f'''$ such that its underlying relation $R_{f'''}$ is either $\\oplus_3$ or $\\neg\\oplus_3$. Furthermore, if triples $x$ and $y$ have the same number of $1$s then $f'''(x)=f'''(y)$.\n\nWe can now simulate an unweighted version of $\\oplus_3$ or $\\neg\\oplus_3$ using $f'''$ and a unary function $U_c$, with $c$ set to a conveniently-chosen value. ", "There are two cases. ", "Suppose first that the affine support of $f'''$ is $\\neg\\xor_3$. Then let $w_{0}$ denote the value of $f'''$ when applied to the $3$-tuple $(0,0,0)$ and let $w_2$ denote $f'''(0,1,1)=f'''(1,0,1)=f'''(1,1,0)$. Recall that $f'''(x)=0$ for any other $3$-tuple $x$. Now let $c={(w_0/w_2)}^{1/2}$. Note from the definition of $f'''$ that $w_{0}$ and $w_{2}$ are squares of rational numbers, so $c$ is also rational. ", "Define a function $g$ of arity 3 by $g(\\alpha,\\beta,\\gamma) =\nU_c(\\alpha)U_c(\\beta) U_c(\\gamma)f'''(\\alpha, \\beta, \\gamma)$. Note that $g(0,0,0)=w_0$ and $g(0,1,1)=g(1,0,1)=g(1,1,0) = c^2 w_2 = w_0$. Thus, $g$ is a pure affine function with affine support $\\neg\\xor_3$ and range $\\{0,w_0\\}$. The other case, in which the affine support of $f'''$ is $\\xor_3$, is similar.", "\n\nWe have established a reduction from either ${\\mbox{\\sf \\#CSP}\\xspace}(\\oplus_{3},U_{\\lambda},\\delta_0,\\delta_1)$ or ${\\mbox{\\sf \\#CSP}\\xspace}(\\neg\\oplus_{3},U_{\\lambda},\\delta_0,\\delta_1)$, which are both -hard by Lemma \\[one\\].", "\n\n\\[five\\] If $f\\in\\mathcal{F}_2$ is not of product type, then ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f,\\delta_0,\\delta_1,U_{\\lambda}\\})$ is -hard for any positive $\\lambda\\neq 1$.\n\nTake an instance $I$ of ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f,\\delta_0,\\delta_1,U_{\\lambda},U_{c}\\})$, from Lemma \\[two\\], with $n$ variables $x_1,x_2,\\ldots,x_n$. We want to compute the partition function $Z(I)$ using only instances of ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f,\\delta_0,\\delta_1,U_{\\lambda}\\})$. That is, instances which avoid using constraints $U_c$. For each $i$, let $m_{i}$ denote the number of copies of $U_{c}$ that are applied to $x_i$, and let $m=\\sum_{i=1}^n m_{i}$. Then we can write the partition function as $Z(I)=Z(I;c)$ where $$Z(I;w) = \\sum_{\\sigma\\in{\\{0,1\\}}^n}\n\\hat{Z}(\\sigma) \\prod_{i:\\sigma_{i}=1}w^{m_{i}}=\\sum_{\\sigma\\in{\\{0,1\\}}^n}\n\\hat{Z}(\\sigma) w^{\\sum_{i=1}^n m_{i}\\sigma_i},$$ where $\\hat{Z}(\\sigma)$ denotes the value corresponding to the assignment $\\sigma(x_i)=\\sigma_i$, ignoring constraints applying $U_c$, and $w$ is a variable. ", "So $\\hat{Z}(\\sigma)$ is the weight of $\\sigma$, taken over all constraints other than those applying $U_{c}$. Note also that $Z(I;w)$ is a polynomial of degree $m$ in $w$. We can evaluate $Z(I;w)$ at the point $w=\\lambda^j$ by replacing each $U_c$ constraint with $j$ copies of a $U_\\lambda$ constraint. ", "This evaluation is an instance of ${\\mbox{\\sf \\#CSP}\\xspace}({\\left\\{f,\\delta_0,\\delta_1,U_{\\lambda}\\right\\}})$. So, using $m$ different values of $j$ and interpolating, we learn the coefficients of the polynomial $Z(I;w)$. Then we can put $w=c$ to evaluate $Z(I)$.\n\n\\[six\\] Suppose $f\\in\\mathcal{F}_2$ is not of product type, and $g\\in\\mathcal{F}_2$ is not pure affine. ", "Then ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f,g,\\delta_0,\\delta_1\\})$ is -hard.", "\n\nIf $g$ does not have affine support we are done by Lemma \\[lem:CH\\]. ", "So suppose that $g$ has affine support. ", "Since $g$ is not pure affine, the range of $g$ contains at least two non-zero values.", "\n\nThe high-level idea will be to use pinning and bisection to extract a non-trivial unary weight function $U_\\lambda$ from $g$. Then we can reduce from ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f,\\delta_0,\\delta_1,U_{\\lambda}\\})$, which we proved -hard in Lemma \\[five\\].", "\n\nLook at the relation $R_g$, viewed as a table. ", "If every column were constant, then $g$ would be pure affine, so this is not the case. ", "Select a non-constant column with index $h$. If there are two non-zero values in the range of $g$ amongst the rows of $R_g$ that are $0$ in column $h$ then we derive a new function $g'$ by pinning column $h$ to $0$. The new function $g'$ is not pure affine, since the two non-zero values prevent this. ", "So we will show inductively that ${\\mbox{\\sf \\#CSP}\\xspace}({\\left\\{f,g',\\delta_0,\\delta_1\\right\\}})$ is -hard. ", "This will give the result since ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f,g',\\delta_0,\\delta_1\\})$ trivially reduces to ${\\mbox{\\sf \\#CSP}\\xspace}(\\{f,g,\\delta_0,\\delta_1\\})$.\n\nIf we don’t finish this way, or symmetrically by pinning column $h$ to $1$, then we know that there are distinct positive values $w_0$ and $w_1$ such that, for every row $x$ of $R_g$ with $0$ in column $h$, $g(x)=w_0$ and, for every row $x$ of $R_g$ with $1$ in column $h$, $g(x)=w_1$. Now note that, because the underlying relation $R_g$ is affine, it has the same number of $0$’s in column $h$ as $1$’s. ", "This is because $R_g$ is the solution of a set of linear equations. ", "Adding the equation $x_h=0$ or $x_h=1$ exactly halves the set of solutions in either case. ", "We now project onto the index set ${\\left\\{h\\right\\}}$. We obtain the unary weight function $U_\\lambda$, with $\\lambda=w_1/w_0$, on using the earlier observation about proportional functions. ", "This was our goal, and completes the proof.", "\n\nLemma \\[seven\\] now follows from Lemma \\[lem:2pinning\\] and Lemma \\[six\\], completing the proof of Theorem \\[thm:main\\].", "\n\n[1]{} G. Brightwell and P. Winkler, Graph homomorphisms and phase transitions, *Journal of Combinatorial Theory (Series B)* **77** (1999), 221–262\n\nA. Bulatov and V. Dalmau, Towards a dichotomy theorem for the counting constraint satisfaction problem, in *Proc. 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", "Annual ACM Symposium on Theory of Computing*, ACM Press, 1978, pp. ", "216–226.", "\n\nA. Scott and G. Sorkin, Polynomial constraint satisfaction: a framework for counting and sampling CSPs and other problems, <http://arxiv.org/abs/cs/0604079>.", "\n\nJ. Schwartz, Fast probabilistic algorithms for verification of polynomial identities, *Journal of the Association for Computing Machinery* **27** (1980), 701–717.", "\n\nL. Valiant, The complexity of enumeration and reliability problems, *SIAM Journal on Computing* **8** (1979), 410–421.", "\n\nD. Welsh, [*Complexity: knots, colourings and counting*]{}, LMS Lecture Note Series, vol.", " 186, Cambridge University Press, 1993.", "\n\nAppendix\n========\n\nThe purpose of this appendix is to prove Lemma \\[lem:pinning\\] for an arbitrary fixed domain $[q]$. We used only the special case $q=2$, which we stated and proved as Lemma \\[lem:2pinning\\]. ", "However, pinning appears to be a useful technique for studying the complexity of , so we give a proof of the general Lemma \\[lem:pinning\\], which we believe will be applicable elsewhere.", "\n\nFor every $\\mathcal{F}\\subseteq \\mathcal{F}_q$, ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F}\\cup \\bigcup_{c\\in[q]}\\delta_c) \\redT {\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$.\n\nIn order to prove the lemma, we introduce a useful, but less natural, variant of . ", "Suppose $\\mathcal{F}\\subseteq \\mathcal{F}_q$. An instance $I$ of ${\\mbox{\\sf \\#CSP}\\xspace}^{\\neq}(\\mathcal{F})$ consists of a set $V$ of variables and a set $\\mathcal{C}$ of constraints, just like an instance of ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$. In addition, the instance may contain a *single* extra constraint $C$ applying the arity-$q$ *disequality* relation $\\chi_{\\neq}$ with scope $(v_{C,1},\\ldots,v_{C,q})$.\n\nThe disequality relation $\\chi_{\\neq}$ is defined by $\\chi_{\\neq}(x_1,\\ldots,x_q)=1$ if $x_1,\\ldots,x_q\\in[q]$ are pairwise distinct. ", "That is, if they are a permutation of the domain $[q]$. Otherwise, $\\chi_{\\neq}(x_1,\\ldots,x_q)=0$.\n\nLemma \\[lem:pinning\\] follows immediately from Lemma \\[lem:first\\] and \\[lem:second\\] below.", "\n\n\\[lem:first\\] For every $\\mathcal{F}\\subseteq \\mathcal{F}_q$, ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F}\\cup \\bigcup_{c\\in[q]}\\delta_c) \\redT{\\mbox{\\sf \\#CSP}\\xspace}^{\\neq}(\\mathcal{F})$.\n\nWe follow the proof lines of Lemma \\[lem:2pinning\\], but instead of subtracting the contribution corresponding to configurations in which some $t_i$’s get the same value, we use the disequality relation to restrict the partition function to configurations in which they get distinct values.", "\n\nSay that $\\mathcal{F}$ is *symmetric* if it is the case that for every arity-$k$ function $f\\in\\mathcal{F}$ and every tuple $x\\in[q]^k$ and every permutation $\\pi:[q]\\rightarrow[q]$, $f(x_1,\\ldots,x_k) = f(\\pi(x_1),\\ldots,\\pi(x_k))$.\n\nLet $I$ be an instance of ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F}\\cup \\bigcup_{c\\in[q]}\\delta_c)$ with variable set $V$. Let $V_c$ be the set of variables $v\\in V$ to which the constraint $\\delta_c(v)$ is applied. ", "Assume without loss of generality that the sets $V_c$ are pairwise disjoint. ", "Let $V_{q}=V \\setminus\\bigcup_{c\\in[q]} V_c$. We construct an instance $I'$ of ${\\mbox{\\sf \\#CSP}\\xspace}^{\\neq}(\\mathcal{F})$. The instance has variables $V_q \\cup \\{t_0,\\ldots,t_{q-1}\\}$. Every constraint $C$ of $I$ involving a function $f\\in\\mathcal{F}$ corresponds to a constraint $C'$ of $I'$. Here $C'$ is the same as $C$ except that variables in $V_c$ are replaced with $t_c$, for each $c\\in[q]$. Also, we add a new disequality constraint to the new variables $t_0,\\ldots,t_{q-1}$.\n\n[**Case 1. ", "$\\mathcal{F}$ is symmetric:**]{}\n\nBy construction, $Z(I')= \\sum_{y_0,\\ldots,y_{q-1}} Z(I'\\mid\n\\sigma(t_0)=y_0,\\ldots,\\sigma(t_{q-1})=y_{q-1})$, where the sum is over all permutations $y_0,\\ldots,y_{q-1}$ of $[q]$. By symmetry, the summands are all the same, so $Z(I') = q! ", "Z(I' \\mid \\sigma(t_0)=0,\\ldots,\\sigma(t_{q-1})=q-1) =\nq! ", "Z(I)$.\n\n[**Case 2. ", "$\\mathcal{F}$ is not symmetric:**]{}\n\nSay that two permutations $\\pi_1:[q]\\rightarrow[q]$ and $\\pi_2:[q]\\rightarrow[q]$ are *equivalent* if, for every $f\\in\\mathcal{F}$ and every tuple $x\\in[q]^k$, $f(\\pi_1(x_1),\\ldots,\n\\pi_1(x_k)) = f(\\pi_2(x_1),\\ldots,\\pi_2(x_k))$. Partition the permutations $\\pi:[q]\\rightarrow[q]$ into equivalence classes. ", "Let $h$ be the number of equivalence classes and $n_i$ be the size of the $i$’th equivalence class, so $n_1 + \\cdots + n_h=q!$.[^6] Let $\\{\\pi_1,\\ldots,\\pi_h\\}$ be a set of representatives of the equivalence classes with $\\pi_1$ being the identity. ", "We know that $n_1\\neq q!$ since $\\mathcal{F}$ is not symmetric.", "\n\nFor a positive integer $\\ell$ we will now build an instance $I'_\\ell$ by adding new constraints to $I'$. For each $\\pi_i$ other than $\\pi_1$ we add constraints as follows. ", "Choose a function $f_i\\in\\mathcal{F}$ and a tuple $y$ such that $f_i(y_1,\\ldots,y_k)\\neq f_i(\\pi_i(y_1),\\ldots,\\pi_i(y_k))$. If $f_i(y_1,\\ldots,y_k)> f_i(\\pi_i(y_1),\\ldots,\\pi_i(y_k))$ then define the $k$-tuple $x^i$ by $(x^i_1,\\ldots,x^i_k)\n=(y_1,\\ldots,y_k)$. Otherwise, let $n$ be the order of the permutation $\\pi_i$ and let $g_r$ denote $f_i(\\pi_i^r(y_1),\\ldots,\\pi_i^r(y_k))$. Since $g_0<g_1$ and $g_n=g_0$ there exists a $\\xi\\in\\{1,\\ldots,n-1\\}$ such that $g_\\xi>g_{\\xi+1}$. Let $(x^i_1,\\ldots,x^i_k)=(\\pi^{\\xi}(y_1),\\ldots,\\pi^{\\xi}(y_k))$ so $f_i(x^i_1,\\ldots,x^i_k)> f_i(\\pi_i(x^i_1),\\ldots,\\pi_i(x^i_k))$.\n\nLet $w_{ij}$ denote $f_i(\\pi_j(x^i_1),\\ldots,\\pi_j(x^i_k))$ so, since $\\pi_1$ is the identity, we have just ensured that $w_{i1}>w_{ii}$. Let $s^i=(t_{x_1^i},\\ldots,t_{x_k^i})$, and let $0\\leq z_i\\leq h$ $(i=2,\\ldots,h)$ be positive integers, which we will determine below. ", "Add $\\ell z_i$ new constraints to $I'_\\ell$ with relation $f_i$ and scope $s^i$. Let $\\lambda_i = \\prod_{\\gamma=2}^h w_{\\gamma i}^{z_\\gamma}$. Note that, given $\\sigma(t_0)=\\pi_i(0),\\ldots,\\sigma(t_{q-1})=\\pi_i(q-1)$, the contribution to $Z(I'_\\ell)$ for the new constraints is $$\\prod_{\\gamma=2}^h f_\\gamma(\\sigma(t_{x^\\gamma_1}),\\ldots,\\sigma(t_{x^\\gamma_k}))^{z_\\gamma \\ell}=\n\\prod_{\\gamma=2}^h f_\\gamma(\\pi_i(x^\\gamma_1),\\ldots,\\pi_i(x^\\gamma_k))^{z_\\gamma \\ell}=\n\\prod_{\\gamma=2}^h w_{\\gamma,i}^{z_\\gamma \\ell}\n= \\bigg(\\prod_{\\gamma=2}^h w_{\\gamma,i}^{z_{\\gamma}} \\bigg)^\\ell=\n{\\lambda_i}^\\ell.$$ So $$Z(I'_\\ell) = \\sum_{i=1}^h n_i\\,\nZ(\\,I'\\mid \\sigma(t_0)=\\pi_i(0),\\ldots,\\sigma(t_{q-1})=\\pi_i(q-1)\\,)\n\\,\\lambda_i^\\ell.$$\n\nWe have ensured that $\\lambda_1>0$, since $w_{i1}>w_{ii}\\geq 0$, so $w_{i1}>0$ for all $i=2,\\ldots,h$. We now choose the $z_i$’s so that $\\lambda_i\\neq \\lambda_1$ for all $i=2,\\ldots,h$. If $w_{\\gamma i}=0$ for any $\\gamma=2,\\ldots,h$, we have $\\lambda_i=0$ and hence $\\lambda_i\\neq\\lambda_1$. Thus we will assume, without loss of generality, that $w_{\\gamma i}>0$ for all $\\gamma=2,\\ldots,h$ and $i=2,\\ldots,h'$, where $h'\\leq h$. Then we have $$\\frac{\\lambda_i}{\\lambda_1}\\ =\\ \\prod_{\\gamma=2}^h \\Big(\\frac{w_{\\gamma i}}{w_{\\gamma 1}}\\Big)^{z_\\gamma}\\ =\\ e^{\\sum_{\\gamma=2}^h\\alpha_{\\gamma i}z_\\gamma}\n\\qquad (i=2,\\ldots,h'),$$ where $\\alpha_{\\gamma i}=\\ln(w_{\\gamma i}/w_{\\gamma 1})$. Note that $\\alpha_{ii}< 0$, since $w_{ii}<w_{i1}$. We need to find an integer vector $z=(z_2,\\ldots,z_h)$ so that none of the linear forms $\\mathcal{L}_i(z)=\\sum_{\\gamma=2}^h\\alpha_{\\gamma i}z_\\gamma$ is zero, for $i=2,\\ldots,h'$. We do this using a proof method similar to the Schwartz-Zippel Lemma. (", "See, for example, [@Schwar80].) ", "None of the $\\mathcal{L}_i(z)$ is identically zero, since $\\alpha_{ii}\\neq 0$. Consider the integer vectors $z\\in[h]^{h-1}$. At most $h^{h-2}$ of these can make $\\mathcal{L}_i(z)$ zero for any $i$, since the equation $\\mathcal{L}_i(z)=0$ makes $z_i$ a linear function of $z_\\gamma$ ($\\gamma\\neq i$). ", "Therefore there are at most $(h'-1)h^{h-2}<h^{h-1}$ such $z$ which make any $\\mathcal{L}_i(z)$ zero. ", "Therefore there must be a vector $z\\in[h]^{h-1}$ for which none of the $\\mathcal{L}_i(z)$ is zero, and this is the vector we require.", "\n\nNow, by combining terms with equal $\\lambda_i$ and ignoring terms with $\\lambda_i=0$, we can view $Z(I'_\\ell)$ as a sum $Z(I'_\\ell) = \\sum_i c_i \\lambda_i^{\\ell}$ where the $\\lambda_i$’s are positive and pairwise distinct and $$c_1 = n_1 Z(I'\\mid \\sigma(t_0)=0,\\ldots,\\sigma(t_{q-1})=q-1).$$ Thus, by Lemma 3.2 of [@DG] we can interpolate to recover $c_1$. Dividing by $n_1$, we get $$Z(I'\\mid \\sigma(t_0)=0,\\ldots,\\sigma(t_{q-1})=q-1)=Z(I).\\qedhere$$\n\nFor every $\\mathcal{F}\\subseteq \\mathcal{F}_q$, ${\\mbox{\\sf \\#CSP}\\xspace}^{\\neq}(\\mathcal{F}) \\redT {\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$. \\[lem:second\\]\n\nWe use Möbius inversion for posets, following the lines of the proof of [@BD Theorem 8].[^7] Consider the set of partitions of $[q]$. Let $\\underline{0}$ denote the partition with $q$ singleton classes. ", "Consider the partial order in which $\\eta\\leq \\theta$ iff every class of $\\eta$ is a subset of some class of $\\theta$. Define $\\mu(\\underline{0})=1$ and for any $\\theta\\neq \\underline{0}$ define $\\mu(\\theta)=-\\sum_{\\eta\\leq \\theta,\\eta\\neq\\theta} \\mu(\\eta)$. Consider the sum $\\sum_{\\eta \\leq \\theta} \\mu(\\eta)$. Clearly, this sum is $1$ if $\\theta=\\underline{0}$. From the definition of $\\mu$, it is also easy to see that the sum is $0$ otherwise, since $$\\sum_{\\eta \\leq \\theta} \\mu(\\eta) = \\mu(\\theta)+\n\\sum_{\\eta \\leq \\theta,\\eta\\neq \\theta} \\mu(\\eta)\n=0.$$\n\nNow let $I$ be an instance of $ {\\mbox{\\sf \\#CSP}\\xspace}^{\\neq}(\\mathcal{F})$ with a disequality constraint applied to variables $t_0,\\ldots,t_{q-1}$. Let $V$ be the set of variables of $I$. Given a configuration $\\sigma:V \\rightarrow[q]$, let $\\vartheta(\\sigma)$ be the partition of $[q]$ induced by of $(\\sigma(t_0),\\ldots,\\sigma(t_{q-1}))$. Thus $i$ and $j$ in $[q]$ are in the same class of $\\vartheta(\\sigma)$ iff $\\sigma(t_i)=\\sigma(t_j)$. We say that a partition $\\eta$ is consistent with $\\sigma$ (written $\\eta \\preccurlyeq \\sigma$) if $\\eta\\leq \\vartheta(\\sigma)$. Note that $\\eta \\preccurlyeq \\sigma$ means that for any $i$ and $j$ in the same class of $\\eta$, $\\sigma(t_i)=\\sigma(t_j)$.\n\nLet $\\Omega$ be the set of configurations $\\sigma$ that satisfy all constraints in $I$ except possibly the disequality constraint. ", "Then $Z(I)=\\sum_{\\sigma\\in\\Omega} w(\\sigma) \\mathds{1}_\\sigma$, where $\\mathds{1}_\\sigma=1$ if $\\sigma$ respects the disequality constraint, meaning that $\\vartheta(\\sigma)=\n\\underline{0}$, and $\\mathds{1}_\\sigma=0$ otherwise. ", "By the Möbius inversion formula derived above, $$Z(I)\\,=\\,\\sum_{\\sigma\\in\\Omega} w(\\sigma)\n\\sum_{\\eta \\leq \\vartheta(\\sigma)} \\mu(\\eta).$$ Changing the order of summation, we get $$Z(I)\\,=\\,\\sum_{\\eta} \\mu(\\eta) \\sum_{\\eta\\leq \\theta}\\hspace{1pt}\n\\sum_{\\sigma\\in \\Omega:\\vartheta(\\sigma)=\\theta} w(\\sigma)\\,=\\,\n\\sum_{\\eta} \\mu(\\eta) \\sum_{\\sigma\\in \\Omega: \\eta \\preccurlyeq \\sigma}\nw(\\sigma).$$\n\nNow note that $\\sum_{\\sigma: \\eta \\preccurlyeq \\sigma} w(\\sigma)$ is the partition function $Z(I_\\eta)$ of an instance $I_\\eta$ of ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$. The instance $I_\\eta$ is formed from $I$ by ignoring the disequality constraint, and identifying variables in $t_0,\\ldots,t_{q-1}$ whose indices are in the same class of $\\eta$. Thus we can compute all the $Z(I_\\eta)$ in ${\\mbox{\\sf \\#CSP}\\xspace}(\\mathcal{F})$. Finally, $Z(I) = \\sum_{\\eta} \\mu(\\eta) Z(I_\\eta)$, completing the reduction.", "\n\n[^1]: Usually $[q]$ is defined to be ${\\left\\{1,2,\\ldots,q\\right\\}}$, but it is more convenient here to start the enumeration of domain elements at 0 rather than 1.", "\n\n[^2]: We assume $0\\in\\mathbb{N}$, so we allow non-negative constants.", "\n\n[^3]: This is not quite the original statement of the theorem. ", "We have chosen here to restrict all inputs to be rational, in order to avoid issues of how to represent, and compute with, arbitrary real numbers.", "\n\n[^4]: A more general disequality function is defined in the Appendix.", "\n\n[^5]: It is not necessary to choose this particular ordering for $J$, but it is convenient to do so.", "\n\n[^6]: In fact, it can be shown that these equivalence classes are cosets of the symmetry group of $f$, and hence are of equal size, though we do not use this fact here.", "\n\n[^7]: Lovász [@Lovasz67] had previously used Möbius inversion in a similar context.", "\n" ]

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[ "Pop Up: Madonna And Elton John Legitimately End Their Feud\n\nAlright people, it seems like the end of the feud between Madonna and Elton John is legit.", "\n\nAccording to The Sun, the two entertainers met at a restaurant in Nice, France and ended their feud. ", "A source said, “Elton goes to the Petite Maison for dinner fairly regularly as it’s near his villa in Mont Boron. ", "But Madonna was also there for a bite before her show, which was down the road and didn’t start until around 10:30 PM.”", "\n\nThe source added, “Just as he was leaving, he went over to talk to Madonna and they were laughing and smiling. ", "They even hugged each other.”" ]

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[ "2019 CALENDAR\n\nMy four-legged friends at their favorite local spots. ", "It was an absolute joy to work with these critters and experience Asheville through their eyes. ", "We all know the camera loves them, and I couldn’t have asked for a more professional group of models. ", "We hope you enjoy the new year with our amazing animals and beautiful city!", "\n\nAvailable now at Pet Vet on Patton, Asheville Humane Society, and Ben’s Tune-Up" ]

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[ "Increased catabolism of nucleic acid in nonalcoholic fatty liver disease patients of different ages.", "\nUric acid is the end product of nucleic acid catabolism for humanity. ", "Serum uric acid level has been suggested to be associated with nonalcoholic fatty liver disease (NAFLD). ", "This study aims to examine the association between serum uric acid levels and NAFLD in subjects of different ages. ", "A cross-sectional study was performed among the patients of different ages with NAFLD (abnormal group) or without NAFLD (normal group) in Lianyungang, Jiangsu, China. ", "The levels of serum uric acid, total cholesterol, triglyceride, and low density lipoprotein in the abnormal group were significantly higher than those in the normal group, while the level of high density lipoprotein in the abnormal group was significantly lower than that in the normal group (p<0.05). ", "The serum uric acid prevalence, total cholesterol, triglyceride, high density lipoprotein, and low density lipoprotein in the abnormal group was significantly higher than those in the normal group (p<0.05). ", "The serum uric acid, total cholesterol, triglyceride, and low density lipoprotein levels and prevalence rates in elderly patients were higher than younger and middle aged patients. ", "Serum uric acid level is highly associated with NAFLD in patients of different ages." ]

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[ "1. ", "Introduction {#sec1-marinedrugs-17-00149}\n===============\n\n*Ecklonia cava*, which is edible seaweed belonging to the Laminariaceae family, is found in Asian countries, such as Japan and Korea. *", "E. cava* is a nutrient-dense food and it contains many types of organic compounds, including fucoidan and phlorotannins \\[[@B1-marinedrugs-17-00149]\\], which have various biological activities, such as antioxidant \\[[@B2-marinedrugs-17-00149],[@B3-marinedrugs-17-00149]\\], anti-allergic \\[[@B4-marinedrugs-17-00149]\\], anti-plasmin inhibitory \\[[@B5-marinedrugs-17-00149]\\], anticancer \\[[@B6-marinedrugs-17-00149]\\], antihypertensive \\[[@B7-marinedrugs-17-00149]\\], and elastase and hyaluronidase inhibitory effects \\[[@B8-marinedrugs-17-00149]\\], as well as strong antiviral activity against porcine epidemic diarrhea virus (PEDV) (a coronavirus) \\[[@B9-marinedrugs-17-00149]\\].", "\n\nInfluenza, similar to coronaviruses, is also a contagious viral infection that causes acute respiratory illnesses and rapidly spreads through outbreaks \\[[@B10-marinedrugs-17-00149],[@B11-marinedrugs-17-00149]\\]. ", "The influenza virus belongs to the Orthomyxoviridae family, and there are three different types: A, B, and C. Among them, types A and B are of great concern as human pathogens and result in seasonal or interpandemic epidemics as well as global pandemics, such as those that are caused by influenza A viruses \\[[@B12-marinedrugs-17-00149],[@B13-marinedrugs-17-00149]\\]. ", "Over the past 100 years, four pandemics have been reported: H1N1 Spanish influenza (1918), H2N2 Asian influenza (1957), H3N2 Hong Kong influenza (1968), and recently, 2009 H1N1 (H1N1pdm09), which are all caused by influenza A viruses \\[[@B12-marinedrugs-17-00149],[@B14-marinedrugs-17-00149]\\]. ", "In each pandemic, the number of novel virus strains arose and spread through human communities, leading to the substantial morbidity and mortality related to bacterial pneumonia \\[[@B11-marinedrugs-17-00149],[@B15-marinedrugs-17-00149]\\]. ", "The influenza A viruses are single-stranded negative-sense RNA viruses that are subtyped on the basis of the composition of their two surface glycoproteins: hemagglutinin (HA) and neuraminidase (NA) \\[[@B16-marinedrugs-17-00149]\\]. ", "The genome of influenza A viruses comprises 8 separated gene segments encoding 16 proteins and is surrounded by a liquid envelope \\[[@B17-marinedrugs-17-00149],[@B18-marinedrugs-17-00149]\\]. ", "Two glycoproteins, HA and NA, are found in the viral envelopes, and these are considered to be the basis of antigenicity. ", "HA initiates the infection of influenza virus by binding to the α-2,6-linked sialic acid and/or α-2,3-linked sialic acid receptors on the surface of the host cell, followed by receptor-mediated endocytosis of the virion into the cell \\[[@B17-marinedrugs-17-00149],[@B19-marinedrugs-17-00149]\\]. ", "The NA protein serves as a sialidase, and it cleaves the link between the sialic acid and the HA protein to release the virus particles \\[[@B11-marinedrugs-17-00149],[@B19-marinedrugs-17-00149]\\]. ", "To date, the Food and Drug Administration (FDA) \\[[@B20-marinedrugs-17-00149]\\] have approved only three drugs: Tamiflu (oseltamivir phosphate), Relenza (zanamivir), and Rapivab (peramivir).", "\n\nDereplication is the term that is used to discover new natural materials through the rapid identification of compounds that are based on new analytical tools, such as MS and NMR spectroscopy, to avoid the re-isolation of known components. ", "In particular, the rapid development of modern MS techniques has accelerated dereplication strategies using high-performance liquid chromatography---quadrupole time-of-flight mass (HPLC-qTOFMS) spectra with tandem database searching to identify the known compounds in the extract \\[[@B21-marinedrugs-17-00149]\\]. ", "By increasing the use of high-resolution mass spectrometry (HRMS) for measuring exact masses, the concept of \"mass defect\" is also increasingly being used in natural products research \\[[@B22-marinedrugs-17-00149]\\]. ", "The mass defect is calculated as the deviation between the exact mass and the nominal mass (integer-rounded mass) of a compound \\[[@B22-marinedrugs-17-00149],[@B23-marinedrugs-17-00149]\\]. ", "Several methods have been developed to simplify the use of mass defects, including Kendrick mass defects, mass defect filtering, and fractional mass, which have played an important role in identifying many previously unknown compounds. ", "In addition, a relative mass defect (RMD), calculated as (mass defect/measured monoisotopic mass) × 10^6^ in ppm, was recently introduced \\[[@B23-marinedrugs-17-00149],[@B24-marinedrugs-17-00149]\\]. ", "The RMD remains constant for classes that share the same fractional hydrogen content, which is useful in classifying the compounds into groups with the same skeleton \\[[@B23-marinedrugs-17-00149]\\]. ", "While fatty acids or hydrocarbons with high hydrogen contents show high RMDs (400 to 600 ppm), the values for polyphenolic metabolites are often less than 300 ppm due to their high oxygen contents \\[[@B23-marinedrugs-17-00149]\\]. ", "In addition to modern techniques, such as HPLC-qTOFMS, classical bio-guided isolation has played an important role in the primary bioactive screening and identification of potential natural products. ", "Pre-fractionation has recently been applied in bioactivity-guided isolations as an important step in obtaining bioactive materials. ", "This is because the activities of minor components can be masked or decreased by more abundant inactive compounds in the total extract \\[[@B25-marinedrugs-17-00149]\\]. ", "The combination of bio-guided isolation and dereplication enables the rapid and effective identification of the potential bioactive compounds \\[[@B26-marinedrugs-17-00149]\\].", "\n\nThe aim of the present study was to develop a new antiviral agent from *E. cava*. ", "Specifically, we want to develop dereplication methods, including classification that is based on RMD and HPLC-qTOFMS, to accurately and rapidly identify bioactive phlorotannin-type compounds in *E. cava*, as well as to easily target new compounds.", "\n\n2. ", "Results and Discussion {#sec2-marinedrugs-17-00149}\n=========================\n\n2.1. ", "Bioactivity-Guided Isolation and Dereplication of the Active Antiviral Fraction {#sec2dot1-marinedrugs-17-00149}\n------------------------------------------------------------------------------------\n\n### 2.1.1. ", "Bioactivity-Guided Fractionation Based on Antiviral Activity Against Human Influenza H1N1 A/PR/8/34 Virus {#sec2dot1dot1-marinedrugs-17-00149}\n\nThe 80% MeOH extract of *E. cava*, which showed antiviral activity at 10 µg/mL in a cytopathic effect (CPE) assay when compared to ribavirin (10 µM) as a positive control ([Figure 1](#marinedrugs-17-00149-f001){ref-type=\"fig\"}A), was divided into five fractions on a Diaion HP-20 column, with a gradient of MeOH and H~2~O (0/100→100/0, *v*/*v*), and finally acetone. ", "The antiviral activities of the five fractions indicated that the sub-fraction eluted with 70% MeOH (EC70) had the strongest antiviral activity against H1N1-infected cells at three different concentrations (5, 10, and 20 µg/mL, [Figure 1](#marinedrugs-17-00149-f001){ref-type=\"fig\"}B). ", "Thus, the EC70 fraction was chosen for further bio-guided isolation and dereplication using an HPLC-qTOFMS to identify the predicted bioactive compounds ([Figure 2](#marinedrugs-17-00149-f002){ref-type=\"fig\"}). ", "Potential components against H1N1 and H9N2 viruses were further tested against viral protein synthesis of the H1N1 A/PR/8/34 virus ([Figure 3](#marinedrugs-17-00149-f003){ref-type=\"fig\"}).", "\n\n### 2.1.2. ", "Dereplication and Isolation of the Phlorotannins from the EC70 Fraction Using HPLC-qTOFMS and RMD {#sec2dot1dot2-marinedrugs-17-00149}\n\nThe dereplication of the secondary metabolites from *E. cava* was accomplished by HPLC-qTOFMS analysis of the active fraction (EC70). ", "The HPLC-qTOFMS data were used for database construction based on the chromatographic retention times, full-scan (high-resolution) mass spectra, and MS/MS spectra in negative electrospray ionization (ESI) mode. ", "The secondary metabolites in the EC70 fraction were identified by comparison with compounds that were previously reported in the *Ecklonia* genus (12 species listed in [Table S1, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}) in online (Scifinder) and in-house databases that are based on exact mass and RMD using MassHunter quantitative B4.00 software. ", "The identified compounds are shown as the base peaks of the MS and MS/MS chromatograms in [Figure 4](#marinedrugs-17-00149-f004){ref-type=\"fig\"}B--E. The MS^1^ and MS^2^ data of all the peaks that were detected in the spectra are listed in [Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"}. ", "The RMD value of each peak was used to determine its chemical skeleton. ", "The structures of the components, which were identified by the comparison of the MS/MS fragmentations of the peaks in [Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"} to those in the in-house library, are shown in [Figure 2](#marinedrugs-17-00149-f002){ref-type=\"fig\"}.", "\n\nThe components on the EC70 fraction could be classified as phlorotannins, fatty acids, and others compounds with different ranges of RMD values. ", "The RMD values of the detected phlorotannins were smaller than those of the other compounds, which are only approximately 10 to 200 ppm due to the nature of the structure of phlorotannins ([Figure S15, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "Positive absolute mass defects are usually observed in compounds that contain a large number of hydrogen atoms (mass defect +7.83 mDa), whereas the lower values are observed in compounds with large numbers of oxygen (mass defect −5.09 mDa) and nitrogen atoms (mass defect +3.07 mDa) \\[[@B23-marinedrugs-17-00149]\\]. ", "Fatty acids contain many hydrogens and they have large RMD values (as suggested by compounds **21**, **22**, **24**, and **26**--**29**, [Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"} and [Figure 2](#marinedrugs-17-00149-f002){ref-type=\"fig\"}), but oxygen is a major elementary building block of phlorotannins, which reduces the RMD values of this group, particularly in this study, to less than 200 ppm (as demonstrated by compounds **6**--**18**, [Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"}). ", "These features are very helpful for further studying the genus *Ecklonia,* which contains phlorotannins as major components. ", "As the first step of screening by HPLC-qTOFMS, the calculation of RMD values for all detected peaks facilitates the identification of phlorotannins and allows researchers to dereplicate the reported compounds, especially in this genus. ", "The RMD values in the list of unidentified peaks in [Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"} are also useful for estimating the types of structures that are associated with detected peaks.", "\n\nFurthermore, polyphenols have lower hydrogen contents and higher oxygen contents when compared to other types of compounds, resulting in lowered RMD values, which is consistent with what is seen in phlorotannins. ", "Thus, in further studies, classification that is based on the RMD value is suitable for other types of polyphenolic compounds and it can be broadly applied in the screening of natural resources. ", "In particular, for most species of brown algae, phlorotannin-type compounds can easily be classified because they are the main components of this species.", "\n\nPhlorotannins, which are composed of phloroglucinol (1,3,5-trihydroxybenzene) units \\[[@B27-marinedrugs-17-00149],[@B28-marinedrugs-17-00149]\\], are well-known as major components of the genus *Ecklonia* \\[[@B9-marinedrugs-17-00149]\\]. ", "Overall, phlorotannins are divided into four classes that are based on the type of linkage that connects the phloroglucinol units, including ether-bound phlorotannins (fuhalols and phlorethols), phenyl-bound phlorotannins (fucols), ether and phenyl-bound phlorotannins (fucophlorethols), and dibenzo-dioxin-bound fucophlorethols (eckols and carmalols) \\[[@B27-marinedrugs-17-00149],[@B28-marinedrugs-17-00149]\\]. ", "Monomers in each class can be linked at various positions; thus, it is possible to have many structural isomers with the same molecular weight \\[[@B27-marinedrugs-17-00149],[@B28-marinedrugs-17-00149]\\]. ", "The fragmentation patterns that were observed for these peaks are therefore key to structural identification, as they indicate different losses of phloroglucinol monomers and hydroxyl groups \\[[@B27-marinedrugs-17-00149]\\]. ", "In most previous studies, it has been necessary to isolate the compounds and use a combination of MS and NMR spectroscopy to elucidate the exact structures of phlorotannins \\[[@B27-marinedrugs-17-00149]\\]. ", "Recently, an attempt was made to classify the types of phlorotannins in *Sargassum fursiforme* without isolation using ultrahigh-performance liquid chromatography triple quadrupole tandem mass spectrometry (UHPLC-QqQ-MS) for the first time. ", "In this study, we proposed an efficient method to dereplicate the various phlorotannins derivatives, which are major constituents in seaweeds, based on preliminary HPLC-qTOFMS accompanied by RMD analysis. ", "Moreover, the hypothesis of dereplication was further validated by the isolation and structural elucidation of phlorotannins using NMR and HPLC-qTOFMS. ", "The MS/MS data of the peaks that were detected in the EC70 fraction and the anticipated components are listed in [Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"}. ", "As shown in [Figure 4](#marinedrugs-17-00149-f004){ref-type=\"fig\"}C and [Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"}, forty-one peaks were observed in the HPLC-qTOFMS spectroscopic data of the active fraction, and compounds **6**--**18** ([Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"}) were identified as phlorotannins based on a comparison of their RMD values, molecular weights, and MS/MS fragments to the literature data. ", "According to their MS/MS data, thirteen compounds were identified as eckol-type phlorotannins, which contain at least one 1,4-dibenzodioxin fragment in their structures and have been mostly found in *Ecklonia* and some other genus of seaweeds \\[[@B27-marinedrugs-17-00149],[@B28-marinedrugs-17-00149]\\]. ", "The fragment peaks that were observed in the MS/MS patterns of these compounds revealed the losses of phloroglucinol (126 amu), pentahydroxyl dibenzodioxin (263 amu) and eckol (371 amu) based on the dominant product ions at *m/z* 229 amu \\[371 (eckol) −126 (phloroglucinol) −16 (oxygen atom) amu\\] and 261 amu \\[263 (pentahydroxyl dibenzodioxin) −2H\\]. ", "For example, bieckol (compound **15**, *m/z* 741.0737, [Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"}) showed product ions at *m/z* 615, 479, 371, and 260 from losses of one phloroglucinol (−126 amu), two phloroglucinols and a water (−249 −18 amu), eckol (−371 amu), and eckol with a phloroglucinol being followed by the attachment of an oxygen atom (−371 --126 +16 amu).", "\n\nAccording to dereplication results, ten known phlorotannins (**6**, **7**, **10**--**16**, and **18**) were directly isolated from the EC70 fraction to validate the accuracy of the identification. ", "These ten compounds were measured HPLC-qTOFMS in negative ion mode at a collision energy of 50 eV to further confirm their retention times ([Figure S16, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "The fragmentation patterns of the isolated phlorotannins are presented in [Figure S17 (Supporting Information)](#app1-marinedrugs-17-00149){ref-type=\"app\"}. ", "The results revealed that HPLC-qTOFMS was a powerful tool for identifying the structure of phlorotannins, especially in the cases of isomers, as it can provide sufficient data for the elucidation of the chemical structures. ", "As an example, the MS/MS fragmentation pattern of compound **18** (*m/z* 974.1038) is presented in detail. ", "As identified in [Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"}, compound **18** was established as compound 974-A due to its low RMD value (107 ppm) and the characteristic fragments that were observed at *m/z* 973, 707, 353, and 229 \\[[@B29-marinedrugs-17-00149]\\]. ", "After isolation, the exact structure of **18** was further confirmed by NMR spectroscopy in conjunction with MS/MS fragmentation ([Figure S17, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "The product ions of **18** are shown in [Figure S17 (Supporting Information)](#app1-marinedrugs-17-00149){ref-type=\"app\"} and they are due to the losses of bifuhalol (--266 amu), three phlorotannins and one water (−375 −18 amu), and four phlorotannin units and one water (−448 −18 amu). ", "Similarly, although compounds **10**, **11**, and **15** showed the same molecular ions (\\[M − H\\]^−^ of 741.0647, 741.0723, and 741.0737, respectively) ([Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"}), they could be distinguished by their NMR data and different fragment ions in their MS/MS patterns ([Figure S17, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "In particular, compound **10** showed fragments at *m/z* 619.0680, 495.0568, and 250.0477 from the losses of phloroglucinol (126 amu) and two equivalents of phloroglucinol (250 amu). ", "Compound **11** generated product ion peaks at *m/z* 565.3355, 437.0064, 369.0253, and 230.9843 from the losses of phloroglucinol and three waters (176 amu) and a dioxinodehydroeckol (369 amu). **", "15** showed product ions at *m/z* 477.0398, 369.0198, 351.0087, and 125.0224, due to the losses of two molecules of phloroglucinol and water (264 amu) and three phloroglucinol units (372 amu).", "\n\n### 2.1.3. ", "Antiviral Activities of the Major Phlorotannins in the EC70 Active Fraction against H1N1 and H9N2 {#sec2dot1dot3-marinedrugs-17-00149}\n\nIn this study, seven major phlorotannins that were isolated from the EC70 fraction using bioactivity-guided isolation were tested for their antiviral activities against two influenza A viral strains (H1N1 A/PR/8/34 and H9N2 virus strains) ([Figure 1](#marinedrugs-17-00149-f001){ref-type=\"fig\"}C,D). ", "The results indicated that six compounds (**6** and **10**--**14**) had moderate to strong antiviral activities against both viral strains at a concentration of 20 µM. Compounds **11**--**14**, which displayed potent antiviral effects, were further tested against viral protein synthesis of the H1N1 A/PR/8/34 virus and compared to ribavirin as a positive control. ", "The results, as shown in [Figure 3](#marinedrugs-17-00149-f003){ref-type=\"fig\"}A and [Table S2 (Supporting Information)](#app1-marinedrugs-17-00149){ref-type=\"app\"}, suggested that compound **12** was the most active compound with an EC~50~ of 13.48 ± 1.93 μM. Thus, the dose-dependent inhibitory effects of compound **12** on viral protein synthesis were evaluated at four different concentrations 5, 10, 20, and 40 μM in Madin--Darby canine kidney (MDCK) cells. ", "As shown in [Figure 3](#marinedrugs-17-00149-f003){ref-type=\"fig\"}B, it is clear that, at higher concentrations, **12** more effectively inhibited protein expression in viral-infected cells. ", "Compound **12** decreased the expression of neuraminidase and hemagglutinin at 10 μM, and the strongest inhibition capacity was observed at 40 µM. The fluorescence images that are shown in [Figure 3](#marinedrugs-17-00149-f003){ref-type=\"fig\"}C also explain the inhibitory effect of compound **12** on neuraminidase protein expression in H1N1 infected MDCK cells. ", "The results suggested that compound **12** showed strong antiviral activity through inhibiting the expression of surface glycoproteins, hemagglutinin, and neuraminidase.", "\n\n2.2. ", "Identification of Two New Phlorotannins from the EC30 Fraction of E. cava using Dereplication and RMD Rules {#sec2dot2-marinedrugs-17-00149}\n----------------------------------------------------------------------------------------------------------------\n\nThe results from the dereplication of the EC70 fraction suggested a quick method to identify phlorotannins. ", "Hence, we attempted to isolate the new phlorotannins from the other fractions of *E. cava* based on their MS/MS fragmentation patterns and RMD values. ", "The HPLC-qTOFMS data from fraction EC30 of *E. cava* was exported and the RMD values of the detected peaks were calculated. ", "Compounds **1** (*m/z* 743.0881, [Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"}) and **2** (*m/z* 1113.1146, [Table 1](#marinedrugs-17-00149-t001){ref-type=\"table\"}) showed RMD values that were between 0 and 200 ppm, and their fragments, which were not presented in the in-house database or online database of the *Ecklonia* genus, suggested that these compounds might be new phlorotannins ([Figure 4](#marinedrugs-17-00149-f004){ref-type=\"fig\"}A). ", "Thus, the two compounds were isolated to confirm their chemical structures through NMR spectroscopy along with molecular modelling and MS/MS fragmentation analysis.", "\n\nCompound **1** was obtained as a brown powder and its molecular formula was determined to be C~36~H~24~O~18~ from its deprotonated ion peak at *m/z* 743.0895 (calcd. ", "for \\[M − H\\]^−^, 743.0890, [Figure S1, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "The chemical shifts observed in the ^1^H and ^13^C NMR spectra ([Figures S3, and 4, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}), proton resonances in the downfield region of the aromatic chemical shift range, suggested a polyphenolic structure. ", "In the ^1^H NMR spectrum, eleven aromatic protons were observed, including one proton singlet at δ~H~ 6.12 (H-3), one 2H overlapped singlet at δ~H~ 5.86 (H-3″″, 5″″), and four sets of *meta*-coupled doublets \\[δ~H~ 6.12 (overlap, H-5″)/5.70 (d, 1.5, H-3″), 6.09 (br s, H-6′)/5.99 (d, 1.5, H-2′), 5.97 (d, 2.7, H-8)/5.81 (d, 2.7, H-6), and 589 (d, 2.7, H-5‴)/5.54 (d, 2.7, H-3‴)\\]. ", "The ^13^C NMR signals corresponded to thirteen unsubstituted (δ~C~ 92.7--100.7 ppm) and twenty-three oxygenated aromatic carbons (δ~C~ 122.1--158.1 ppm), which were assigned based on their HSQC and HMBC correlations ([Figures S5 and S6, respectively, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "These data suggested that the compound contained six phloroglucinol units. ", "The coupling constants of the proton signals and HMBC data further confirmed the pattern of the substituents on **1** ([Figure S20A, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "The signal at δ~H~ 6.12 (1H, s) and the *meta*-coupled doublets at δ~H~ 5.89 (d, 2.7) and 5.81 (d, 2.7) were similar to those of eckol \\[[@B30-marinedrugs-17-00149]\\], and they were thus assigned to H-3 and H-8/H-6 of rings A and B, respectively. ", "Similarly, the structure of **1** was presumed to include the C, D, E, and F rings of fucodiphloroethol G by comparison with the reported NMR data \\[[@B31-marinedrugs-17-00149]\\]. ", "The two carbon signals at δ~C~ 100.7 and 101.4 ppm also indicated that the structure of **1** contained one C−C bond similar to that of fucodiphloroethol G \\[[@B31-marinedrugs-17-00149]\\]. ", "However, the asymmetric structure of the C ring revealed that fucodiphloroethol and eckol moieties of **1** were linked through the C ring. ", "The ROESY correlation of H-3 (A ring) with 5′-hydroxyl (C ring) ([Figure S7, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}) suggested a phenyl linkage connecting the C and A rings. ", "The long-range couplings between the proton of rings A and C, as well as rings B and E, were explained by their positions derived from three-dimensional structure optimization using molecular modelling ([Figure S20B, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "A comprehensive explanation of the structure was obtained from the MS/MS data. ", "As shown in [Figure S17 (Supporting Information)](#app1-marinedrugs-17-00149){ref-type=\"app\"}, the fragment ion corresponding to C~24~H~15~O~11~ (*m/z* 479.0608, calcd 479.0614, Δ −0.6 mmu) was the main fragment of **1**, and the other detected fragments were explained by the losses of phloroglucinol units (−126 amu) or pentahydroxyl dibenzodioxin (263 amu). ", "In particular, the fragments of **1** at *m/z* 583.0716, 479.0608, 353.0284, and 265.0342 amu were from the losses of one phloroglucinol and two water molecules (−126 −34 amu), two phloroglucinol units and one oxygen atom (−248 −16 amu), three phloroglucinol and one water (−372 --18 amu), and pentahydroxyl dibenzodioxin and two protons (−263 −2 amu). ", "Therefore, the chemical structure of **1** was determined to be dibenzodioxin-fucodiphloroethol (DFD).", "\n\nCompound **2** was obtained as a brown powder and the molecular formula of **2** was established to be C~54~H~34~O~27~ on the basis of its molecular ion in the negative mode (HPLC-qTOFMS, *m/z* 1113.1215, calcd. ", "for \\[M − H\\]^−^, 1113.1215) ([Figure S8, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "The ^1^H NMR spectrum of **2** ([Figure S10, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}) showed a total of sixteen proton signals, including three proton singlets at δ~H~ 6.21 (1H, s, H-5″″″), 6.14 (1H, s, H-4″″‴), and 5.92 (1H, s, H-3); two sets of *meta*-coupled doublets at δ~H~ 6.01 (1H, d, 2.2 Hz, H-6), 5.88 (1H, d, 2.2 Hz, H-8), 5.89 (1H, d, 2.4 Hz, H-5‴), and 5.55 (1H, d, 2.4 Hz, H-3‴); two sets of *meta*-coupled 2H signals at δ~H~ 5.87 (2H, s, H-3″, 5″), 5.74 (2H, d, 1.8 Hz, H-2′, 6′), and 5.81 (1H, br t, H-4′); and, two sets of *meta*-coupled singlets at δ~H~ 6.14 (1H, br s, H-5″″), 5.71 (1H, br s, H-3″″), 6.11 (1H, br s, H-5″‴), and 6.00 (1H, br s, H-3″‴). ", "The ^13^C NMR spectrum of **2** ([Figure S11, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}) showed 54 signals in the chemical shift range of aromatic carbons. ", "The thirteen protonated carbon signals from δ~C~ 92.7 to 98.7 ppm, along with thirty-six oxygenated aromatic carbon signals that were located in the range of 121.4--163.5 ppm, suggested the presence of nine aromatic rings in the structure of **2**. ", "All of the carbon signals in ^13^C NMR spectrum of **2** were assigned based on their HSQC and HMBC correlations ([Table S3 and Figures S12 and S13, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "When compared with the reported phlorotannins, the chemical shifts of the atoms in rings A, B, C, and D of **2** were similar to those belonging to 7-phloroeckol \\[[@B32-marinedrugs-17-00149]\\]. ", "Similarly, the chemical shifts of the atoms in rings D, E, F, and G were assigned based on the structure of fucodiphloroethol G \\[[@B31-marinedrugs-17-00149]\\]. ", "The C−C bond between rings F and G of **2** was confirmed from the chemical shifts at δ~C~ 100.7 and 101.5 ppm. ", "In addition, the remaining two aromatic rings of **2** were suggested to be rings H and I based on the MS/MS fragmentation analysis of **2**. ", "The fragment at 265 amu was suggested by the vicinal trihydroxylated structure of compound **2** ([Figure S17, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}), which was characteristic of a bifuhalol unit that contained an extra hydroxyl group on the terminal phlorotannin unit \\[[@B27-marinedrugs-17-00149]\\]. ", "The connectivity between all of the fragments was further determined by the ROESY spectrum ([Figure S14, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "The correlations of the signals corresponding to H-2′/6′ in ring A with H-3 (B ring) and H-6 (C ring) and between H-6 (C ring) with H-3 (D ring) confirmed the connectivity of these rings, as shown in [Figure S20 (Supporting Information)](#app1-marinedrugs-17-00149){ref-type=\"app\"}. ", "The 7-phloroeckol moiety was connected to the 1‴ position of the ring E based on the ROESY correlation between the H-3″ (D ring) and H-6‴ (E ring). ", "Similar to compound **1**, the long-range correlations of the protons on rings A and D, as well as on rings B and G of **2**, were further supported by the optimized three-dimensional structure obtained using molecular modelling ([Figure S20B, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "The fragment information from HRMS/MS data of **2,** as described in [Figure S17 (Supporting Information)](#app1-marinedrugs-17-00149){ref-type=\"app\"}, also supported the chemical structure of compound **2**. ", "The fragments that were observed in [Figure S17 (Supporting Information)](#app1-marinedrugs-17-00149){ref-type=\"app\"} at *m/z* 989.0983, 879.0299, 556.0534, and 265.0334 amu were explained by the losses of one phloroglucinol unit (−126 amu); two phloroglucinol units and the addition of one water (−252 and +18 amu); one eckol, one bifuhalol, the addition of one acetic acid, and one water (−372 −263 +60 +18 amu); and, a bifuhalol fragment at 265.0334 amu. ", "Finally, the exact structure of **2** was elucidated as dibenzodioxin-fucodiphloroeckol (DFE).", "\n\n2.3. ", "Discussion of the Potential Applications of the Developed Dereplication Strategy and Phlorotannins {#sec2dot3-marinedrugs-17-00149}\n-------------------------------------------------------------------------------------------------------\n\nIn this paper, the developed dereplication strategy using RMD values is based on the general dereplication strategy that determines the similarities between the MS/MS fragmentation patterns of standard compounds and those of the unknown compounds. ", "However, unlike the general dereplication method, RMD can be used to group compounds of the same type to show the overall phytochemical trends by the types of secondary metabolites. ", "The extracts of *E. cava* or phlorotannins (dieckol, phloroglucinol, eckol, phlorofucofuroeckol, and 7-phloroeckol, etc.) ", "have been reported to have antiviral activities, such as PEDV \\[[@B9-marinedrugs-17-00149]\\], viral hemorrhagic septicemia virus (VHSV) \\[[@B33-marinedrugs-17-00149]\\], human papilloma virus (HPV) \\[[@B34-marinedrugs-17-00149]\\], and murine norovirus (MNV) \\[[@B35-marinedrugs-17-00149]\\]. ", "However, in this study, a method of producing a bioactive fraction was developed, and this fraction could be used to obtain antiviral materials from a total extract. ", "Additionally, seven lead compounds were identified and purified from the bioactive fraction. ", "Therefore, the fraction of the *E. cava* extract and the isolated phlorotannins have been shown to be potentially industrially applicable in the prevention or treatment of influenza A viruses.", "\n\n3. ", "Conclusions {#sec3-marinedrugs-17-00149}\n==============\n\nThe present study discussed using MS/MS fragmentation analysis coupled with RMD filtering as a dereplication method to identify phlorotannins, which are bioactive substances in the *Ecknolia* genus, especially in *Ecknolia cava*. ", "To the best of our knowledge, this is the first attempt to apply and post-evaluate the accuracy of the RMD values in conjunction with MS/MS analysis for the identification of phlorotannins. ", "The results of the dereplication study provide a quick initial screening for the phlorotannin motif, as these compounds show RMD values of less than 200 ppm, typically with fragments from losses of phloroglucinol units. ", "In addition, different ranges of RMD values suggest the presence of different types of structures that are present in the compound. ", "MS/MS was not only useful for dereplication but it also provided strong evidence to confirm the structures of phlorotannins by their fragments. ", "Based on this dereplication method, two new compounds were identified and further elucidated by spectroscopy. ", "In addition, this study was conducted to identify the antiviral lead compounds from *E. cava* using bio-guided isolation. ", "Our results suggested that phlorofucofuroeckol A (**12**) from *E. cava* plays a key role in the antiviral activities of this seaweed against H1N1 and H9N2 virus. ", "The strong inhibitory effects of this compound against neuraminidase and hemagglutinin suggested that it could be a potential agent for the further development of antiviral drugs.", "\n\n4. ", "Materials and Methods {#sec4-marinedrugs-17-00149}\n========================\n\n4.1. ", "General Experimental Procedures {#sec4dot1-marinedrugs-17-00149}\n------------------------------------\n\nOptical rotations were determined on a JASCO P-2000 polarimeter (JASCO International Co. Ltd., Tokyo, Japan). ", "The IR spectra were recorded on a Nicolet 6700 FT-IR spectrometer (Thermo Electron Corp., Waltham, MA, USA). ", "NMR data were measured in DMSO-*d*~6~ and recorded on 300, 800, and 850 MHz spectrometers at the College of Pharmacy, Seoul National University, Korea. ", "ESI-MS data were recorded on an Agilent 1100 series LC/MSD TRAP (Agilent Technologies, Waldbronn, Germany). ", "The active fractions were analysed by TLC (thin-layer chromatography) on silica gel 60 F254 and RP-18 F254 plates from Merck (Darmstadt, Germany). ", "Spraying with a vanillin reagent that contained 0.5 g of vanillin, 80 mL of sulfuric acid, and 20 mL of ethanol developed the plates, and the compounds on the TLC plates were detected at 254 and 365 nm. ", "Column chromatography separations were carried out using various resins, including Diaion HP-20 (Mitsubishi Chemical Co. Tokyo, Japan), Sephadex LH-20 (Sigma-Aldrich Corp, St. Louis, Missouri, USA), silica gel (Merck, 40--63 µm particle size), and C~18~-RP silica gel (Merck, 40--63 μm particle size). ", "Preparative HPLC separations were performed by using a Gilson system with an Optima Pak C~18~ column (10 mm × 250 mm, 10 µm in particle size, RS Tech, Seoul, Korea) and a UV detector at wavelengths of 205 and 254 nm. ", "All of the solvents that were used for extraction and isolation were of analytical grade. ", "The most stable configuration of compounds **1** and **2** ([Figure S20, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}) were calculated by CONFLEX 8 (Conflex Corp., Tokyo, Japan) using molecular mechanics force-field (MMFF94s) calculations with a search limit of 1.0 kcal/mol.", "\n\n4.2. ", "Plant Material {#sec4dot2-marinedrugs-17-00149}\n-------------------\n\n*E. cava* was collected from Cheongsando Island, Republic of Korea in July 2016. ", "Prof. T. O. Cho botanically identified the plant sample. ", "A voucher specimen (TCMBRB0019/TC9484) was deposited at the Algae Systematics laboratory of Chosun University, College of Natural Sciences, Gwang-ju, Republic of Korea.", "\n\n4.3. ", "Extraction and Isolation {#sec4dot3-marinedrugs-17-00149}\n-----------------------------\n\nDried powder of *E. cava* (100 g) was extracted with 80% MeOH (5 L) at room temperature under ultra-sonication for two days. ", "After filtration, the combined MeOH solution was concentrated under reduced pressure to yield dry extract (20.4 g). ", "The crude extract was suspended in distilled water (2 L) and then fractionated using Diaion HP-20. ", "Four litres of water was used to elute the salts and then two litres each of 30% MeOH, 70% MeOH, 100% MeOH, and 100% acetone were used as the elution systems to obtain five fractions. ", "The fraction eluted with 30% MeOH was chromatographed on a reversed-phase silica gel column with a gradient of MeOH/H~2~O from 30% to 100% to yield five subfractions (F1--F5) ([Scheme S1, Supporting Information](#app1-marinedrugs-17-00149){ref-type=\"app\"}). ", "Sub-fraction F2 was subjected to a Sephadex LH-20 column that was eluted with 100% MeOH to afford four fractions (F2.1--F2.4). ", "Fraction F2.3 was further purified on a Sephadex column with 100% MeOH to give three subfractions (F2.3.1--3). ", "Subfraction F2.3.2 was further purified by semi-preparative HPLC \\[Optimapack C~18~ column (10 mm × 250 mm, 10 µm particle size, RS Tech, Seoul, Korea); mobile phase MeCN in H~2~O containing 0.1% HCO~2~H (0--40 min: 35 to 100% MeCN, 41--51 min: 100% MeCN); and, flow rate: 2 mL/min\\] to yield compounds **1** (6.0 mg) and **2** (5.0 mg). ", "The 70% MeOH fraction was separated by reversed-phase chromatography (40--63 µm particle size) with a stepwise elution gradient of MeOH/H~2~O (2:3--1:0) to afford four subfractions (F70.1--F70.4). ", "The fraction F70.1 was subjected to HPLC \\[Optimapack C~18~ column (10 mm × 250 mm, 10 µm particle size, RS Tech, Seoul, Korea); mobile phase MeCN in H~2~O containing 0.1% HCO~2~H (0--65 min: 20% MeCN, 65 min: 100% MeCN); flow rate: 2 mL/min\\] to afford compounds **11** (dieckol, 15.0 mg), **13** (dibenzo\\[1,4\\]dioxine-2,4,7,9-tetraol, 3.0 mg), **6** (eckol, 2.0 mg), and **10** (6,6′-bieckol, 2.0 mg). ", "Subsequently, fraction F70.2 was separated by HPLC \\[Optimapack C~18~ column (10 mm × 250 mm, 10 µm particle size, RS Tech, Seoul, Korea); mobile phase MeCN in H~2~O containing 0.1% HCO~2~H (0--65 min: 20% MeCN, 65 min: 100% MeCN); flow rate: 2 mL/min\\] to give compounds **12** (phlorofucofuroeckol A, 15.0 mg) and **14** (dioxinodehydroeckol, 4.0 mg). ", "Fraction F70.3 was further purified by HPLC (Optimapack C~18~ column (10 mm × 250 mm, 10 µm particle size, RS Tech, Seoul, Korea); mobile phase MeCN in H~2~O containing 0.1% HCO~2~H (0--60 min: 30% MeCN, 65 min: 100% MeCN); flow rate: 2 mL/min) to yield compound **16** (fucofuroeckol A, 4.0 mg). ", "All of the isolated materials and related information were kept in the Korea Bioactive Natural Material Bank (KBNMB).", "\n\n### 4.3.1. ", "Dibenzodioxin-fucodiphloroethol (DFD) (**1**) {#sec4dot3dot1-marinedrugs-17-00149}\n\nBrown powder; \\[*α*\\]${}_{D}^{20}$ +1.5 (*c* 0.1, MeOH); UV(MeOH) λ~max~ (log ε) 230 (2.15); IR *ν*~max~ 3863, 3840, 3734, 3647, 3272, 2675, 2348, 1748, 1620, 1540, 1508, 1487, 1396, 1269, 1198, 1150, 1090, 1039, 1024 cm^−1^; ^1^H and ^13^C NMR data (800/200 MHz; DMSO-*d*~6~), see [Table S3](#app1-marinedrugs-17-00149){ref-type=\"app\"}; HRFABMS *m/z* 743. ", "0895 (calcd. ", "for \\[M − H\\]^−^, 743.0890).", "\n\n### 4.3.2. ", "Dibenzodioxin-fucodiphloroeckol (DFE) (**2**) {#sec4dot3dot2-marinedrugs-17-00149}\n\nBrown powder; \\[*α*\\]${}_{D}^{20}$ −22.7 (*c* 0.1, MeOH); UV(MeOH) λ~max~ (log ε) 230 (2.15); IR *ν*~max~ 3840, 3223, 2348, 1748, 1620, 1508, 1474, 1396, 1259, 1088, 1039, 1023, 997, 827 cm^−1^; ^1^H and ^13^C NMR data (850/212.5 MHz; DMSO-*d*~6~), see [Table S3](#app1-marinedrugs-17-00149){ref-type=\"app\"}; HRFABMS *m/z* 1113.1215 (calcd. ", "for \\[M − H\\]^−^, 1113.1215).", "\n\n4.4. ", "HPLC-qTOFMS Measurement {#sec4dot4-marinedrugs-17-00149}\n----------------------------\n\nHPLC-qTOFMS was performed on an Agilent 6500 series instrument (Agilent technologies, USA) with a dual ESI interface at the College of Pharmacy, Seoul National University, Korea. ", "A YMC-Triart C~18~ column (150 mm × 3.0 mm i.d., ", "5 μm) was used for separation. ", "The elution gradient consisted of H~2~O (A) and MeCN (B) (both buffered with 0.01% formic acid) increased from 10% B to 60% B (for fraction EC30) and 100% B (for fraction EC70) in 40 min (0.3 mL/min), where it was held in 12 min (1 mL/min), returned to 10% B in 0.1 min, and then maintained for 5 min (0.3 mL/min). ", "Nitrogen was used as the drying gas and collision gas in the ESI source. ", "The electrospray ion source parameters were as follows: drying gas flow rate, 10 L/min; heated capillary temperature, 350 °C; sheath gas temperature of 350 °C and flow of 12 L/min; nebulizer pressure, 30 psi; VCap, fragmentor, skimmer, and octopole RF peak voltages that were set at 4000, 180, 60, and 750 V, respectively. ", "The detection was carried out in positive and negative electrospray ionization modes, and the spectra were recorded by MS scanning in the range of *m/z* 100--1000. ", "The MS/MS analyses were carried out by targeted fragmentation and the collision energy was set at 20--80 eV for fractions EC30 and EC70 and at 50 eV for single compounds. ", "MassHunter software version B.06.01 (Agilent Technology) was used to control the LC--MS/MS system. ", "Version B.07.00 was used for data acquisition, analysis, and processing, including the prediction of chemical formula and exact mass calculation.", "\n\n4.5. ", "Cell Cultures and Viruses {#sec4dot5-marinedrugs-17-00149}\n------------------------------\n\nMDCK cells were provided by American Type Culture Collection (ATCC, Manassas, VA 20108, USA) and cultured in Dulbecco's modified Eagle's medium (DMEM) containing 10% FBS, 100 U/mL penicillin, and 100 μg/mL streptomycin. ", "Influenza A viruses (H1N1 A/PR/8/34 or H9N2 A/chicken/Korea/01210/2001) were obtained from Choong Ang Vaccine Laboratory, Korea, and they were stored at −80 °C.", "\n\n4.6. ", "Cytotoxicity Assay {#sec4dot6-marinedrugs-17-00149}\n-----------------------\n\nThe cell viabilities were evaluated by a MTT (3-(4,5-dimethyl-2-thiazolyl)-2,5-diphenyl-2H-tetrazolium bromide) assay. ", "Briefly, MDCK cells were seeded in 96-well plates at 1 × 10^5^ cells per well and then incubated for 24 h. The cultures were then replaced with DMEM-free serum and treated with the test compounds for 48 h. To avoid solvent toxicity, the final concentration of dimethyl sulfoxide (DMSO) was maintained at 0.05% (*v*/*v*). ", "Subsequently, 20 μL of a 2 mg/mL MTT solution was added to each well and the plates were incubated for 4 h at 37 °C. ", "After removing the medium, the formazan crystals were dissolved with 100 μL of DMSO and the absorbance was measured at 550 nm using a microplate reader (VersaMaxTM, Randor, PA, USA). ", "All of the experiments were performed in triplicate. ", "Regression analysis was applied to calculate the 50% cytotoxic concentration (CC~50~).", "\n\n4.7. ", "Cytopathic Effect (CPE) Assay {#sec4dot7-marinedrugs-17-00149}\n----------------------------------\n\nMDCK cells were seeded onto 96-well plates and incubated for 24 h. After washing twice with phosphate-buffered saline (PBS), the cells were infected with influenza viruses using DMEM that contained 0.15 µg/mL trypsin and 5 µg/mL BSA for 2 h. The cells were then washed with PBS, and new media containing the fractions or test compounds at different concentrations was added. ", "After incubation for three days at 37 °C under a 5% CO~2~ atmosphere, the medium was replaced with DMEM, 20 µL of 2 mg/mL MTT was added to each well, and the plates were incubated for an additional 4 h. The next steps followed a typical cytotoxicity assay procedure and the 50% effective concentration (EC~50~) was calculated by regression analysis.", "\n\n4.8. ", "Western Blot Analysis {#sec4dot8-marinedrugs-17-00149}\n--------------------------\n\nMDCK cells were grown into 6-well plates and infected with H1N1 A/PR/8/34 virus for 2 h. The cells were then washed with PBS and the media was replaced with new media with various concentrations of the test compounds. ", "After 24 h of incubation, the cells were washed with cold PBS and then lysed with 100 µL of lysis buffer \\[50 mM Tris-HCl (pH 7.6), 120 mM NaCl, 1 mM EDTA, 0.5% NP-40, and 50 mM NaF\\]. ", "The supernatants were collected, and the protein concentrations were determined using a protein assay kit (Bio Rad Laboratories Inc., USA). ", "The aliquots of the lysates were boiled for 5 min and loaded onto 10% or 12% SDS-polyacrylamide gels. ", "The gels were then electrophoresed and electrotransferred to polyvinylidene fluoride membranes (PVDF 0.45 µm, Immobilon-P, USA). ", "After blocking with a 5% skim milk solution, the membranes were incubated overnight at 4 °C with primary antibodies, namely, neuraminidase (Gene Tex, San Antonio, TX, USA), hemagglutinin (Sigma, St Louis, MO, USA), or mouse monoclonal actin (Abcam, Cambridge, UK). ", "After washing a few times with PBS, the membranes were incubated with secondary antibodies for 2 h and then detected by a chemiluminescence Western blotting detection kit (Thermo Fisher Scientific., ", "Rockford, IL, USA) using an Image Quant^TM^ LAS4000 imaging system.", "\n\n4.9. ", "Immunofluorescence Assay {#sec4dot9-marinedrugs-17-00149}\n-----------------------------\n\nThe MDCK cells were maintained on sterile glass coverslips and then infected with H1N1 A/PR/8/34 virus for 2 h. After that, the cells were transferred to new media containing the test compounds at various concentrations. ", "The cultures were incubated at 37 °C under a 5% CO~2~ atmosphere for 24 h. After fixation and permeabilization, the cells were blocked with 1% bovine serum albumin (BSA) solution (Sigma, St. Louis, MO, USA) for 1 h and then incubated overnight with monoclonal neuraminidase antibody (Gene Tex) that was diluted 1:50 in PBS. ", "The cells were then incubated with the secondary antibody FITC (fluorescence isothiocyanate)-conjugated goat anti-Rb IgG (Abcam, Cambridge, UK) for 1 h, and the nuclei were stained with 500 nM 4′,6-diamidino-2-phenylindole (DAPI, Thermo Fisher Scientific, USA) solution for 5 min at room temperature. ", "The slides were mounted with mounting medium for fluorescence (Vectashield, Vector Lab, Inc., USA). ", "The immunofluorescence images were acquired with fluorescence microscopy (Olympus ix70 Fluorescence Microscope, Olympus Corporation, Tokyo, Japan).", "\n\n4.10. ", "Statistical Analysis {#sec4dot10-marinedrugs-17-00149}\n--------------------------\n\nThe results are presented as the mean ± SD of three independent experiments. ", "The significant differences between the groups were calculated by one-way analysis of variance (ANOVA) using Tukey's or Duncan's post hoc tests, which were conducted with SPSS Statistics 23 software (SPSS, Inc., Chicago, IL, USA). ", "Statistical significance was accepted at \\* *p* \\< 0.05, \\*\\* *p* \\< 0.01, and \\*\\*\\* *p* \\< 0.001. ", "ImageJ software was applied for Western blot analysis.", "\n\nThe following are available online at <https://www.mdpi.com/1660-3397/17/3/149/s1>, Figure S1: HRESIMS spectrum of compound **1**; Figure S2: IR spectrum of compound **1**; Figure S3: ^1^H NMR spectrum of compound **1** (800 MHz, DMSO-*d*~6~); Figure S4: ^13^C NMR spectrum of compound **1** (200 MHz, DMSO-*d*~6~); Figure S5: HSQC spectrum of compound **1** (800 MHz, DMSO-*d*~6~); Figure S6: HMBC spectrum of compound **1** (800 MHz, DMSO-*d*~6~); Figure S7: ROESY spectrum of compound **1** (800 MHz, DMSO-*d*~6~); Figure S8: HRESIMS spectrum of compound **2**; Figure S9: IR spectrum of compound **2**; Figure S10: ^1^H NMR spectrum of compound **2** (850 MHz, DMSO-*d*~6~); Figure S11: ^13^C NMR spectrum of compound **2** (212.5 MHz, DMSO-*d*~6~); Figure S12: HSQC spectrum of compound **2** (850 MHz, DMSO-*d*~6~); Figure S13: HMBC spectrum of compound **2** (850 MHz, DMSO-*d*~6~); Figure S14: ROESY spectrum of compound **2** (850 MHz, DMSO-*d*~6~); Figure S15: Relationships between ion masses (*m/z* value) in negative ion mode and RMD values for compounds detected by HPLC-qTOFMS in the EC70 fraction; Figure S16: HPLC-qTOFMS measurement for twelve isolated compounds in negative ion mode at collision energy of 50 eV; Figure S17: MS/MS spectra and fragment ions analysis of isolated compounds. ", "HPLC-qTOFMS spectroscopic data of these single compounds were measured in negative mode at collision energy of 50 eV; Figure S18: Effects of compound **11**, **12**, **13**, and **14** on the viral protein synthesis at a concentration of 20 µM; original uncropped blots; Figure S19: Inhibitory effects of compound **12** on the viral proteins synthesis in a concentration-dependent manner; original uncropped blots; Figure S20: (A) Key HMBC correlations of compounds **1** and **2**. (", "B) Key ROESY correlations on 3D structure of compounds **1** and **2;** Figure S21: Physicochemical properties of isolated known compounds from *E. cava*; Table S1: List of species in *Ecklonia* genus from website World Register of Marine Species (<http://www.marinespecies.org>); Table S2: The inhibitory effects of compounds **11**, **12**, **13**, and **14** against the H1N1 A/PR/8/34 virus in a cytopathic effect and cytotoxicity assays; Table S3. ", "^1^H and ^13^C NMR spectroscopic data of compounds **1** and **2** in DMSO-*d*~6~ (*δ* in ppm); Scheme S1: Extraction and fractionation procedure of *E. cava.*", "\n\n###### \n\nClick here for additional data file.", "\n\nH.M.C. and T.P.D. performed phytochemical experiments and analysis experiments. ", "T.K.Q.H. designed and performed biological experiments. ", "H.W.K., B.W.L. and H.T.T.P. performed phytochemical experiments. ", "T.O.C. collected and authenticated the plant. ", "W.K.O. provided experimental ideas, designed the research and analyzed the data. ", "W.K.O. contributed to the supervision and discussion and reviewed/edited the manuscript. ", "All authors reviewed the manuscript.", "\n\nThis work was supported financially in part by grants from the Marine Biotechnology Program of the Ministry of Oceans and Fisheries (PJT200669) and the Korea Institute of Planning and Evaluation for Technology in Food, Agriculture and Forestry (IPET) through the Animal Disease Management Technology Development Program funded by the Ministry of Agriculture, Food and Rural Affairs (MAFRA) (318031031SB010).", "\n\nThe authors declare no conflicts of interest.", "\n\n{#marinedrugs-17-00149-f001}\n\n{#marinedrugs-17-00149-f002}\n\n{#marinedrugs-17-00149-f003}\n\n{#marinedrugs-17-00149-f004}\n\nmarinedrugs-17-00149-t001_Table 1\n\n###### \n\nMS/MS fragmentation of the peaks detected in the total ion chromatograms of the EC30 and EC70 fractions of *E. cava* in negative ionization mode.", "\n\n No Compound Name RT (min) MS ^1^ MS ^2^ \\[M − H\\]^−^ Chemical Formula Calcd. ", "Mass Dif. ", " RMD\n -------- --------------------------------------------------------------- ---------- ------------------------------------ ---------------------------------------- -------------- ------------------ ------------- -------- ------\n **1** Compound **1** 21.793 195, 247, 389, 479, 583, 743 139, 231, 285, 353, 447, 493, 643, 743 743.0881 C~36~H~23~O~18~ 743.0884 −1.73 119\n **2** Compound **2** 25.279 195, 359, 479, 556, 663, 982, 1113 231, 353, 461, 705, 925, 1104, 1113 1113.1146 C~54~H~33~O~27~ 1113.1209 −7.94 103\n **3** Benzo\\[1,2-b:3,4-b′\\]bis\\[1,4\\] benzodioxin-1,3,6,9,11-pentol 24.015 195, 211, 246, 292, 369 193, 230, 267, 285, 310, 369 369.0224 C~18~H~9~O~9~ 369.0247 −4.80 61\n **4** Unknown 2.268 162, 197 163, 821 162.8392 \\- 5154\n **5** Unknown 2.631 122, 139, 168, 195 111, 139, 171 195.9504 \\- 4850\n **6** Eckol 15.543 154, 371 217, 246, 282, 371 371.0446 C~18~H~11~O~9~ 371.0403 6.56 120\n **7** 7-Phloroeckol 15.845 154, 495 263, 297, 387, 488, 495, 616 495.0511 C~24~H~15~O~12~ 495.0564 −7.28 103\n **8** 2-Phloroeckol 16.003 154, 263, 495 201, 229, 283, 346, 495 495.0528 C~24~H~15~O~12~ 495.0564 −6.00 107\n **9** Dibenzo\\[b,e\\]\\[1,4\\]dioxin-1,2,4,7,9-pentol 16.209 137, 155, 263 207, 218, 263 263.0157 C~12~H~7~O~7~ 263.0192 −5.92 60\n **10** 6,6′-Bieckol 18.854 154, 741 229, 261, 371, 479, 585, 666, 741 741.0647 C~36~H~21~O~18~ 741.0728 −9.00 87\n **13** Dibenzo\\[1,4\\]dioxine-2,4,7,9-tetraol 19.274 123, 155, 247 141, 195, 247, 385, 479 247.0243 C~12~H~7~O~6~ 247.0243 0.00 98\n **15** 6,8′-Bieckol 19.569 741 260, 371, 479, 615, 741 741.0737 C~36~H~21~O~18~ 741.0728 3.00 99\n **11** Dieckol 19.917 321, 741 229, 261, 371, 430, 545, 714, 741 741.0723 C~36~H~21~O~18~ 741.0728 −2.24 98\n **14** Dioxinodehydroeckol 20.370 123, 196, 325, 369 123, 161, 173, 199, 261, 369 369.0269 C~18~H~9~O~9~ 369.0247 4.69 73\n **12** Phlorofucofuroeckol A 21.796 155, 601 245, 385, 447, 493, 601 601.0640 C~30~H~17~O~14~ 601.0618 4.69 106\n **17** 2,7″-phloroglucinol 6,6′-bieckol (PHB) 22.018 155, 973 229, 353, 427, 493, 707, 806, 973 973.1153 C~48~H~29~O~23~ 973.1100 7.28 118\n **18** 974-A 22.050 113, 601, 973 229, 353, 393, 605, 707, 805, 941 973.1038 C~48~H~29~O~23~ 973.1100 −7.87 107\n **19** Unknown 22.44 155, 369, 551 223, 304, 551, 583, 710 551.1816 \\- 329\n **16** Fucofuroeckol A 22.897 155, 477, 2551, 352, 477, 545 477.0425 C~24~H~13~O~11~ 477.0457 −5.66 89\n **20** Unknown 23.799 242, 310 201, 271, 348 242.1758 \\- 726\n **21** 6,9,12-Octadecatrienoic acid, (6Z,9Z,12Z) 29.966 277, 527 264, 353, 481, 277.2125 C~18~H~29~O~2~ 277.2168 −6.56 767\n **22** 9,12-Octadecadienoic acid (9Z,12Z) 30.496 279 218, 248, 279, 346, 380 279.2168 C~18~H~31~O~2~ 279.2324 −12.49 776\n **23** Ecklonialactone B (R/S) 32.281 265, 291 201, 291 291.2020 C~18~H~27~O~3~ 291.1960 7.75 694\n **24** Eicosanoic acid 34.206 311 225, 311, 349 311.2848 C~20~H~39~O~2~ 311.2950 −10.10 915\n **25** Ecklonialactone B (S/R) 34.499 291, 555, 623 251, 291, 411,651 291.1938 C~18~H~27~O~3~ 291.1960 −4.96 666\n **26** 11,14,17-Eicosatrienoic acid, (11Z,14Z,17Z)- 34.771 305 211, 284, 305, 248, 583, 804 305.2410 C~20~H~33~O~2~ 305.2481 −8.43 790\n **27** Tricosanoic acid 36.294 353 257, 333, 353, 529 353.3407 C~23~H~45~O~2~ 353.3420 −3.61 964\n **28** Heneicosanoic acid 36.667 325 225, 239, 248, 267, 282, 325 325.3131 C~21~H~41~O~2~ 325.3107 4.90 962\n **29** Unknown 37.806 293, 325 281, 325, 386, 449, 674 325.1812 \\- 557\n **30** Unknown 37.869 239, 293 207, 239, 243, 383 239.0709 \\- 297\n **31** Unknown 38.947 321 248, 321, 399, 572, 815 321.2178 \\- 678\n **32** Docosanoic acid 39.599 339 226, 339, 433, 660, 809 339.3292 C~22~H~43~O~2~ 339.3263 5.39 970\n **33** Unknown 41.028 346 206, 254, 330, 346, 642 346.1092 \\- 316\n **34** Unknown 41.59 346, 485 280, 421, 485 485.2671 \\- 550\n **35** 5,8,11,14,17-Eicosapentaenoic acid, (5Z,8Z,11Z,14Z,17Z) 43.93 301, 369 205, 269, 301, 440 301.2099 C~20~H~29~O~2~ 301.2168 −8.31 697\n **36** Triacontanoic acid 47.116 346, 451, 535, 691, 775 295, 387, 451, 456 451.4499 C~30~H~59~O~2~ 451.4515 −4.00 997\n **37** Dotricontanoic acid 47.575 346, 479, 609, 691, 775 461, 479, 648, 866 479.2903 C~32~H~63~O~2~ 479.4828 −5.00 606\n **38** Unknown 49.963 223, 297, 441, 535, 701, 849 254, 333, 441, 673 441.2074 \\- 470\n **39** Unknown 51.118 149, 223 149, 448 149.0021 \\- 14\n **40** Unknown 53.856 149, 223, 297 149, 221, 350, 630.7708 149.0086 \\- 58\n **41** Unknown 58.796 135 135.9698, 287.5036 135.9751 \\- 7171\n\n[^1]: These authors contributed equally.", "\n" ]

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