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"\nWe and others have previously observed that immunologic activity can often be restored to both lymphocytes and neutrophils by removing them from the burn environment, leading to the conclusion that burn serum contains substances capable of suppressing immunologic function. ",
"The present studies were initiated to better define the serum component(s) responsible for this immunosuppression. ",
"The majority of immunosuppressive activity vs. both neutrophil chemotaxis and mixed lymphocyte cultures contained in large-volume serum samples obtained from three patients with greater than 40% body surface area flame burns was found to reside in a less than 25,000 mw fraction of serum obtained by Amicon ultrafiltration. ",
"A single suppressive serum component was isolated by precipitation and resuspension, followed by ion-exchange chromatography using an SP Sephadex C-25 column. ",
"Purity of the samples was verified by SDS slab-gel electrophoresis, and immunosuppressive activity was confirmed vs. both lymphocytes and neutrophils. ",
"Analysis of this isolated burn-associated suppressor indicates: a) a molecular weight of between 1,000 and 5,000 daltons; b) a complex composition containing a protein component, a lipid component, and a carbohydrate component; c) a structure which is heat stable, pH stable and unaffected by treatment with trypsin, proteinase K, DNAse, and RNAse; and d) a noncytotoxic immunosuppressive mode of action. ",
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"Q:\n\nFailed to resolve: com.android.support:appcompat-v7:jar\n\nI am newbie in Android Studio Platform. ",
"I am getting this error \n\" Failed to resolve: com.android.support:appcompat-v7:jar \"\nCurrently my version Android Studio is 3.5.0.",
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"There is no jar in this dependency.",
"\nUse stable\n implementation 'com.android.support:appcompat-v7:28.0.0' //28.0.0-alpha1\n\nFYI\n\nPlease fix the version conflict either by updating the version of the\n google-services plugin or or updating the version of\n com.google.android.gms to 11.4.2\n\nYou should upgrade below version\nimplementation 'com.google.android.gms:play-services-auth:17.0.0'\n\nIt will be good approach if you use upgraded classpath\nclasspath 'com.google.gms:google-services:4.2.0'\n\n"
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"BRIAN FERGUSON\n\nEDINBURGH faces being toppled from its position as the global festival capital unless it can attract substantial new funding, create new venues, catch up on the digital revolution and make it easier to stay in the city, new research has found.",
"\n\nIts festivals are being urged to save money by working together on projects, embrace new technology to allow audiences to experience events in their own homes, help develop new air routes to key festival markets, and target wealthy diaspora and arts philanthropists to shore up projected funding gaps.",
"\n\nEdinburgh and Glasgow are also being urged to work together to promote their “complementary” high-profile festivals and agree a “memorandum of understanding” for future joint working in the wake of a Commonwealth Games pilot project.",
"\n\nA 10-year blueprint to protect the status of events like the Edinburgh International Festival, the Fringe, the Tattoo and the Hogmanay celebrations warns they risk relegation from the “premier division” if current levels of backing are not maintained.",
"\n\nThe majority of the £9 million in public sector support comes from Edinburgh City Council and Creative Scotland, with the Scottish Government providing £2.25 million a year through its “Edinburgh festivals expo fund.” ",
"Consultants brought in by a body overseeing the 12 main festivals say “radical” solutions must be found to cope with a looming “fiscal cliff”.",
"\n\nTheir report states: “The combination of ticket sales, private finance and public funding is at the heart of the festivals’ success. ",
"The erosion of public investment is likely to have a disproportionate effect on earned income, with investment in innovative product and investor confidence affected.”",
"\n\nPrevious research in 2006, which looked at the growing competition from events across the UK and overseas, had warned that the city’s pre-eminent position was “vulnerable.” ",
"Entitled Thundering Hooves 2.0, the new study found “Edinburgh’s place as the foremost festival city is, for the moment, secure.”",
"\n\nBut the report - which benchmarked the city against Austin, in Texas, Montreal, Venice and Manchester - make 38 different recommendations and is intended as a “spur to action to make sure Scotland, the city and its festivals do not rest on their laurels and become complacent.”",
"\n\nThe £50,000 study found infrastructure improvements and investment in some of Edinburgh’s competitor cities were exposing significant “weaknesses” in the Scottish capital, which has lost a number of cultural venues in recent years.",
"\n\nA new concert hall and the creation of a digital arts centre at Edinburgh University are recommended for exploration to address gaps in provision, while the study also reports accommodation shortages at both the “low cost” and “luxury” ends of the market. ",
"The festivals were also said to be “behind the curve” in their use of new technology and have been urged to become “leaders in the digital sphere.”",
"\n\nCreative Scotland chief executive Janet Archer said the report showed the festivals’ “huge potential,” but was also honest about its looming challenges.",
"\n\nShe added: “It’s important we recognise and are realistic about the continued pressures on public funding, and that we all work together, be imaginative and think hard to protect the important vitality of the festivals.”",
"\n\nSandy Howat, deputy council leader, said the authority was fully aware the festivals were “very much dependent” on their public funding.",
"\n\nHe added: “The cuts that are coming are going to be a considerable challenge, but the council also has to recognise that our cultural budget at the moment is reasonably small considering the benefit that culture as a whole gives the city.”",
"\n\nScottish culture secretary Fiona Hyslop said: “In the past 10 years the environment has evolved beyond anything that could have been imagined and Edinburgh’s festivals face new challenges and challengers.",
"\n\n“Our festivals need to be dynamic and brave to embrace future opportunities and to protect Edinburgh’s position as the world’s leading festival city.”",
"\n\nCONSULTANTS BOP and Festivals and Events International have recommended the biggest ever global marketing campaign for the festivals is created to coincide with the 70th anniversary of the city’s first major events being held in 1947.",
"\n\nThey have urged the festivals to use the model of programmes developed to coincide with the London Olympics and Glasgow’s Commonwealth Games to inspire a five-year celebration.",
"\n\nThe festivals are also being urged to develop Olympics and Commonwealth Games-style volunteering programmes, and explore how to offer more apprenticeships and internships.",
"\n\nThe report recommends the creation of new initiatives to bridge the gap between residents who feel “great pride” in the city hosting major events and those who buy tickets.",
"\n\nIt says leadership is needed to overcome “siloed thinking” in the city, while new accommodation is recommended to match audience demographics and the “world-class quality” of competing cities."
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"Reduction of acute-phase proteins with tenidap sodium, a cytokine-modulating anti-rheumatic drug.",
"\nFour independent studies have investigated and compared the effects of tenidap sodium, naproxen and placebo on CRP in patients with active RA. ",
"One of these studies also investigated the effects of tenidap and naproxen on serum amyloid A (SAA) concentrations and ESR. ",
"The duration of the four studies ranged between 2 weeks and 24 weeks, and depending on the study, tenidap sodium was administered orally in doses of 40-120 mg/day and naproxen in doses of 1000 mg/day. ",
"In all four studies serum CRP concentrations in tenidap-treated patients had decreased significantly from baseline at the time of final assessment. ",
"The decrease in CRP concentration in tenidap-treated patients was observed as early as 1 week after initiation of therapy and was sustained for up to 6 months, the last assessment timepoint. ",
"CRP concentrations in naproxen-treated and placebo patients were essentially unchanged. ",
"The decreases from baseline observed in tenidap-treated patients were significantly greater than the changes observed in naproxen-treated or placebo patients. ",
"After 24 weeks of tenidap treatment the decrease in CRP was paralleled by significant decreases in SAA concentration and ESR. ",
"The finding that tenidap sodium rapidly, consistently and significantly lowered CRP serum concentrations differentiates tenidap sodium from the NSAID, naproxen. ",
"This could possibly have important therapeutic implications given that other long-term investigations have shown that reducing serum CRP and SAA concentrations correlates with a reduction in radiographically-assessed disease progression."
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"“I rarely take my nut allergic daughter out for meals but our friend had booked their for her daughters birthday.",
"\n\nI rang in advance and explained about my allergy concerns and the manager came on the phone to reassure me that there would be sage food available for my daughter to eat. ",
"She asked me to identify ourselves to staff and they would give me their allergy book. ",
"Upon arrival I did what they had asked and I was given their comprehensive allergy book that listed allergy information for every single item they sell. ",
"The chefs were also on hand to talk to about any food item. ",
"My daughter thoroughly enjoyed the reaction free meal and we will definitely go back again.”",
"\n\nLaura\n\nHi there My son is 4 years old and has a peanut allergy which is level 4 he has Epi… Continue Reading\n\nNutmums Article Archive\n\nNutmums Article Archive\n\nVisit Allergy UK\n\nDisclaimer\n\nThe contents of Nutmums.com are for information purposes only. ",
"I am not a medical professional. ",
"Please see your doctor for medical advice. ",
"I am not warranting the nut free status of any food products. ",
"Please always check the label each time & contact the manufacturer if you are ever unsure."
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"6/11/2010 @ 6:00AM\n\nEuro Bargains\n\nTo many American tourists France is the center of the universe, but to U.S. investors it’s more like Mars. ",
"It’s an alien, inhospitable place where costs are high, labor laws rigid and economic growth paltry. ",
"To top it off, the land of Bordeaux and brie hasn’t exactly gone out of its way to make it easy for Yanks to own a few shares. ",
"Only seven French companies trade on the NYSE (compared with about two dozen British firms).",
"\n\nThat’s too bad, say Isabel Levy and François-Marie Wojcik. ",
"Over the past eight years they have built Metropole Gestion, situated steps from the Paris Bourse, into a firm managing $2.5 billion, mostly in Sicavs (French mutual funds).",
"\n\nDabbling in a broad swath of European stocks, Metropole’s value yardsticks differ by industry. ",
"For banks Levy and Wojcik typically get interested when the price of a share is close to or below its book value. ",
"In pharmaceuticals they’ll pay up to 2 times annual sales but sell at 3 times. ",
"With retailers like Carrefour they like to buy at 0.3 times sales and sell at a multiple of 0.7 (\nWal-Mart\ngoes for 0.5 times). ",
"Often such practices put Metropole France, their all-French fund, far away from the action of the moment.",
"\n\n“We are the grandchildren of the 18th-century philosophers Voltaire and Diderot,” says Wojcik, seated in Metropole Gestion’s offices inside an ornate Belle Epoque building that once housed a Paris bank. “",
"We doubt everything. ",
"Doubt, doubt, doubt.” ",
"As an example, he says if a chief executive boasts of his plans to become his industry’s leader, “We don’t say, ‘Merveilleux!’ “",
"\n\nLevy, 48, and Wojcik, 50, are a rather odd pair of business partners. ",
"A friend of both introduced the two in 1997 when tech was booming. ",
"They discovered a shared skepticism about the prospects for then high-flying telecom and set up Metropole Gestion in 2002. (",
"Their funds are unavailable in the U.S., but for a list of French companies and a fund that do trade in the U.S., see the table above.)",
"\n\nLevy, who oversees the company’s portfolio managers, is hyperkinetic. ",
"Wojcik, the chief executive officer, is a portly and slow-talking Frenchman of Polish descent. ",
"They are one, however, in the view that there’s often a disconnect between the way the market values companies and their intrinsic value, which can be defined as what a rational corporate acquirer would be willing to pay.",
"\n\nSince buying something cheap isn’t profitable if it stays that way, Levy and Wojcik look for a catalyst that will lead to an upward reappraisal over the following 18 months. ",
"That might be an acquisition or other event, or the higher profit margin they believe will result from a merger. ",
"To U.S. investors talk of “catalysts” is clichéd; in France it’s still somewhat novel.",
"\n\nHow’s the shopping these days? ",
"Excellent, says Levy, given the tumble European stocks have taken at the hands of Greek deadbeats. “",
"You can buy companies with very good pricing power at a discount,” she says. ",
"For U.S. buyers of European stocks, there is the additional bargain element of the euro’s fall to a four-year low versus the dollar.",
"\n\nLevy likes Schneider Electric, a Rueil-Malmaison maker of gear like circuit breakers. ",
"Metropole France first invested in January 2009 when Schneider was trading at $51.24, or 0.9 times sales, versus a long-term average of 1.6. ",
"The fund dumped Schneider this March when it hit 1.4 times sales and $106 (at current exchange rates), then bought it back in May at 1.2 times sales.",
"\n\nMost of Metropole France’s investments are more long term, and it holds up to 10% of its assets in a single stock. ",
"Over the past five years the fund has returned 6% annually, net of fees, versus –0.49% for the CAC-40.",
"\n\nIts holdings include some companies not generally regarded as value plays, like ad agency Publicis and luxe-brand purveyor LVMH Moët Hennessy Louis Vuitton. ",
"Metropole first bought into LVMH seven years ago, after it had acquired distributor DFS, fallen out of favor with investors and was trading for a 34% discount to Metropole’s estimate of its breakup value.",
"\n\n“We don’t believe there are value stocks and growth stocks,” explains Levy. “",
"Every company at some point can be undervalued or overvalued.”",
"\n\nMetropole France watched Publicis for four years before its price fell to 1.2 times revenue, and the fund began buying at $34. ",
"It has taken a similar approach to banks.",
"\n\nMany investors shun BNP Paribas for what by European standards is a large retail banking operation relative to its investment banking business. ",
"Levy, by contrast, was attracted to the commercial side’s stability and began buying shares at one times book, or $49, in November 2008.",
"\n\nBNP Paribas looks likely to fare better than many rivals in the current environment, thanks to the stability of its branch network. ",
"It should also enjoy a boost from its recent purchase of Fortis, a banking, insurance and money management outfit, Levy adds.",
"\n\nLevy and Wojcik are willing to venture even into industries whose fundamentals they loathe–if the price is right. ",
"That’s what they did with Air France in March 2009, when they bought in at 1.2 times operating cash flow (in the sense of cash flow from operations minus costs associated with long-term investments and capital spending), or $9 a share at the present exchange rate. ",
"Recent price: $12.",
"\n\nSome companies they wouldn’t buy at any price.",
"\nBP\nfits that category, with its open-ended liabilities from the Gulf of Mexico spill. ",
"Not to mention the fact that it’s British.",
"\n\nEuro Treasure?",
"\n\nThe French fund and stocks below are among the few available in the U.S. They may be worth a look amid the recent tumble in European stock prices and the euro.",
"\n\n.",
"\n\nPrice\n\n.",
"\n\n.",
"\n\nCompany/Fund\n\nRecent\n\n52-week\n\nHigh\n\nMarket\n\nValue\n\n($Bil)\n\n2010\n\nEstimated\n\nPE\n\niShares MSCI France Index ETF\n\n$20.21\n\n$27.32\n\nNA\n\nNA\n\nBNP Paribas\n\n28.24\n\n43.55\n\n$66.9\n\n8\n\nSanofi-Aventis\n\n29.76\n\n41.59\n\n78.0\n\n7\n\nSchneider Electric\n\n10.16\n\n12.00\n\n27.1\n\n16\n\nSociete Generale\n\n8.40\n\n16.40\n\n31.1\n\n8\n\nAll figures are in U.S. dollars. ",
"Prices as of June 1. ",
"All companies trade in the U.S. as ADRs. ",
"NA: Not available.",
"\nSources: Bloomberg; Interactive Data and Thomson Reuters IBES via FactSet Research Systems."
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"DJ Music Mixer is a full-featured DJ and beat-mixing system to create your own custom DJ-style music. ",
"The applications features include Audio and Video player with two independent decks and professional sound quality, as well as beat-sync display, pitch control, and fast-seek mode. ",
"There is a crossfader for manual mixing and an autofade functionality. ",
"Play multiple files with configurable number of sampler, each of which has its own player and volume control. ",
"Mix your favorite video clips (DivX, or MPEG) together with audios to have a professional mixing output. ",
"Also consist of 10-band equalizer and more than 12 presets to make your work easier and faster."
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"This is a special guest post form Leslie Newton, founder of Timi & Leslie Diaper bags. ",
"Planning a baby shower can be exciting but intimidating. ",
"Sometimes it’s a good idea to have a few people plan it together so you can really have fun rather than be too stressed. ",
"The first thing to figure out is your [...][Continue Reading...]"
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"Q:\n\nWhy is Canon 60D considered 'better' than Canon 700D?",
"\n\nSo Canon 60D is considered a semi-pro machine, but it's like 3 years old, while the 700D is a year old and can pack the same punch it seems from the specs.",
"\nDoes the 60D have anything that the 700D doesn't?",
"\nIs it really better?",
"\n\nA:\n\nBoth the Canon 60D and the 700D/T5i are built around the same basic sensor: Canon's 18MP APS-C sensor with 4.3µm pixel pitch. ",
"It has also appeared in the T2i/550D, T3i/600D, T4i/650D,SL1/100D, EOS M, and 7D. When shooting RAW and editing on a computer any of these cameras can use the latest updates to Canon's demosaicing algorithms and image processing that have improved over the time this sensor has been in use.",
"\nBut the sensor is not the only thing that makes a camera a consumer grade, prosumer grade, or professional level camera. ",
"There are other things to consider:\n\nHandling speed. ",
"It doesn't matter how good the image quality of your sensor is if your camera handles too slow to get the shot you need. ",
"This applies to the placement of controls, how many clicks it takes to change a setting, the frame rate the camera can burst, and startup and shutter lag times.",
"\nDurability. ",
"The ability to stand up to day in and day out heavy use in challenging environments is a larger consideration when your livelihood depends on the reliability of your gear.",
"\n\nThe biggest difference in my mind is the faster handling of the 60D versus the 600D. The 60D has two control wheels and more direct controls that allow for changing settings on the fly without removing your eye from the viewfinder. ",
"This allows an experienced shooter who intimately knows his camera to get certain shots in challenging situations he couldn't get with the 600D.\nIn some ways the 60D was a step back from the 50D as a lower end pro grade camera. ",
"At the same time the 7D was a step up from the 50D. The magnesium alloy body of the 50D (and 7D) was replaced by a polycarbonate one for the 60D. The 60D is not quite as weather sealed as the 50D and 7D. The 50D (and 7D) included Auto Focus Micro Adjustment, the 60D does not have AFMA. ",
"But the 60D still has more features that pros demand that are lacking in the Rebel series.",
"\n\nA:\n\nAs already pointed out none of them would be considered more \"pro\" than the other by working photographers. ",
"There are differencies though and I list some of the advantages of the 60D below:\n\n60D has a pentaprism (instead of a pentamirror in the 700D) which is a lot better (it's brighter) albeit heavier.",
"\n60D has a larger viewfinder which makes omposing and manual focusing easier\nThe sensor of the 60D is according to DxOMark better (lower noise and better image quality and more dynamic range)\n60D is said to be better wheather sealed than the 700D\n60D's battery will produce 1100 shots on one charge vs 550 shots for the 700D\n\nThere are also some (in my opinion) lesser advantages such as less start up delay and shutter lag, higher maximum shutter speed, faster continious shooting and higher viewfinder coverage.",
"\nAdditionally the 60D is available in the 60Da variant which is a very specialised version that is considered \"pro\" since it's unique even when compared to the conventional \"pro\" bodies.",
"\nWhen deciding of it's better or not it's so subjective that I can't give an answer since price comes into play. ",
"It all comes down to what the user is prepared to pay, but if money where of no concern and the touch screen unimportant i would choose the 60D.\n\n"
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"Q:\n\nHow to read specific lines of codes in another sub?",
"\n\nI was wondering if there is a way to run specified lines of code in another sub , say execute only line 1 through line 10 of Sub_2 before executing the code in say, Sub_1.",
"\nI have done some background research and found up a code suggested by @Andrew below.",
"\nPrivate Sub Sub_1(sender As System.",
"Object, e As System.",
"EventArgs) Handles Button1.Click\n\n 'Call the Save Sub\n Call Sub_2(sender, e) \n 'Proceed to execute the rest of the this Sub_1\n\nWhat the above code does at the moment is execute the entire sub I have called, which is not what I want to achieve.",
"\nI have also read about the using a shared routine common to both Subs as suggested by @Henry , but the problem for me is that variables are declared in the shared routine which are required at various stages in Sub_2, but which cannot be used as the program says they are not declared (in Sub_1).",
"\nAgain, what I want to achieve is execute only a few lines of code from Sub_2 in Sub_1.",
"\nI am using Visual Basic 2010 Express.",
"\n\nA:\n\nWhat you are trying to do is technically possible (in various really hackish ways), but would easily become a code structure nightmare. ",
"Instead, consider an alteration to the way you are building your routines. ",
"\nSubs and Functions are intended to be independent 'islands' of code, which can execute (generally) without knowing anything about the rest of the program. ",
"You can pass variables into them, and get variables back, but they should operate the same time after time. ",
"\nConsider this theoretical code from the scenario you are proposing:\nDim variable_1\n\nPrivate Sub Sub_1()\n Dim variable_2\n Call Sub_2()\nEnd Sub\n\nPrivate Sub Sub_2()\n 'lines 1-10\n 'lines 11-20\nEnd Sub\n\nHere we have the setup for what you're looking at. ",
"Both Subs can 'see' variable_1, but Sub_2 can't see variable_2. ",
"This is called scope. ",
"Finally, there is the issue of only wanting to execute lines 1-10 inside Sub_2 when being called from Sub_1.",
"\nThe first thing to do is to separate our code into logical groups. ",
"According to what you're describing, lines 1-10 of Sub_2 make sense when looked at on their own, so this automatically means that they should be grouped together. ",
"Lines 11-20 may be a different story, but since they are sometimes relevant and others not, we should separate them as well.",
"\nPrivate Sub Sub_2()\n 'lines 1-10\nEnd Sub\n\nPrivate Sub Sub_3()\n Call Sub_2()\n 'lines 11-20\nEnd Sub\n\nThe change looks simple, but is actually quite powerful. ",
"When we only want to execute lines 1-10, we call Sub_2. ",
"If we want to execute lines 1-20, we call Sub_3. ",
"Sub_3 will automatically call Sub_2 (lines 1-10) to execute first, and then execute lines 11-20. ",
"Easy.",
"\nNow we just have to figure out what to do about the variables. ",
"There are quite a few ways you can handle this situation, but the easiest thing to do is to expand their scope. ",
"If all the variables you need to access are declared outside of these Subs (one \"level up\"), then they can all access them. ",
"For example:\nDim variable_1\nDim variable_2\n\nPrivate Sub Sub_1()\n Dim variable_3\n 'can access 1, 2, 3 but NOT 4, 5\n Call Sub_2()\nEnd Sub\n\nPrivate Sub Sub_2()\n Dim variable_4\n 'can access 1, 2, 4 but NOT 3, 5\nEnd Sub\n\nPrivate Sub Sub_3()\n Dim variable_5\n 'can access 1, 2, 5 but NOT 3, 4\nEnd Sub\n\nThis should get you started. ",
"Once you've got this down, you can read about the various ways to pass variables into Subs and Functions (which also gives some hints about how to get them out, as well). ",
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"The invention relates to a cryogenic fluid offloading system comprising: an offshore mooring structure, connected to the seabed, a connecting member that is attached to the mooring structure with a first end to be displaceable around a vertical axis, a tanker vessel for loading cryogenic fluid at a first location, transporting it and offloading the cryogenic fluid at a second location, the tanker vessel being connected to the mooring structure via the connecting member, a first fluid duct connected to the mooring structure, for supplying fluid away from the mooring structure, a second fluid duct connected to the mooring structure, for transporting fluid coming from the tanker vessel to the mooring structure, a processing unit for receiving a cryogenic fluid in liquid phase from the tanker vessel and for supplying a gaseous phase of the fluid to the first fluid duct, and fluid supply means for controlling supply of cryogenic fluid from the tanker vessel to the processing unit. ",
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"Discovery of Betatrophin: A New Hope for Beta Cell Replication\n\n*Update on November 2, 2014: Researchers no longer believe betatrophin induces beta cell growth. ",
"For more see here.",
"\n\nBeta cells are persnickety little cells. ",
"They are the only cells in the body that produce insulin, life-preserving hormone that it is, and yet they are killed off by the immune system in type 1 diabetes, and it is thought that they die off from over-taxation following insulin resistance in type 2 diabetes. ",
"To make matters worse, unlike many cells of the body, beta cells are poor at replicating– which is to say, the set of beta cells you have by the time you are an adult is not going to expand or grow very much. ",
"So when a type 1 or type 2 diabetic starts to lose beta cells, the body can not adequately replenish them, and thus diabetics become hyperglycemic as their dwindling stores of beta cells become unable to produce enough insulin.",
"\n\nAs a result of this limited supply of natural beta cells, there is much to be gained from having a renewable source of new beta cells. ",
"A number of advances have been made towards transplanting donor beta cells or creating new beta cells from stem cells, but now scientists have found a promising new way to stimulate the body’s own beta cells to replicate. ",
"Dr. Doug Melton’s group at Harvard, which has made many advances in our understanding of stem cell and beta cell biology over the years, has identified a new hormone, which they called betatrophin, that, when produced by the liver, induces beta cell growth in the islets of the pancreas.",
"\n\nHow did the researchers find betatrophin? ",
"Ironically enough, they started with a new model of insulin resistance in mice. ",
"They infused a drug that blocks the insulin receptor, S961, into mice, and thus blocked insulin signaling in the mice. ",
"The mice became hyperglycemic as expected, but something else happened too: the beta cells of the insulin resistant mice replicated and expanded, achieving a replication rate that was 12 times the normal rate at the highest dose of S961.",
"\n\nNow, this expansion of beta cell mass in response to insulin resistance is not unheard of; other researchers have shown that beta cells increase their replication rates in certain types of insulin resistance, in response to glucose, and even in pregnancy. ",
"However, the growth seen with S961 was much more dramatic, and so the researchers decided to figure out what might be causing the beta cell growth.",
"\n\nThe researchers first checked whether the drug was causing beta cells to grow directly, but the drug alone had no effect on beta cells. ",
"The researchers next looked at the levels of genes in the liver, fat tissue, and skeletal muscle of the mice. ",
"They compared the mice treated with the drug to untreated mice, and were able to find one particular gene that was present at a very high rate in the treated livers and fat, but not in the untreated. ",
"This gene had previously been predicted and identified as Gm6484, but the authors of the study have decided to name it betatrophin.",
"\n\nThis gene betatrophin is abundant in liver tissue for many mammals, including humans, and gets turned into a protein that the researchers showed can be excreted from the liver into the bloodstream. ",
"The researchers next had to check whether betatrophin was actually the gene that mattered with the mice that were treated with S961. ",
"To do this, they injected mouse livers with the gene such that the livers would begin to make the betatrophin protein. ",
"Even in the absence of insulin resistance, this production of betatrophin by the liver resulted in rapid expansion of the beta cells in the mice, with a replication rate 17 times higher than normal. ",
"Further, the betatrophin-expressing mice had lower fasting blood glucose values and improved glucose tolerance compared to normal mice.",
"\n\n[youtube]http://youtu.be/Lk4DDlct__4[/youtube]\n\nTo summarize, then: researchers identified a protein, betatrophin, produced by the liver that leads to increased rates of beta cell replication, and thus more beta cells and more insulin capacity. ",
"If you’re still with me, you’re probably asking, “Well, wait, but what is betatrophin actually doing?” ",
"The answer to that important question is that the researchers don’t know yet. ",
"Based on the fact that they had to make the liver produce betatrophin, and they couldn’t just inject betatrophin into circulation in the mice or put betatrophin directly on beta cells, it seems that the effect of betatrophin is indirect, and likely there are other proteins or hormones between betatrophin and beta cell replication.",
"\n\nEven so, the researchers here have taken a very interesting first step in identifying betatrophin, and it is exciting to think what might come of this over the next few years as the scientists try to figure out how exactly betatrophin is working, and how we might be able to leverage its activity (and the activity of the other proteins it effects) to increase beta cell replication in humans who have a shrinking supply of beta cells. ",
"Clearly, there is quite a bit to do before this betatrophin pathway is useful in human treatment, but this first step certainly introduces new and intriguing possibilities.",
"\n\nThe Diabetes Media Foundation (DMF) is a 501(c)(3) tax-exempt nonprofit media organization, registered in New York and the IRS, under EIN #46-3338815. ",
"DMF is devoted to informing, educating, and generating community around, living a healthy life with diabetes."
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"\n\nThis transcript has been automatically generated and may not be 100% accurate.",
"\n\nI ... the the the the ... you'll find a lot of people still ... and you don't go ... to ... before going outdoors ... and doing on the movement is phenomenal brokerage ... firm is ... to memos letters ... lolx people ... orders will use for traders to write books ... my name is ... Michael ... and the sales manager general manager of Splinter Corporation ... we manufacture typewriters distribute them to all types of industries ... were here ... to stay ... case in the ... meantime traders start first with calculators electronic calculators ... electronic driver aids ... it's ... a series of different actors ... moving on from one mall to another ... winning features a display in the memory and style ... now all we've moved on to make clear that Cyprus would use in ... prisons for present and makes it ... very ... comforting thing to know that if they make an old typewriter ... not to want to hide contraband inside ... so when you want Sir chart I ... just a matter of looking ... and looking through it ... this year's ... critically transparent ... Stephen King's it inside ... even the agreement to set its terms ... so you can see through ... some teams to ... military to ... Treasury to many different from offices to require use of pipers ... and most people do require a news conference ... as ... for some reason have one in a corner some points ... usually can affect you can do it in ... words or you just type the characters and try ... to make a mistake ... on one meetings with display have spell check ... still Coca-Cola ... you don't earn a typewriter on ... and go to the intervention to Facebook ... we don't chat with the ... known ... the wheels with a pipers attire for years two hundred percent productivity ... what the doctor ... shortage ... is faced with two computers store didn't have to go someplace where profit is printed out ... so what's old typewriters are ... just ... just just a ... basic tool from for putting words on paper ... the late ... eighteen hundreds when indebted and back incline ... Synergos general business"
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"Q:\n\nQuestions that are pretty much on topic but the answers are maybe found elsewhere\n\nWith most Off Topic questions it is easy to comment saying where they should go within the network or elsewhere with their answer, but what about when a question is on topic but perhaps even more relevant somewhere else (and there isn't already such a question in place).",
"\nCase in point - Math behind multivariate testing for website optimization\nIn my mind it definitely belongs as a webmaster question, but the proposed solution (which appears to have been accepted) was to ask the question on another SE website.",
"\nIf this question should be closed, what should it be closed with?",
"\n\nA:\n\nClose > Off Topic > Belongs on... > Statistics SE\nIn this case, the question was already posed with Math behind multivariate testing for website optimization so I'm going to close as off topic rather than migrate a duplicate, but please do flag for migration over telling the asker to ask again wherever possible.",
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"I will definitely recommend them in the future. ",
"Executed a terrific work for a cheap cost.\"",
"\n5 out of 5 stars\n\nChance Daniel:\n\"I was very surprised with how much knowledge the specialists had. ",
"They achieved to fix every issue we had in our flat. ",
"They even repaired small issues for no added fee. ",
"I am totally content.\"",
"\n5 out of 5 stars\n\nIrvin Hatfield:\n\"Many thanks professional plumbers for the speedy outstanding work. ",
"I connected with them for emergency rescue. ",
"They were there in less than 4 hours. ",
"They worked hard and solved the problem. ",
"They were exceptionally good natured. ",
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"1. ",
"Field of the Invention\nThe present invention relates to an apparatus and a method for recording and reproducing video and sound for providing a \"time-shift reproduction\" function and a \"time-shift fast-forward reproduction\" function.",
"\n2. ",
"Description of the Related Art\nIn recent years, the popularization of satellite broadcasting, CATVs and the like has caused a considerable increase in the number of broadcasting channels. ",
"As a result, very frequently TV audiences want to watch several TV programs broadcasted in the same time period. ",
"Moreover, home-use video apparatuses have also been popularized. ",
"Therefore, it is desirable to develop a method for utilizing such apparatuses more efficiently.",
"\nFIG. ",
"16 shows an exemplary conventional apparatus for recording and reproducing video and sound, in which a TV set is connected with a video cassette recorder (VCR).",
"\nHereinafter, the respective components shown in FIG. ",
"16 will be described.",
"\nBroadcast receiving sections 1 and 2 receive a broadcast. ",
"Typically, the broadcast receiving section 1 is a tuner incorporated into a TV set, and the broadcast receiving section 2 is a tuner incorporated into a VCR.",
"\nA video/sound recording section 3 converts the video and the sound output from the broadcast receiving section 2 into a recording signal so as to record the recording signal on a magnetic tape. ",
"The magnetic tape is driven by a magnetic tape driving section 4.",
"\nA video/sound reproducing section 5 converts the recording signal recorded on the magnetic tape, thereby reproducing the video and the sound. ",
"The video and the sound reproduced by the video/sound reproducing section 5 are supplied to a selective output section 6.",
"\nThe selective output section 6 selectively outputs one of the output from the broadcast receiving section 1 and the output from the video/sound reproducing section 5. ",
"The selection in the selective output section 6 is manually determined by a user.",
"\nA video display section 7 displays the video selected by the selective output section 6. ",
"A sound output section 8 outputs the sound selected by the selective output section 6.",
"\nHowever, in order to reproduce a program now being recorded, a conventional apparatus having the above-described configuration is required to suspend the recording operation once, rewind the magnetic tape and then start the reproducing operation. ",
"Therefore, such an apparatus has the following problems.",
"\n(1) During recording of a program which is now being broadcasted, it is impossible to reproduce the program from the beginning while continuing recording of the program.",
"\n(2) In the case where watching and listening of a program now being broadcasted must be suspended, it is impossible to reproduce the program from the point at which watching and listening of the program was suspended while continuing recording of the program.",
"\n(3) In the case where watching and listening of a program now being broadcasted must be suspended, it is impossible to fast-forward reproduce the program from the point at which watching and listening of the program was suspended while continuing recording of the program.",
"\nIn addition, it is impossible for a conventional apparatus to simultaneously record a plurality of programs on one and the same magnetic tape. ",
"Therefore, in order to simultaneously record a plurality of programs, it has been necessary to provide the same number of recording and reproducing apparatuses as the number of programs."
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"The medial breast margin sign: A new sign of pectus excavatum in adult females and comparison with classic signs.",
"\nPectus excavatum (PX) denotes a concavity of the sternum and is the most common chest wall deformity. ",
"Many characteristic signs have been described on frontal chest radiographs including increased density of the inferomedial right lung, blurring of the right heart border, steeply angled anterior ribs, horizontal posterior ribs, straight or convex left heart border and loss of silhouette of the descending thoracic aorta. ",
"These signs are often subtle and rarely all present in any individual. ",
"We describe a further sign in women with PX, the medial breast margin sign. ",
"This sign consists of more sharply defined and more vertically oriented medial breast borders on frontal chest radiographs than in those of women with normal chest wall morphology. ",
"This study aims to document the sensitivity, specificity and inter-observer reliability of the new sign, and compare with that of the previously described signs in patients with varying degrees of PX. ",
"The frontal chest radiographs of 185 women with PX and 50 women with normal chest wall morphology were assessed by two reviewers for the presence of the new sign and previously described signs. ",
"All the signs had their sensitivity, specificity and inter-observer reliability calculated. ",
"The new sign had a sensitivity of 0.48, a specificity of 0.96 and inter-observer reliability of κ = 0.79, ranking third, second and first among all other signs, respectively. ",
"The medial breast margin sign is useful in diagnosing PX and is comparable to the previously described classic signs."
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"Showdown on the Rio Alamar\n\n“The company [is] owned by the engineers and Eduardo Aranda and Armando Miranda, brothers of the Secretary of State for Social Development, Jorge Aranda. [",
"It] has in its [history] a series of works defects, among [them] the collapsed parking [lot] of the State Government in Mexicali, and a cracked bridge in Chaparral in Tijuana.”",
"\n\nMagdalena Cerda, the Environmental Health Coalition’s campaign director (and my driver during the site inspection in Tijuana), says, “They [the Ark of the Pacific] do work all over the state. ",
"There are many pending lawsuits, because they don’t deliver quality.”",
"\n\nIn 1983, president Ronald Reagan…\n\n…and Mexican president Miguel de la Madrid promised to cooperate in eliminating cross-border pollution. ",
"They worked out a progressive cleanup plan called the La Paz Agreement. ",
"Some of the results were impressive: in Tijuana, the agreement mobilized the removal of a veritable mountain of toxic lead left behind by a maquila car-battery recycler. ",
"In other areas of Mexico, air pollution was reduced and sewage treatment plants were constructed. ",
"But in spite of best intentions and the building of treatment plants in Tijuana, even more prodigious amounts of sewage continued to run downhill into the U.S.\n\nIn answer to the worsening conditions, the International Boundary Water Commission opened a sewage-treatment plant at San Ysidro in 1997. ",
"Steve Smullen is the area operations manager for the plant. ",
"He explains how it is that Tijuana sewage still infects the estuary and the Pacific. ",
"The commission’s plant, he says, treats TJ sewage only, but with this caveat: sometimes, during the dry season, water is diverted from the Tijuana River and sent to the treatment plant.",
"\n\nImage by Leticia Ayala\n\nNearly completed river channel\n\n“But in the winter, after a big storm, the diversion capacity is exceeded and the water flows right out into the Pacific Ocean. ",
"Part of this is urban runoff,” Smullen says, addressing the fact that there are still numerous homes in Tijuana without sewer hookups. “",
"At times, the water is fairly clean, and at other times, it is fairly dirty.” ",
"Still, he claims the situation has improved. “",
"Ten years ago it was bad. ",
"The Mexican government built more sewage-treatment facilities, and now, a lot of what’s running down the Tijuana River is secondary effluent.”",
"\n\nEffluent is liquid waste or sewage discharged into a river or the sea. ",
"Secondary effluent is the term for treated effluent. ",
"It is not drinkable. ",
"It requires additional treatment just to make it usable for non-potable applications, such as irrigation.",
"\n\nPaloma Aguirre agrees that the water coursing through the Tijuana River channel is indeed secondary effluent. ",
"Aguirre is the U.S. Mexico Border Coastal Program manager at WILDCOAST. ",
"She calls from her office in Imperial Beach. “",
"The problem is this,” she says. “",
"There’s no place to put all the treated water. ",
"And there is no way to redirect it, either,” meaning various gray-water applications, “because the plumbing infrastructure doesn’t exist.” ",
"The solution for now, she says, is that municipal water trucks carry the effluent off-site and dump it.",
"\n\n“But they don’t have enough trucking capacity, so the majority of the effluent flows into the Tijuana River and down the channel, and that’s what you’re seeing.” ",
"There’s more: “I’ve done [maquila] site inspections myself, and I’ve seen water coming out of the factories being piped right into the Alamar. ",
"What is in it is anybody’s guess. ",
"We do know that the river is being impacted by industrial contaminants, but we don’t know by what. ",
"We don’t really know what’s in the water.”",
"\n\nThis is a statement that I will hear over and over from the environmentalists in Mexico — that they don’t know what’s in the water, but they suspect that it’s bad.",
"\n\nAguirre says there are only eight government inspectors for the maquilas. (",
"Diane Takvorian will later confirm this number.) ",
"Aguirre has petitioned the Mexican water commission Conagua for water-quality information and for Rio Alamar project updates. “",
"They just say no.” ",
"She says there is only one way left to get answers. “",
"We will have to go to El Paso, where their headquarters are.”",
"\n\nRick Gersberg, PhD, is a professor at San Diego State. ",
"He teaches courses in water quality, risk assessment, and hazardous-waste management. ",
"In the late 1990s, he was involved in water-testing in Tijuana and along Tecate Creek. ",
"He still handles heavy-metals testing for Coastkeeper. “",
"If I ran a few tests on Rio Alamar and a few on the Tijuana River,” he says, “that could cost 10,000 dollars. ",
"That’s just the testing. ",
"Someone’s then gotta write the report.”",
"\n\nToxicity tests are even more expensive. “",
"That’s where you’re putting little critters into the water, like water fleas, and seeing how many of them die.”",
"\n\nIn Mexico, the high costs of such testing means it doesn’t get done.",
"\n\n“I have seen fumes coming from the ground…\n\n…at the Rio Alamar where toxic materials have been dumped,” says Oscar Romo, 64, the watershed coordinator of the Tijuana River National Estuary Research Reserve. ",
"He has been interviewed many times over the years about pollution and flooding and the dark side of Tijuana’s municipal sewers. ",
"He is aware of the original plans to restore habitat along the Alamar and to eco-channelize the creek. “",
"But that project was abandoned,” he says. “",
"The desire to control flooding, and to gain access to land to develop the highway, and to gain land to develop to make money for the city and the state replaced the old plan. ",
"Eco-channelization required a bigger footprint, and it cost more money.”",
"\n\nHe describes a politicized hotbed fraught with complexities. “",
"Different people were claiming ownership of the [Rio Alamar] land, and in many cases, they were bringing many loads of trash to fill in the riverbed. ",
"The Paleo-channel was being filled at a pretty fast rate by industry, and by developers.” ",
"This is a big business, he says, in particular for a notoriously radical Mexican movement known as antorcha campesina, or the “farmers’ torch.”",
"\n\n“This group is famous for what they’ve been doing. ",
"They take land from the government by invading. ",
"This is a huge business. ",
"This group is probably the most wealthy in the country. ",
"They claim they’re turning the land into viable real estate. ",
"They have been taking land in the Alamar River by that process, by dumping. ",
"And they were making money by charging other people to dump. ",
"If you go to the landfill, it costs what, 60 dollars per ton? ",
"At the Rio Alamar, they were charging only 5 dollars per ton.”",
"\n\nComments\n\nInteresting stuff. ",
"It's difficult to write about Tijuana issues without drawing direct comparisons to how things operate in the U.S., everything is totally different here. ",
"In both countries, the government will get its way, except that the path to that end is entirely different. ",
"This creek will go away, economics is more important than ecology, this is what happens. ",
"That would happen in the U.S. as well, just differently, with the same exact concerns and the same ignorance by the government.",
"\n\nBy the way, the vast majority of the Tijuana River was not paved in the 1960's. ",
"From the Rodriguez Dam to 5 Y 10 was all dirt when I came to live here in 1992 - in fact, most roads went through the River and their were no bridges East of 5 Y 10. ",
"During the big El Niño storms in 1993, the Dam overflowed and the floodgates were eventually opened for safety reasons and the flooded River killed countless people illegally camping/living in the way of the water. ",
"Directly afterwards, the government paved the river all of the way back to the Dam, many believe that bodies of victims still lie entombed underneath the concrete.",
"\n\nBut to summarize, all governments do what all governments do, which is whatever they want to do.",
"\n\nAbout six years ago, the president of our College of Architects was on the air to lobby for the remediation of the Tijuana River canalization project. ",
"In his professional opinion, he said, all that cement was engineering overkill but, now that it's been built, at least we might make its floor porous, to help our water table, and maybe cover over some of the bleakness with public architecture.",
"\n\nThere has been no remediation. ",
"Instead, we now use the canalization to house people kicked out of California prisons, to park police cars, and to shoot off city-sponsored fireworks.",
"\n\nWhat a shame it is to see that the Alamar has already been canalized in that same over-engineered way. ",
"Doesn't the International Boundary and Water Commission have anything to say in the matter?",
"\n\nI'm surprised there hasn't been more commentary on this very important story. ",
"What happens with the Alamar has a major impact on San Diego as well. ",
"With the Reader having taken a (somewhat specious) ribbing recently in San Diego CityBeat for its cover articles, it's all the more gratifying to see such a detailed and comprehensive cover story that tackles an important issue affecting all who live in southern SD and TJ --"
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"People say Mary sues are feminism at work.",
"\n\nBut what the hell is Feminist about a white girl with a Japanese name whom appropriates other cultures while speaking broken Japanese thinking she sounds cool and has a poorly researched mental illness attached to them?",
"\n\nAnd what is Feminist about White ocs meant to represent Greenland but Ignore the Inuits that are the Majority of the population and the reason Greenland’s culture exists?",
"\n\nNot all Mary sues and Gary stus are power Fantasies.",
"\n\nSome are actually offensive as hell"
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"College Hockey:\n\nEAST LANSING, Mich. — Heading into the third period tonight, anyone watching this game might have thought it was already over. ",
"No. ",
"11 Michigan State was leading 4-1 and had outshot Alabama-Huntsville 29-12 through two periods, and the Chargers haven’t exactly established themselves as a comeback team in this young season, having been outscored 25-9 in five previous losses.",
"\n\nWhen Justin Cseter scored to make it 4-2 at 16:12 in the third, however, things changed.",
"\n\nA shoving match at the goal sends Michigan State's Kevin Walrod into the air (photo: Erica Treais).",
"\n\nQuickly.",
"\n\nThe Chargers netted three goals in the final four minutes of regulation to force overtime, and came close to winning in OT with the man advantage. ",
"The game ended tied at four.",
"\n\nIt’s not surprising that no one in green and white was happy when all was over, but on the other side of the rink, UAH was just as displeased. ",
"Chargers head coach Chris Luongo said that the slow starts are a puzzle UAH is trying to solve and he made it clear that there’s no such thing as a moral victory.",
"\n\n“It’s not a two period game,” said Luongo outside of a Chargers locker room that was dead silent.",
"\n\n“Really, it’s not ‘we.’ ",
"I’m not necessarily pointing a finger, but our [locker] room needs to solve that. ",
"As a group, they need to solve it. ",
"When it comes down to it, they’re the guys that provide the emotion. ",
"Ultimately, it is up to them.”",
"\n\n“They only had two scoring chances the first two periods, and they scored on one of them,” said\n\nMSU head coach Rick Comley. ",
"In the final minutes, the Chargers went “from protecting to attacking,” said Comley. “",
"We were sloppy.”",
"\n\nZach Golembiewski opened the scoring for the Spartans in the first period at 9:39, a soft goal on an essentially empty net after the puck squeaked through the legs of UAH goaltender C.J. Groh, who was turned around and facing the net at the left post. ",
"All Golembiewski had to do was tap it home.",
"\n\nMSU led 2-1 after one on goals by Daultan Leveille and Jake Chelios that sandwiched Charger Keenan Desmet’s second marker of the season. ",
"Leveille’s goal was a wrap-around at 4:43.",
"\n\nDesmet answered when he converted a turnover in front of the MSU net at 10:26. ",
"Chelios’ goal at 19:03 looked like an innocent shot through traffic from the left point. ",
"Spartans 4, Chargers 1.",
"\n\n“I think a bunch of guys thought it was kind of in the bag, third period coming around and being up three goals,” said Leveille. “",
"I think we sat back a bit too much and got away from our systems.",
"\n\n“We were trying to play a bit too defensive and let them have too much ice and that caught up to us in the end.”",
"\n\nIn the end is right. ",
"Cseter started the third-period flurry for the Chargers, finishing Desmet’s hard work behind the goal. ",
"Desmet fought to keep the puck close to the MSU net and handed off to Cseter, who skated across the crease and lifted the puck up and over Drew Palmisano on the glove side.",
"\n\nJust over a minute later at 17:30, Neil Ruffini made it 4-3 when he scored on the fly, shooting past a Spartan defender in the slot to beat Palmisano clean and high.",
"\n\nMichigan State sophomore Zach Golembiewski celebrates after scoring his first goal of the season (photo: Erica Treais).",
"\n\nThen at 19:56 and with a two-man advantage — an empty net plus a power play with Leveille in the box for slashing — Joey Koudys hit the Spartan post, the puck hit Palmisano’s back and bounced into the MSU net, and the game was tied 4-4.",
"\n\nLuongo said that the way the Chargers ended the game came from “making a decision to go 100 percent.”",
"\n\n“It’s difficult to maintain a pace like that for 60 minutes,” said Luongo. “",
"We have to find a way to do it for more than 40.”",
"\n\n“We played okay for 50 minutes,” said Comley. “",
"Weakest game of the year, for sure.”",
"\n\nThe following is a self-policing forum for discussing views on this story. ",
"Comments that are derogatory, make personal attacks, are abusive, or contain profanity or racism will be removed at our discretion. ",
"USCHO.com is not responsible for comments posted by users. ",
"Please report any inappropriate or offensive comments by clicking the “Flag” link next to that comment in order to alert the moderator.",
"\n\nPlease also keep “woofing,” taunting, and otherwise unsportsmanlike behavior to a minimum. ",
"Your posts will more than likely be deleted, and worse yet, you reflect badly on yourself, your favorite team and your conference."
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"A Music Review Collective\n\nFirst Listen Review: Echodrone – “Five”\n\nWhat I Like: This band feels like something between a Mood Rings and Slowdive. ",
"A lot of the shoegaze revival the past few years has incorporated the indie/punk speed that came in the music era directly proceeding this one in 2004-2010. ",
"This album goes back to the roots of the genre with a slower and louder sound. ",
"Often this, especially with the vocals, sounds a little like earlier Smashing Pumpkins. ",
"The sound is extraordinary. ",
"This is definitely on “When Two Ends Meet” when a quiet interlude is suddenly interrupted by a loud and powerful storm of sound. ",
"I also admire this band for taking the genre to new levels with a little addition of electronic keyboards. ",
"They aren’t too much, but certainly create a mood reminding me of something desolate like the planet Mustafar. ",
"I also like how these songs are mostly over 5 minutes, really allowing the atmosphere of the guitars to grab the listener.",
"\n\nWhat Could Be Improved: Not too much at the moment! ",
"They have mastered a sound and peaked it."
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"Q:\n\nComo exibir 2 arrays em um foreach?",
"\n\n$array1 = ['Hello', 'World'];\n$array2 = ['Hello Girl', 'Hello Boy'];\n\nforeach($array1 as $text) \n{ \n echo $text;]\n}\n\nA:\n\n<?",
"php\n\n$array = ['Hello', 'World'];\n$array2 = ['Hello Girl', 'Hello Boy'];\n\nforeach($array as $indice => $text) \n{ \n echo $text;\n echo $array2[$indice];\n}\n\n?",
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"Q:\n\nDisplay somthing different if condition is met I using PHP/MySQL\n\nI have an over time sheet that gets printed when there is over time from an employee. ",
"The overtime format goes like \"B.Eng.\" ",
"And then the name of the employee. ",
"Now I need it to check the name of the employee (or id) to print either \"B.Eng.\" ",
"Or \"MR.\", ",
"This because there is an employee (just one) that does not have a degree. ",
"I would think the answer would be an IF condition.",
"\nHere is my code:\n $db = mysql_select_db (\"over_time\");\n $strqry = \"SELECT emp_name FROM contr_acces where id_emp='\".$vp_idemp.\"';\"; ",
"\n $qry2 = mysql_query ($strqry);\n $row2 = mysql_fetch_object ($qry2);\n $vl_emp_name= \"B.Eng. \"",
".$row2->emp_name;\n print $vl_emp_name;\n\nA:\n\nyou can do something like this\n $db = mysql_select_db (\"over_time\");\n $strqry = \"SELECT emp_name FROM contr_acces where id_emp='\".$vp_idemp.\"';\"; ",
"\n $qry2 = mysql_query ($strqry);\n $row2 = mysql_fetch_object ($qry2);\n $vl_emp_name= $row2->emp_name;\n if($vl_emp_name == 'name_without_the_degree){\n $vl_emp_name= \"Mr. \".$vl_emp_name;\n }else{\n $vl_emp_name= \"B.Eng \".$vl_emp_name;\n }\n print $vl_emp_name;\n\nIt is not the best solution since you are hardcoding the condition but without knowing more about the db structure is not possible to give you a better solution. ",
"The most efficient one would be to add a field to the db with the degree type for the users and retrieve it together with the name.",
"\nSee my comment under your question for the api used and the sql injection risk that this soultion doesn't address\n\n"
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"\n\nParent company CMA CGM, which bought APL four years ago, will shortly stop using the APL brand on trades covering the Indian subcontinent, Middle East and Latin America, according to Alphaliner.",
"\n\nThe move follows the earlier exit of the APL brand from the Oceania trade in January this year and from the Asia-Europe and transatlantic trades in October last year. ",
"APL’s intra-Asia short sea services have been operated via another CMA CGM acquisition, CNC, since December 2018.",
"\n\nA spokesperson for CMA CGM declined to comment when contacted by Alphaliner as to whether the APL name might also disappear on the transpacific too soon.",
"\n\nSingapore-based APL, formerly called American President Lines, features a distinctive eagle in its logo. ",
"It has had many incarnations during its 172-year history, starting out as the Pacific Mail Steamship Company."
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"This invention is generally concerned with envelope flap moistening apparatus and more particularly with a mailing machine through which envelopes may be fed at different rates and including envelope flap moistening apparatus including a flap moistening brush overlaying a pad of wicking material mounted in a fluid supply well which is constructed and arranged to facilitate the appropriate transfer of fluid to and from the well over a wide range of envelope feeding rates.",
"\nIn U.S. Pat. ",
"No. ",
"4,643,123 for Envelope Flap Moistening Apparatus, issued Feb. 17, 1987 to Auerbach and assigned to the assignee of the present invention, there is disclosed an envelope feeding machine with envelope flap moistening structure including a source of supply of fluid, having a pad of wicking material mounted therein, and including a brush disposed in overlaying relationship with the supply, and thus with the pad, for transferring fluid from the supply via the pad to the brush bristles, and, when an envelope flap is fed beneath the brush bristles, for transferring fluid from the brush bristles to the envelope flap for moistening the same.",
"\nIn U.S. Pat. ",
"No. ",
"3,905,325 for Envelope Flap Moistening Apparatus, issued Sep. 16, 1975 to Labore, et. ",
"al, there is disclosed envelope flap moistening structure including a moistening blade having a fluid supply cavity formed in its upper surface, for receiving fluid from a remotely located source of supply of fluid, and having a slot extending downwardly from the fluid supply cavity to the lower surface thereof for transferring fluid from the cavity via the slot to an envelope flap when the flap is fed into engagement with the lower surface of the blade and thus across the slot, to moisten the gummed surface on the flap.",
"\nFurther, on pages 110, 448-451, 454 and 455 of the International Mailing Machines (IMS)/ Hasler Technical Manual, dated January 1987, there is disclosed envelope flap moistening structure including a source of supply of moistening fluid having a well and a pad of wicking material mounted therein, wherein the pad is held in place in the well by a pad retaining member having a screen-shaped upper wall and having depending side walls connectable to the structure forming the well, and wherein the brush is removably connected to the moistening structure.",
"\nNotwithstanding the aforesaid disclosures, there has been a long felt and as yet unsatisfied need to provide envelope flap moistening structure for use in envelope feeding apparatus, which is operable at variable speeds, for moistening envelope flaps at different envelope throughput rates, wherein the flap moistening structure is constructed and arranged for appropriately, but not excessively, wetting the envelope flaps to substantially the same extent independently of the throughput rate of the envelopes. ",
"Accordingly:\nan object of the invention is to provide an improved mailing machine including means for feeding an envelope, having body and flap portions, in a path of travel, and means for supplying moisture to the flap portions;\nanother object of the invention is to provide improved apparatus for supplying moisture to an envelope; and\nyet another object is to provide a mailing machine comprising envelope flap moistening apparatus including a fluid supply well, having fluid inlet and outlet channels and which is constructed and arranged for removable connection to the mailing machine."
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"\nSubstance use disorder is one of the important mental health problems related to suicide, nearly equal with depressive disorder. ",
"However, it is unclear how differences of abused substances influence the suicide risk of individuals with substance use disorder in Japan. ",
"The purpose of the present study is to compare an estimated suicide risk among patients with alcohol, amphetamine, and sedative, hypnotic or anxiolytic use disorders. ",
"Subjects were 1082 outpatients with alcohol use disorder (AUD), 191 with amphetamine-like use disorder (AMUD), and 63 with sedative, hypnotic or anxiolytic use disorder (SUD), all of whom consecutively had consulted seven medical facilities specialized for treatment of substance use disorder during a month of December 2009. ",
"A self-reporting questionnaire including the items of Kessler 10 (K10) and the Mini International Neuropsychiatric Interview (M.I.N.I.) was administered, and scores of the K10 and M.I.N.I., and the other clinical information were compared among the patients with AUD, AMUD, and SUD. ",
"Patients with AMUD and SUD showed significantly higher score on the M.I.N.I. than those with AUD, while those with SUD showed highest score on the K10, followed by those with AMUD. ",
"In either of patients with AUD, AMUD, or SUD, approximately 60% of the subjects who attempted suicide within a month reported to take any psychoactive substance in the attempted suicide. ",
"Additionally, a remarkable difference was found in histories of general psychiatric treatment before consulting medical facilities specialized for treatment of substance use disorder between these three groups. ",
"Most of outpatients with SUD reported such histories. ",
"Outpatients with SUD appeared to involve the highest risk for suicide of all patients with substance use disorder, probably because of the comorbid-severe depression. ",
"In our speculation, SUD which our subjects suffered from might be induced or accelerated by pharmacotherapy performed in general psychiatric facilities. ",
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"Air France has cancelled two flights between Paris and Sofia scheduled for Sunday, 12 June, due to strike of pilots at the French flag carrier.",
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"\n\nAir France also cancelled its flights between Paris and Bulgaria's capital on Saturday as part of a 30% cut in the number of destinations for the day.",
"\n\nSeveral French pilot unions have called for strike action to protest government plans to reform the labour market. ",
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"\nThe genome of Escherichia coli K12 contains 9 open reading frames encoding aldo/keto reductases (AKRs) that are differentially regulated and sequence diverse. ",
"A significant amount of data is available for the E. coli AKRs through the availability of gene knockouts and gene expression studies, which adds to the biochemical and kinetic data. ",
"This together with the availability of crystal structures for nearly half of the E. coli AKRs and homologues of several others provides an opportunity to look at the diversity of these representative bacterial AKRs. ",
"Based around the common AKR fold of (β/α)8 barrel with two additional α-helices, the E. coli AKRs have a loop structure that is more diverse than their mammalian counterparts, creating a variety of active site architectures. ",
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"Lyon have won back-to-back Women's Champions League titles - and four in total since 2011\n\nUefa Women's Euro 2017 When: 16 July - 6 August Where: The Netherlands BBC Coverage: Live commentaries on BBC Radio 5 Live, Scotland games on Radio Scotland and online, plus live text coverage on the BBC Sport website\n\nIt is only four years since the previous Women's Euros ended with Germany as champions. ",
"Yet between then and now, and with Euro 2017 about to start, the European women's game has changed to the point that it is scarcely recognisable.",
"\n\nRemember Swedish side Tyreso, German club Turbine Potsdam or French outfit Juvisy?",
"\n\nAll those teams reached European semi-finals within 12 months of Euro 2013 - a tournament that came in a year when Manchester City did not have a Women's Super League side and a crowd of less than 5,000 watched the Women's FA Cup final at Doncaster's Keepmoat Stadium.",
"\n\nFast forward to 2017 and Barcelona, Paris Saint-Germain and Manchester City were all in the last four of the Women's Champions League, arriving fashionably late to a party that is just getting started.",
"\n\nAnd with the well-financed emergence of those clubs, the European women's game, dominated for so long by Scandinavian and German trailblazers, now feels closer to mirroring the men's game - on a branding level - than ever before.",
"\n\n\"When Chelsea, Manchester City, Barcelona, PSG and Lyon took the decision to put effort into the women's game, that was a huge decision for women's sport,\" Iceland Women head coach Freyr Alexandersson told BBC Sport.",
"\n\n\"Just marketing-wise, it's very, very important, but also sports-wise, bringing their facilities and knowledge to the women's game is just vital. ",
"The game has grown hugely in the past five years.",
"\n\n\"In the Women's Champions League, and also in the biggest leagues - in France, Germany and England - the best teams are getting better every year.\"",
"\n\nSo, as some of the supposed bigger names are starting to take women's football seriously, how far has the game really progressed on the continent, and just how healthy is the state of women's football in some of the nations competing in Women's Euro 2017?",
"\n\nItaly - The Old Lady emerges as newest team\n\nItaly held England to a 1-1 draw in an international friendly at Vale Park in April\n\nOf the 12 teams in the women's Serie A in 2016-17, just three were run on a professional budget: Verona, Brescia and new champions Fiorentina.",
"\n\nHowever, men's Champions League finalists Juventus announced their intention to launch a women's team in June, and they will land straight in the top tier.",
"\n\nOn 1 July, the Turin club revealed they had bought external-link the rights to existing top-flight club Cuneo Calcio Femminile, meaning they can compete in Serie A in 2017-18.",
"\n\nPreviously, Fiorentina - who lifted their first title in May after adopting ACF Firenze in 2015 - were the only top-flight side to be formerly affiliated with a men's Serie A namesake, with the rest relying on third-party sponsorship.",
"\n\nItaly's top two sides qualify for the Women's Champions League each year, although the country has never produced a finalist, with only Verona reaching the semi-finals, in 2007-08.",
"\n\nSpain - Barca in talks over unprecedented US move\n\nBarca a perfect move for me - Duggan\n\nBarcelona became Spain's first women's side to reach the last four in Europe this term.",
"\n\nGreater domestic competition in recent years has prompted the Catalan club to bolster their investment significantly since 2016, re-signing Spain midfielder Vicky Losada from Arsenal in November.",
"\n\nEarlier this month, they grabbed headlines across the continent with the signing of England striker Toni Duggan from Manchester City, making a clear statement of their intent to win European silverware.",
"\n\nReal Madrid have also revealed their intention to get a women's team started.",
"\n\nBut whether that will lead to a female El Clasico remains unclear on both sides, as Barcelona are in talks with the American National Women's Soccer League about entering a team for 2018.",
"\n\nHaving opened a permanent office in New York in September, Barca want a team in the US to expand their women's football pedigree.",
"\n\nFrance - crowds soar at dominant big two\n\nA crowd of 19,214 saw Lyon host Manchester City in their Women's Champions League semi-final, while 19,192 were at Parc des Princes (right) for Paris St-Germain v Barcelona\n\nAnother rapidly-changing aspect of the women's game in Europe is attendances, with Lyon and Paris Saint-Germain leading the way in France, playing games at the regular homes of their men's counterparts - in contrast to the non-league men's grounds commonly used in England.",
"\n\nNevertheless, the American NWSL - with average attendances over 5,000 in 2016 - remains far better-supported overall than any league on this side of the Atlantic.",
"\n\nWith a reported annual budget external-link of eight million euros, Lyon are the best-funded women's team in Europe and rewarded their financial backers with their fourth Champions League title in Cardiff in June.",
"\n\nLyon and France captain Wendie Renard said it was \"no coincidence\" that two French sides made that final, with the investment being made. ",
"However, questions remain about the resulting competitiveness of the French top flight overall, as Lyon have claimed 11 consecutive domestic titles and they won an ominous seven league matches by six or more goals in 2016-17.",
"\n\nGermany - Bayern and Wolfsburg enjoying Bundesliga reign\n\nNearly 70% of games in the 2016-17 Frauen Bundesliga were either drawn or settled by a one-goal margin, compared to just 50% in the English WSL 1 Spring Series\n\nThe Frauen Bundesliga, previously dominated by Turbine Potsdam, FFC Frankfurt and FCR 2001 Duisburg, has seen Bayern Munich and Wolfsburg occupy the top two spots since 2014-15.",
"\n\nCompared to other European countries, German sides have long been backed with greater finances and, of the past 16 women's European champions, nine have been German clubs.",
"\n\nHowever, the emergence of Lyon in recent years has seen the Germans challenged from outside Scandinavia for the first time - with the exception of Arsenal Ladies' 2007 quadruple-winning side - and the power-shift has attracted players across the Franco-German border.",
"\n\nGermany captain Dzsenifer Marozsan now plays her club football for Lyon after leaving Frankfurt in 2016, while Norway striker Ada Hegerberg - winner of the Uefa Best Women's Player in Europe Award for 2016 and the BBC Footballer of the Year for 2017 - left Turbine Potsdam for the French champions in 2014.",
"\n\nNetherlands - Part-time clubs the norm for Euro 2017 hosts\n\nFC Twente's stadium in Enschede, with a capacity of 30,000, will host the Euro 2017 final on 6 August\n\nTickets have sold out for all of the Netherlands' group-stage games at the Euros, as the country hosts the tournament for the first time.",
"\n\nBut Dutch top-flight sides do not yet train on a completely full-time basis, prompting some of the nation's best players to move abroad, including star striker Vivianne Miedema, who now plays for Arsenal.",
"\n\n\"In Holland, our league is OK, but the English league - that's the next level for our players,\" prominent Dutch women's football agent Leoni Blokhuis told BBC Sport.",
"\n\nTwo of Blokhuis' clients - Siri Worm and Marthe Munsterman - recently joined promoted WSL 1 side Everton Ladies from Dutch side FC Twente.",
"\n\n\"In Holland, players train in an amateur complex,\" Blokhuis added. \"",
"But when they're at Everton, they are training at the same complex as the men. ",
"That's amazing for the girls.",
"\n\n\"They can be fully professional in England, arriving at the club in the morning, eating breakfast and lunch at the training ground. ",
"But in Holland, the salary is not enough for you to live off it alone.\"",
"\n\nSo when might the Women's Eredivisie reach a stage when clubs can train on a full-time basis?",
"\n\n\"That's quite a long way off,\" Blokhuis replied. \"",
"The support level from the men in the Dutch clubs isn't always there. ",
"It's not important for the club. ",
"They don't give the girls priority.",
"\n\n\"But this is the quickest-growing sport in Holland. ",
"There are more than 153,000 girls and women playing football in the country. ",
"When the clubs' boards are more enthusiastic, that's the starting point.\"",
"\n\nEngland - WSL 'is the league of the future'\n\nIn 2016, Manchester City became Britain's first women's club to average crowds of more than 2,000 across a season\n\nThe Women's Super League is attracting more overseas stars than ever before - including the Fifa World Player of the Year Carli Lloyd - as it emerges as a destination of choice for world-class players.",
"\n\n\"Germany is still the best league because all the clubs can beat each other, but the league in England is the league of the future,\" Blokhuis continued.",
"\n\n\"The Football Association is paying the English international players so, with a club salary as well, some players can earn a lot of money overall.\"",
"\n\nHowever, only seven WSL sides can claim to have absolutely all of their players training on a full-time basis, and some clubs have struggling financially, most notably with top-flight side Notts County folding on the eve of the 2017 Spring Series.",
"\n\nIn terms of television deals, wage packets and sponsorship, the women's game remains men's football poorer relative. ",
"Yet in branding terms - with Chelsea, Manchester City, Arsenal and Liverpool the top four in the WSL 1 Spring Series - the likeness is now undeniable after rapid changes over the past four-year international tournament cycle."
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[
"/*\n * Copyright 2017-2019 B2i Healthcare Pte Ltd, http://b2i.sg\n * \n * Licensed under the Apache License, Version 2.0 (the \"License\");\n * you may not use this file except in compliance with the License.",
"\n * You may obtain a copy of the License at\n *\n * http://www.apache.org/licenses/LICENSE-2.0\n *\n * Unless required by applicable law or agreed to in writing, software\n * distributed under the License is distributed on an \"AS IS\" BASIS,\n * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.",
"\n * See the License for the specific language governing permissions and\n * limitations under the License.",
"\n */\npackage com.b2international.snowowl.snomed.cis.request;\n\nimport com.b2international.snowowl.core.",
"ServiceProvider;\nimport com.b2international.snowowl.core.events.",
"Request;\nimport com.b2international.snowowl.snomed.cis.domain.",
"SctIds;\n\n/**\n * @since 5.5\n */\npublic final class SnomedIdentifierRegisterRequestBuilder extends AbstractSnomedIdentifierEnumeratedRequestBuilder<SnomedIdentifierRegisterRequestBuilder> {\n\n\t@Override\n\tprotected Request<ServiceProvider, SctIds> doBuild() {\n\t\treturn new SnomedIdentifierRegisterRequest(componentIds);\n\t}\n\n}\n"
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"(734) 561-4176 - Lock Rekeying Lincoln Park\n\nIn 24/7 Wyandotte Locksmith we are always here to assist you in any difficult situation. ",
"We are open all day, every day. ",
"Our trained team of professionals specializes in ignition keys, lock picking services, lockout assistance, safe opening, we also install alarm systems. ",
"What sets us apart from the competition is not only our excellent customer service but also our extremely affordable prices. ",
"We beat our competitors’ prices by 20%. ",
"Call us today to talk to one of our specialist and learn more about the services we offer."
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"Q:\n\nZoom delphi TPanel contents (changing DPI)\n\nI created a Report Designer years ago.",
"\nAll components (labels, expressions, etc) of the report structure are represented by TLabels. ",
"\nBands of the report are represented by TPanels....\nThe component designer currently uses default pixels per inch (96). ",
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"\nMy question is, how can I do zoom/scale on the contents of an TPanel? ",
"Is this possible?",
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"This R01 application aims to strengthen capacity for locally-relevant research and policy in the area of movie smoking. ",
"The National Cancer Institute concluded that movie smoking is one cause of youth smoking in 2008. ",
"Since then, directives aimed at reducing youth exposure to movie smoking have been promulgated as part of the implementation guidelines for Article 13 of the World Health Organization's Framework Convention on Tobacco Control. ",
"The proposal pairs researchers who developed the science to demonstrate a causal link between movies and smoking with key scientists in Argentina and Mexico who have begun studying the same link in their respective countries, with the overarching aim of enhancing research on movie and marketing risk factors for youth smoking. ",
"Notably, we involve another researcher from Germany who has successfully completed a similar study in 6 European Union (EU) countries. ",
"In partnership with our Latin American colleagues, we will adapt the EU protocols for content coding domestic movies and surveying young adolescents to test a novel hypothesis-that smoking by Latino characters in domestic films is more potent than in foreign produced films (in which characters are mostly non-Latino). ",
"Additionally, the team will assess the prevalence of adolescent ownership of tobacco branded merchandise (e.g., t-shirts and collectable packs) to determine if receptivity to these products acts in synergy with movie exposures. ",
"The aims of the study are to: 1) develop the capacity for researchers to track tobacco use and brand placement in the domestic film market (which accounts for about 10% of movies viewed in each country); 2) to develop the capacity for researchers to conduct longitudinal school-based surveys of adolescents to determine the importance of media and marketing risk factors in promoting smoking; and 3) to identify leverage points for policy development and for research dissemination efforts that aim to reduce media and marketing exposures. ",
"Aim 2 involves fielding a two-wave school-based survey of young adolescents (12 or 13 at baseline) in both countries, in which exposure is assessed to popular contemporary movies released in the past 10 years. ",
"The movie sample frame will be enriched with domestic movies, in order to test the domestic vs. foreign movie influence hypothesis. ",
"We will also assess the prevalence of ownership of tobacco branded merchandise, including the types of articles, the brand, and where it is displayed. ",
"We will determine whether the branded merchandise is being worn in the school setting and how often such articles are seen by others. ",
"Finally, we will conduct a diagnosis of the structure, financing and policy-making processes of the film industry in each country and analyze the results with a consultant who was instrumental in developing the Smoke Free Movies campaign, a successful campaign to reduce youth exposure to movies in the U.S. In combination with country-specific research results, these efforts aim to develop locally-relevant approaches for reducing youth exposure to tobacco content in films. ",
"Thus, this proposal addresses very significant risk factors for youth smoking in these countries and offers unique and innovative approaches to assessing such exposures, as well as the ability to test novel hypotheses about how race and ethnicity of movie characters and branded merchandise might modify the strength of the association between exposure to movie smoking and youth smoking."
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"Pgrep\n\npgrep is a command-line utility initially written for use with the Solaris 7 operating system by Mike Shapiro. ",
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"Alternatives include pidof (finds process ID given a program name) and ps.",
"\n\nExample usage\nThe default behaviour of pgrep (returning the process identifier of the named tasks) simplifies an otherwise complex task and is invoked with:\n$ pgrep 'bash'\n\nWhich is roughly equivalent to:\n$ ps ax | awk '{sub(/.*\\//, \"\", $5)} $5 ~ /bash/ {print $1}'\n\nAdditional functionality of pgrep is listing the process name as well as the PID (-l Lists the process name as well as the process ID) of all processes belonging to the group alice (-G Only match processes whose real group ID is listed. ",
"Either the numerical or symbolical value may be used):\n$ pgrep -l -G alice\n\nshowing all processes that do not belong to the user root (-u euid Only match processes whose effective user ID is listed. ",
"Either the numerical or symbolical value may be used) by inverting the matching (-v Negates the matching):\n$ pgrep -v -u root\n\nand only matching the most recently started process (-n Select only the newest (most recently started) of the matching processes):\n$ pgrep -n # The most recent process started\n$ pgrep -n -u alice emacs # The most recent `emacs` process started by user `alice`\n\nSee also\n\n List of Unix commands\n pidof — find the process ID of running programs \n pkill — signal processes based on name and other attributes\n ps — display the currently running processes\n grep — search for lines of text that match one or many regular expressions\n\nReferences\n \n \n\nCategory:Unix process- and task-management-related software"
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"2017 Dutchmen 3805\n\nNew Fifth Wheel\n\nLet us help you\nwith your search!",
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"The present invention relates to an electrical circuit arrangement for the protection of presses, in which the current input of the driving motor is proportional or nearly proportional to the torque or moment of load, especially for the protection of high-speed presses with a power control stage, a current-regulating member, a rotational speed regulating member, a device for the continuous detection of a magnitude proportional to the actual value of the torque or moment of load and a device for the continuous detection of a magnitude proportional to the actual value of the rotational speed as well as pre-setting device for presetting or predetermining the magnitude corresponding to the desired rotational speed value.",
"\nThe current carried by the driving motor during the starting phase of the press is larger than the current carried during normal operation by a multiple, owing to the need of overcoming inertia moments of the load coupled to the motor, such as flywheel, slide carriages and the like and static friction in the bearings and slide guidances. ",
"With rapidly operating presses, for example, automated punching or stamping machines for sheet metal parts of electric motors, the predominant proportion of the output produced by the driving motor is absorbed by the friction in bearing- and guide-elements so that a nearly constant power output of the driving motor and consequently also a nearly constant power or current input will establish itself during the operation. ",
"During the operation, the press or at least parts thereof may be destroyed, for example, by bearing damages or by movement in collision position, which may lead to an increased current carried by the motor, over and above the operating current.",
"\nA motor protection circuit is disclosed in the DE-OS No. ",
"26 24 784 with a circuit for determining the motor current input, with a circuit for turning off the motor in case of an excessive current input, and with a circuit for bypassing the starting phase of the motor. ",
"The bypass circuit includes a timing element consisting of an RC-member which is connected to the output voltage of the motor current input circuit by way of a transistor or Zener diode circuit and which keeps the voltage produced at the output of this circuit to a low level. ",
"After a lapse of time--discharge of the condenser of the timing element--the detection of a current which goes beyond the operating current, leads to a disconnection of the motor current circuit.",
"\nAn installation for monitoring the motor current input which relates to the press operation and the protection of the motor, is described in the DE-OS No. ",
"20 13 272, with an actual current value transmitter, a number of desired current value transmitters adjusted to certain angular ranges of the press and a comparator device for producing a warning or disconnecting signal always when the actual current value exceeds the desired current value of predetermined magnitude for the angular range of rotation. ",
"The selection of the respectively responding desired value transmitter and the time interval of the comparison takes place by way of a pulse producer. ",
"The circuit configuration of the DE-OS does not provide a disconnect bypass in the starting phase of the press motor.",
"\nThe present invention starts with the circuit disclosed in the DE-OS No. ",
"15 63 709. ",
"This publication discloses a rotational speed regulating installation for direct current shunt-wound motors with a power stage in the armature circuit and a power stage in the field regulating circuit. ",
"A tachometer device is coupled to the motor shaft to form a voltage proportional to the rotational speed of the motor. ",
"The voltage value is compared in a comparator element with the desired rotational speed value. ",
"The comparison result is present as guide magnitude at the input of a speed-regulating device. ",
"The current flowing in the armature circuit produces, by way of a current converter, an actual current value which is fed to a further comparator connected in the input of an armature current-regulating device operating on the power stage, which receives its second reference magnitude from the rotational speed regulating device.",
"\nThe present invention is concerned with the task of further developing such a prior art circuit arrangement for disconnect purposes in presses in case of current input of the press driving motor going beyond the operating current input in order to limit the destructions occurring in the case of damage to the machine part bringing about the increased current input. ",
"For a continuous comparison between the actual value of the torque or moment of load, respectively, a magnitude (voltage) proportional thereto, with a preset value, the preset value is formed after reaching the desired rotational speed of the motor and is stored in a memory.",
"\nThe underlying problems are solved according to the present invention in that the magnitudes for the desired rotational speed value and the actual rotational speed value are connected to a comparator stage in the input of the rotational speed regulating device and additionally are connected to a second comparator stage for producing a signal when the motor reaches the preset or predetermined desired rotational speed value. ",
"The magnitude for the actual value of the torque or moment of load is connected to a third comparator stage in the input of the current-regulating element and additionally to the input of a memory element and to a first input of a fourth comparator stage. ",
"The output of the memory element is connected to a further input of the fourth comparator stage for producing a signal in the output of the fourth comparator stage when the magnitude for the actual value of the torque or moment of load exceeds the memory content of the memory element. ",
"The output of the second comparator stage is connected to an enabling input of this memory element for enabling the memory element to store the actual value of the torque or moment of load for a continuous access to the memory content.",
"\nSuch a circuit configuration is realizable in an advantageous manner by analog, digital or mixed circuits. ",
"The circuit arrangement of the present invention is not limited to monitoring direct current shunt-wound motors. ",
"Instead, the circuit arrangement of the present invention is applicable to all motors in which the motor current is at least nearly proportional to the torque or moment of load. ",
"The circuit arrangement according to the present invention is applicable in an advantageous manner to motors with high load changes, especially in high-speed presses. ",
"Further significant advantages result from the fact that the fading out of the starting operation of the motor together with the drive system, characterized by considerable mass which is coupled thereto, does not take place as a function of time after the engagement to the preselected operating rotational speed--the desired operating values are not always attained thereby--but instead on the basis of a pure rotational speed interrogation for the operating rotational speed.",
"\nFor a proper operation with load changes of small number, the circuit arrangement may be constructed according to the present invention for attaining a necessary spacing between load current as a consequence of load peaks and of a disturbance magnitude to be controlled by connecting a current limiter in the input of the memory element and/or by connecting a current smoothing stage in the common line section to the input of the memory element and to the first input of the further comparator stage so that in an advantageous manner load peaks can also be detected at the instant of the formation of the desired current value, respectively, of a magnitude proportional thereto.",
"\nThese and other objects, features and advantages of the present invention will become more apparent from the following description when taken in connection with the accompanying drawing which shows, for purposes of illustration only, one embodiment in accordance with the present invention, and wherein:"
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"Alys Clare\n\nAlys Clare is a pseudonym of Elizabeth Harris, author of English historical whodunnit novels primarily set in medieval times, featuring the characters of Abbess Helewise and Josse d'Acquin. ",
"Brought up in the countryside close to where the Hawkenlye Novels are set, she went to school in Tonbridge and later studied English at the University of Keele. ",
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"\n Fortune Like the Moon (1999)\n Ashes of the Elements (2000)\n The Tavern in the Morning (2000)\n The Chatter of the Maidens (2001)\n The Faithful Dead (2002)\n A Dark Night Hidden (2003)\n Whiter Than the Lily (2004)\n Girl in a Red Tunic (2005)\n Heart of Ice (2006)\n The Enchanter's Forest (2007)\n The Paths of the Air (2008)\n The Joys of My Life (2009)\n The Rose of the World (2011)\n The Song of the Nightingale (2012)\n The Winter King (2013)\n A Shadowed Evil (2015)\n The Devil's Cup (2017)\n\nAelf Fen mysteries\nSet in 11th-century England and featuring the healer Lassair, published by Severn House.",
"\n Out of the Dawn Light (2009)\n Mist Over the Water (2009)\n Music of the Distant Stars (2010)\n The Way Between the Worlds (2011)\n Land of the Silver Dragon (2013)\n Blood of the South (2014)\n The Night Wanderer (2016)\n The Rufus Spy (2017)\n\nGabriel Taverner mysteries \nSet in Stuart England featuring physician Dr Gabriel Taverner\n A Rustle of Silk (2016)\n An Angel in the Glass (2018)\n\nWorld's End Mysteries \nSet in Victorian England\n The Woman Who Spoke to Spirits (2019)\n\nOther novels\n The Herb Gatherers (1991) (writing as Elizabeth Harris)\n The Egyptian Years (1992) (writing as Elizabeth Harris)\n The Sun Worshippers (1993) (writing as Elizabeth Harris)\n Time of the Wolf (1994) (writing as Elizabeth Harris)\n The Quiet Earth (1995) (writing as Elizabeth Harris)\n The Sacrifice Stone (1996) (writing as Elizabeth Harris)\n The Twilight Child (1997) (writing as Elizabeth Harris)\n\nReferences\n\nCategory:1944 births\nCategory:Living people\nCategory:English historical novelists\nCategory:Writers of historical fiction set in the Middle Ages\nCategory:Writers of historical mysteries\nCategory:English women novelists\nCategory:Alumni of Keele University\nCategory:Women mystery writers\nCategory:Women historical novelists\nCategory:Pseudonymous women writers\nCategory:Pseudonymous writers"
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[
"1. ",
"Introduction {#sec1-sensors-19-04069}\n===============\n\nHeavy-duty machine tools are characterized by the ability to process large and heavy components. ",
"This working characteristic requires that the machine must have high safety, stability and reliability. ",
"Working load and the inertia of the workpiece are very big under this working condition; therefore, once the machine tool component failure causes the sudden stop of the machine, this damages the rotary parts in the spindle box of machine tool \\[[@B1-sensors-19-04069]\\]. ",
"Thus, there has been widespread concern regarding a guarantee of the normal working state of machine tools and the early diagnosis of faults. ",
"Faults in heavy duty machine tool components usually occur in rotating parts because, under this kind of heavy load condition, the tiny flaw of the rotating part will expand rapidly. ",
"This can cause parts to fail and even have a major impact on other parts. ",
"Therefore, early fault diagnosis and determination is of great significance for the use and maintenance of heavy duty machine tools \\[[@B2-sensors-19-04069]\\].",
"\n\nMethods used for gear box analysis include spectral analysis, cepstrum analysis, demodulation analysis, statistical analysis, and envelope analysis. ",
"The appearance of fast Fourier transform (FFT) \\[[@B3-sensors-19-04069]\\] brings the fault diagnosis method based on Fourier analysis to a new stage. ",
"However, people gradually find that the traditional FFT method cannot effectively display the transient signal of fault characteristics. ",
"Although envelope spectrum analysis can show fault characteristics, it cannot effectively process unsteady state signals \\[[@B4-sensors-19-04069]\\]. ",
"Some methods have been developed and applied to deal with such unsteady problems. ",
"One of them is the short-time Fourier transform (STFT) \\[[@B5-sensors-19-04069]\\]. ",
"This method analyzes the signal at equal intervals. ",
"However, since its time-domain resolution cannot be changed, it has problems in analyzing signals with high time-domain resolution. ",
"Wigner--Ville distribution (WVD) \\[[@B6-sensors-19-04069]\\] can be used to analyze specific fault signals with instantaneous energy due to its high resolution in time domain and frequency domain. ",
"However, the defect of this method is that it will generate cross terms. ",
"The empirical mode decomposition (EMD) \\[[@B7-sensors-19-04069]\\] method and the Hilbert spectrum transform method are also used to analyze the nonlinear and unstable signals caused by the local faults of the gearbox. ",
"Wu used the EMD method to process the de-noising signal and decomposed it into IMF components adaptively, and the corresponding noise and false components were eliminated by the correlation number criterion. ",
"However, the disadvantage of the EMD method is that it consumes a lot of computing resources to decompose the computing steps. ",
"In recent years, wavelet transform has been successfully applied to the processing of unsteady signals and the extraction of embedded features \\[[@B8-sensors-19-04069],[@B9-sensors-19-04069]\\]. ",
"McFadden points out that continuous wavelet transform is able to achieve a series of resolutions when analyzing signals due to the use of variable scaling functions, and it has been proved in experiments that it has advantages in analyzing the abrupt wavelet transform in gear vibration \\[[@B10-sensors-19-04069]\\]. ",
"Purushotham adopts the method of discrete wavelet transform to decompose the vibration signals of normal bearing and bearing with complex faults respectively, and compares the extracted characteristic data \\[[@B11-sensors-19-04069]\\]. ",
"Although the wavelet analysis method has been adopted by many scholars, there is no accurate conclusion about which wavelet basis function is the most suitable to analyze the data.",
"\n\nThe working state of gear or bearing can be identified by analyzing the characteristic parameters of fault. ",
"From the current research, the development trend is that the machine can recognize and judge the fault type by itself. ",
"Many scholars also focus on the application of artificial intelligence, machine learning and other technologies to achieve automatic fault identification. ",
"Among all artificial intelligence methods, artificial neural networks (ANNs) are the most widely used. ",
"Rafiee \\[[@B12-sensors-19-04069]\\] used the multi-layer perceptron neural network to deal with the identification of gear faults and bearing defects, and found in the application that feature vectors play a crucial role in the identification effect of the trained neural network. ",
"In addition to ANNs, some new methods such as support vector machine (SVM), relevance vector machine (RVM), and hidden Markov model (HMM) are proposed for fault diagnosis and classification. ",
"Samanta compared the recognition effect of ANNs and SVM for bearing faults under different loads and speeds, and SVM was better than ANNs in terms of diagnosis accuracy and training rate. ",
"In addition, the author also used a genetic algorithm (GA) \\[[@B13-sensors-19-04069]\\] to optimize the structural characteristics of ANNs, such as the number of layers and nodes, and the kernel function of SVM. ",
"Both of them performed better after optimization than before optimization, which also provided a good guidance for better application of intelligent diagnostic tools. ",
"The advantage of RVM is that it gives up the decision function, which can obtain better relaxation characteristics than SVM \\[[@B14-sensors-19-04069]\\]. ",
"When processing the same data, the RVM method with a linear feature extraction feature can give more successful judgment conclusion than SVM. ",
"Purushotham introduced an HMM model that is successfully used in speech recognition. ",
"Based on the signal data measured in the experiment, the components with unknown faults are diagnosed and good judgment results are obtained. ",
"Ge also studied the HMM method and pointed out that it could classify shock vibration. ",
"However, the judgment effect largely depends on the quality of the training vector, so the autoregressive model (AR) is introduced to assist the extraction of feature vectors \\[[@B15-sensors-19-04069]\\]. ",
"Although the above methods can achieve fault classification, how to select a method that can guarantee both the robustness of classifier and the recognition accuracy still needs to be studied.",
"\n\nIn view of the shortcomings of existing literature, this paper provides a new model by combining frequency domain analysis and pattern recognition to realize the intelligent fault diagnosis of a machine tool spindle box. ",
"To effectively realize fault feature extraction, the wavelet packet is used to decompose the signal into different frequency bands and the power spectrum analysis is carried out on the reconstructed signal. ",
"The structure of the feature vector used for fault diagnosis is established based on the detected vibration signals. ",
"The intelligent pattern recognition model based on a support vector machine is used to identify the faults in the spindle box of a heavy-duty horizontal lathe.",
"\n\n2. ",
"Fault Detection and Classification of Rotary Parts {#sec2-sensors-19-04069}\n=====================================================\n\n2.1. ",
"Fault Classification of Spindle Box {#sec2dot1-sensors-19-04069}\n----------------------------------------\n\nAccording to the different parts of the gear box, the failure can be divided into gear failure, shaft failure, and bearing failure. ",
"Among them, gear faults mainly include tooth surface wear, tooth surface abrasion, tooth surface contact fatigue and tooth fracture. ",
"Bearing faults can be divided into inner ring fault, rolling body fault, cage fault and outer ring fault according to the location. ",
"Shaft failures include shaft bending and shaft imbalance and misalignment. ",
"In addition, as an assembly box, the main assembly faults include too large tooth backlash and loose assembly.",
"\n\nAs it is a new machine tool to be delivered, there is no failure such as tooth wear and bearing failure. ",
"For the spindle box of heavy duty horizontal lathe, the faults mainly include: (1) misalignment existing between motor shaft and axis; (2) misalignment existing between transmission gears in the box; (3) loose assembly of the transmission shaft in the gearbox; and (4) the backlash of two pairs of meshing gears being too large.",
"\n\n### 2.1.1. ",
"Coaxiality out of Tolerance of The Connecting Shaft {#sec2dot1dot1-sensors-19-04069}\n\nThe motor of the machine tool is connected to shaft I through the shaft coupling. ",
"The installation of this transmission structure will be accompanied by the connection shaft coaxiality out-of-tolerance, which is usually manifested as axis parallel misalignment and angle misalignment coexistence, as shown in [Figure 1](#sensors-19-04069-f001){ref-type=\"fig\"}a. ",
"In the case of coaxiality error between connecting shafts, the radial vibration frequency of slave shaft is twice that of master shaft, and with the increase of misalignment angle \"$\\alpha$\". ",
"In addition, when the misalignment phenomenon exists, it is generally the axis parallel misalignment and the angle misalignment appearing at the same time. ",
"The radial vibration of slave shaft appears to have double frequency, and the more serious the misalignment, the greater the proportion and amplitude of double frequency. ",
"In addition, in general, the amplitude of double frequency is higher than the fundamental frequency. ",
"The general form of its spectrum diagram is shown in [Figure 1](#sensors-19-04069-f001){ref-type=\"fig\"}b, where *f* is the frequency of master shaft.",
"\n\n### 2.1.2. ",
"Misalignment of Gears {#sec2dot1dot2-sensors-19-04069}\n\nIn the process of machining and assembly of gear box body, it often occurs that the position accuracy between the holes does not meet the requirements due to the machining error of the box body, which leads to the misalignment of the two axes equipped with meshing gears after assembly. ",
"In addition, some gears in the spindle box of the heavy horizontal lathe are connected to the shaft by spline. ",
"Because the coaxiality of the shaft is out of tolerance and the coaxiality of the gear is out of tolerance, the gear and the shaft will be misaligned. ",
"These two phenomena will lead to the uneven load of the gear in the direction of the width of the tooth, the stability of power transmission decreases, and the load on the local area of the tooth surface increases. ",
"The assembly error diagram of different axes is shown in [Figure 2](#sensors-19-04069-f002){ref-type=\"fig\"}a.",
"\n\nThe misaligned gear will produce meshing frequency vibration with side frequency band at meshing frequency. ",
"Moreover, there are harmonics of meshing frequency, and the peak value of harmonics at double and triple meshing frequency is very high, even higher than the peak value of meshing fundamental frequency. ",
"In addition, the modulation phenomenon appears around the peak value of each order harmonic frequency, and the modulation frequency is consistent with the rotation frequency of the shaft where the gear is located. ",
"The frequency spectrum of gear misalignment is shown in [Figure 2](#sensors-19-04069-f002){ref-type=\"fig\"}b.",
"\n\n### 2.1.3. ",
"Excessive Backlash of Gear Tooth {#sec2dot1dot3-sensors-19-04069}\n\nIn the assembly of the spindle box, the tooth backlash of the meshing wheel teeth will be too large after assembly. ",
"Because the tooth backlash will affect the load change of the gear transmission, the vibration characteristic of the gear system will change and the peak frequency will change. ",
"When the tooth backlash is too large, the fractional multiple harmonics of meshing fundamental frequency will appear in the spectrum diagram in addition to the high harmonics of meshing fundamental frequency. ",
"The spectrum diagram is shown in [Figure 3](#sensors-19-04069-f003){ref-type=\"fig\"}.",
"\n\n### 2.1.4. ",
"Loose Mounting of Bearing Seat {#sec2dot1dot4-sensors-19-04069}\n\nDuring the installation of the drive shaft, the bearing supporting the shaft is mounted on the bearing seat in the box body. ",
"In the process of installation, there may be loose installation of bearing pedestal due to loose installation of connecting bolts of bearing pedestal. ",
"Once it appears, it will cause strong vibration and noise when the box works. ",
"Although the frequency of such faults is lower in practice than those mentioned above, it still needs to be accurately diagnosed and eliminated in view of its serious interference to normal working conditions.",
"\n\nIn the case of installation looseness, in addition to the basic frequency of axis rotation *f*, there will be high order harmonic frequency of 2*f*, 3*f*, 4*f* with equal axis rotation frequency, and fractional frequency components of basic frequency such as *f*/2, 2*f*/3, and *f*/3 in the frequency spectrum. ",
"In addition, the amplitude of these frequencies is significantly increased compared with the normal operation, but the rotational fundamental frequency is still the dominant one. ",
"Therefore, the amplitude of each characteristic frequency should be comprehensively analyzed when diagnosing this kind of fault. ",
"The spectrum diagram is shown in [Figure 4](#sensors-19-04069-f004){ref-type=\"fig\"}.",
"\n\n2.2. ",
"Fault Signal Acquisition {#sec2dot2-sensors-19-04069}\n-----------------------------\n\nThis paper mainly analyzes the vibration signal accompanied by the fault, so it is necessary to collect the vibration signal of the gearbox in the time domain. ",
"Sensors suitable for the acquisition system of this paper should be able to pick up vibration signals caused by assembly faults such as misalignment of shaft and loose installation. ",
"In addition, in order to better analyze the signal in the time domain and frequency domain, the shaft speed should also be measured.",
"\n\nAccording to the corresponding relationship between acceleration and force, the piezoelectric effect can be used to determine the corresponding relationship between acceleration and voltage signal. ",
"Moreover, the piezoelectric effect is characterized by good linearity and high sensitivity, and its frequency measurement range can be from 0.0001 Hz to 1 MHz. ",
"Therefore, the piezoelectric acceleration sensor has the advantages of high sensitivity and spectral bandwidth. ",
"In addition, most of the installation of the acceleration sensor only needs to be fitted with the measured object. ",
"The fixing method includes bonding it with the surface to be tested by adhesive, or fixing it by the external force of magnetic seat. ",
"In this paper, an acceleration sensor (PCB type 356A16) is adopted, as shown in [Figure 5](#sensors-19-04069-f005){ref-type=\"fig\"}b. ",
"The sensitivity is 100 mV/g. The corresponding frequency range is 0.5 to 5000 Hz, the measuring range is 100 g, and the weight is 7.4 g. In the application, the magnetic seat can be bonded on the end cover of the bearing seat.",
"\n\nIn order to better filter out the fault information contained in the vibration signal, it is necessary to know the actual rotation speed of the shaft during the test. ",
"In this way, the possible failure frequency of each drive shaft and the parts on the shaft can be calculated. ",
"By combining this information with the processed original vibration signal, the fault of the spindle box can be diagnosed. ",
"Since the number of gear teeth on each drive shaft of the gearbox can be obtained by consulting the design manual of the corresponding machine tool, it is only necessary to understand the rotation speed of the power input motor.",
"\n\nIn this paper, rotation speed tester (mvp-2c) is used to measure the shaft speed. ",
"During the shaft rotation, the optical signal will be reflected by the reflector when it irradiates the reflector, and the appearance and disappearance of such reflection will trigger the photoelectric sensor to generate pulse signal. ",
"By analyzing the pulse signal, the instrument can output the value of rotation speed and achieve the test purpose. ",
"This method is simple, fast and suitable for complex environment operation. ",
"The instrument and test environment are shown in [Figure 5](#sensors-19-04069-f005){ref-type=\"fig\"}c.",
"\n\nIn this paper, the LMS test instrument is used for data collection, which can realize data collection in continuous time with different sampling frequencies under multiple channels. ",
"In addition, its parameter setting and adjustment are implemented by software installed on the PC. ",
"The tester is easy to operate and can adjust the sampling frequency from 0 to 10 kHz. ",
"The hardware configuration and software interface of the LMS signal acquisition system are shown in [Figure 5](#sensors-19-04069-f005){ref-type=\"fig\"}d,e.\n\nThe signal line of the sensor is connected to the front acquisition box with a multi-channel interface. ",
"There is a module for amplifying the original signal in the pre-acquisition cabinet, which can ensure the signal strength. ",
"The prE--Signal acquisition box is connected with the computer, and the parameter setting, start and end of the acquisition are controlled by the computer. ",
"The three curves in the acquisition software interface correspond to the *x*-, *y*- and *z*-direction input signals of the vibration sensor, respectively. ",
"The collection time, sampling frequency and other parameters can be set in the toolbar on the right side of the software. ",
"After the sampling starts, the system continuously collects the signals from the sensor. ",
"After the set acquisition time arrives, the sampling stops. ",
"Set parameters, sensor information and collected data information are recorded and stored in the same file after the completion of sampling. ",
"These files provide sample data for subsequent data processing.",
"\n\nThe vibration sensor is attached to the bearing seat or the end cover of the bearing seat in the experiment. ",
"In this way, the signal source is relatively close to the transmission medium, and the exchange surface of the transmission medium is relatively small, which can ensure the signal strength of the characteristic information: signal sampling in order for the motor, axis I, axis II, and axis III. ",
"The layout of the sensor is shown in [Figure 6](#sensors-19-04069-f006){ref-type=\"fig\"}.",
"\n\n3. ",
"Wavelet Packet Transform Based Fault Feature Extraction {#sec3-sensors-19-04069}\n==========================================================\n\n3.1. ",
"Determination of Wavelet Basis Function {#sec3dot1-sensors-19-04069}\n--------------------------------------------\n\nBecause wavelet transform can map the signal to time--frequency domain, and a series of wavelet basis functions are used to represent the signal, the hidden features in the signal can be effectively displayed. ",
"The assembly failure of spindle box studied in this paper is a part of rotating machinery failure. ",
"In order to better extract the characteristics of the collected samples, the wavelet analysis method is also used to process the fault signal.",
"\n\nThe basic method of wavelet transform is to take the inner product of the basic wavelet function with the signal $x\\left( t \\right)$ to be analyzed at different scales *a*. ",
"The basic expression of continuous wavelet transform is shown as Equation ([1](#FD1-sensors-19-04069){ref-type=\"disp-formula\"}):$$WT_{x}\\left( {a,b} \\right) = \\left\\langle {x,\\psi_{a,b}} \\right\\rangle = \\frac{1}{\\sqrt{a}}\\int_{- \\infty}^{+ \\infty}{x\\left( t \\right)}\\overline{\\psi}\\left( \\frac{t - b}{a} \\right)dt,$$ where $WT_{x}\\left( a,b \\right)$ is Wavelet coefficients, *a* is scale that represents the frequency parameter, and *b* is time or space position parameters.",
"\n\nFor continuous wavelet transform, scale *a*, time *t* and offset *b* are continuous. ",
"When using the computer to realize the calculation of wavelet transform, they need to be discrete processing, which is the discrete wavelet transform. ",
"Binary discretization of scale *a* and offset *m* is usually carried out, and the corresponding binary wavelet is shown in Equation ([2](#FD2-sensors-19-04069){ref-type=\"disp-formula\"}):$$\\psi_{j,k}\\left( t \\right) = 2^{- \\frac{j}{2}}\\psi\\left( {2^{- j}t - k} \\right).$$\n\nIn the calculation procedure of binary discrete wavelet transform, only the approximate part (low-frequency sub-band) of the signal is divided by exponential interval from scale 2, and the detailed part (high-frequency sub-band) is not processed. ",
"Therefore, the resolution of the high frequency part is poor. ",
"In order to process and analyze the high frequency part, Coifman, Meyer and Wickerhauser put forward the concept of wavelet packet decomposition. ",
"Wavelet packet decomposition divides the frequency band into several levels, so as to further decompose the high frequency part and realize more detailed analysis of the signal. ",
"The decomposition structure obtained by decomposing signal *S* into three-layer wavelet packet transform is shown in [Figure 7](#sensors-19-04069-f007){ref-type=\"fig\"}.",
"\n\nIt can be seen from the analysis structure diagram that wavelet packet analysis can decompose the original signal into each frequency band, so that the frequency band matching the concerned frequency can be selected for analysis, and the time-frequency resolution of signal analysis can be improved.",
"\n\nThe fault characteristic frequency studied in this paper is not limited to the rotation frequency or meshing base frequency of a rotating part. ",
"In order to realize fault diagnosis, their high harmonic frequencies are also analyzed. ",
"Therefore, the wavelet packet transform method can be used as a filter. ",
"A node only reflects the information related to the frequency range of the node but excludes the interference of other frequency components. ",
"In contrast, discrete wavelet transform usually ignores some high frequency information. ",
"In this paper, the advantages and disadvantages of wavelet transform and wavelet packet transform in signal representation will be compared quantitatively.",
"\n\nThe evaluation criteria of wavelet transform method are mainly as follows:\n\n(a) The energy content of the signal\n\nThe energy content of a signal is a parameter directly related to the characteristics of the signal, so it is usually used to describe the signal. ",
"The energy contained in the signal $x\\left( t \\right)$ can be calculated by the wavelet coefficient, as expressed by Equation ([3](#FD3-sensors-19-04069){ref-type=\"disp-formula\"}) \\[[@B16-sensors-19-04069]\\]:$$E_{energy} = \\sum\\limits_{s}\\sum\\limits_{i}\\left| {W\\left( s,i \\right)} \\right|^{2}.$$\n\nIf a major frequency component of a signal corresponds to a specific scale, the wavelet coefficients in that scale will have a relatively large amplitude when the major frequency occurs. ",
"This allows the energy associated with a particular frequency component to be extracted from the signal. ",
"Therefore, the energy content can be used as a criterion to evaluate the applicability of wavelet basis. ",
"For the same signal, the larger the calculated value of the energy content, the more effective and appropriate processing method is considered to show the characteristics of the signal, which is the maximum energy content standard.",
"\n\n(b) Shannon entropy of the signal\n\nThe wavelet coefficients are obtained by wavelet transform. ",
"The coefficients reflect the similarity between wavelet and signal. ",
"If the coefficient matrix obtained by wavelet transform is regarded as a distribution, the sparser the coefficient matrix is, the higher the similarity between signal and wavelet will be. ",
"Shannon entropy can be used as an indicator to evaluate sparsity. ",
"Shannon entropy is defined as Equation ([4](#FD4-sensors-19-04069){ref-type=\"disp-formula\"}):$$\\left\\{ \\begin{array}{l}\n{E_{entropy}\\left( s \\right) = - \\sum\\limits_{i = 1}^{N}p_{i}log_{2}p_{i},} \\\\\n{p_{i} = \\frac{\\left| {wt\\left( s,i \\right)} \\right|^{2}}{E_{energy}\\left( s \\right)},\\sum\\limits_{i = 1}^{N}p_{i} = 1,} \\\\\n\\end{array} \\right.$$ where $p_{i}$ is the energy probability distribution of the wavelet coefficients.",
"\n\nThe probability distribution uniformity can be evaluated by calculating the Shannon entropy of the distribution. ",
"For example, the distribution is the most uncertain when it has equal probability property. ",
"This distribution has the highest Shannon entropy and the lowest sparsity. ",
"This evaluation method is applied to the calculation and evaluation of wavelet coefficient matrix. ",
"The smaller the Shannon entropy is, the more the corresponding wavelet analysis method matches with the signal. ",
"This is also the minimum Shannon entropy standard to evaluate the applicability of wavelet basis function and wavelet analysis method.",
"\n\n(c) The ratio of energy content to Shannon entropy (E--S ratio)\n\nBoth maximum energy content standard and minimum Shannon entropy standard can be used to evaluate wavelet analysis. ",
"These methods are based on the description of the original signal information by wavelet coefficients. ",
"The evaluation standard derived from the combination of these two is called energy--Shannon entropy ratio (E--S ratio), and its definition is shown in Equation ([5](#FD5-sensors-19-04069){ref-type=\"disp-formula\"}). ",
"Corresponding to the maximum energy content standard and the minimum Shannon entropy standard, the wavelet transform method with the maximum E--S ratio has the strongest ability to characterize the signal. ",
"The above three evaluation criteria will be verified and compared in the analysis of actual signals later:$$R\\left( s \\right) = \\frac{E_{energy}\\left( s \\right)}{E_{entropy}\\left( s \\right)}.$$\n\nIn different calculation methods of wavelet transform, discrete transform and wavelet packet transform can be implemented by the computer, and they can decompose signals into different frequency scales. ",
"Regarding which method is more suitable for the fault diagnosis of spindle box, the above-mentioned criteria (as shown in Equations (3)--(5)) are used for comparison. ",
"Using the same wavelet basis function, the methods of discrete wavelet decomposition and wavelet packet transform are compared. ",
"By analyzing the advantages and disadvantages of these two methods for decomposition of original signals, Shannon entropy, energy content and E--S ratio under different decomposition modes are calculated, respectively.",
"\n\nThe wavelet basis function $coif1$ in the Coiflet wavelet system is used, and the discrete wavelet transform is applied to carry out 3-layer decomposition of the original signal. ",
"For the wavelet packet, the decomposition of the original signal with the scale of 3 is also carried out. ",
"In addition, this comparative analysis was carried out for different samples. ",
"Data collected on each axis are analyzed and calculated, and the results are shown in [Table 1](#sensors-19-04069-t001){ref-type=\"table\"}.",
"\n\nBy comparing the data values of each row in the table, it can be seen that Shannon entropy value of signal through wavelet packet transformation is lower than that of discrete wavelet transform. ",
"The energy value is larger than the energy obtained by the discrete wavelet transform, and the E--S ratio is correspondingly larger than the discrete wavelet transform. ",
"It can be verified that wavelet packet decomposition can better show the change rule of the signal, characterizing the characteristics of signal and the information contained in it. ",
"In order to demonstrate the universality of the above verification, the wavelet basis function was replaced to process the same data, and the calculation results of the two analysis methods were still compared. ",
"Using wavelet basis function sym4, which is different from wavelet basis waveform coif1, the calculation result is similar to that of coif1 wavelet, which proves that wavelet packet analysis can reflect the characteristics of signal more precisely. ",
"From the perspective of principle, wavelet packet analysis not only decomposes the low-frequency signal, but also decomposes the high-frequency signal that has not been decomposed in the discrete wavelet transform, so the description of signal features is more appropriate. ",
"Therefore, better calculation results can be obtained when various indexes are introduced for evaluation. ",
"In addition, some of the characteristic frequencies studied in this research are at the higher harmonic frequency of the meshing frequency, so the high-frequency part of the signal also needs to be paid attention to. ",
"Therefore, wavelet packet decomposition can better meet the research needs.",
"\n\nBased on Shannon entropy, energy content and E--S ratio, the wavelet basis suitable for the analysis of spindle box vibration signal is evaluated and selected. ",
"Different wavelet basis functions are used for wavelet packet decomposition transformation, and the results are shown in [Table 2](#sensors-19-04069-t002){ref-type=\"table\"}.",
"\n\nIt can be seen from the calculation results that the values of each index obtained after the processing of different wavelet basis functions are different. ",
"Among these wavelet families, the calculated values of Bior and R-bior wavelet families are superior to other wavelet families. ",
"The results of wavelet functions of different orders in the same family of wavelets are different. ",
"The basic rule is that the higher the order, the better the result of each index. ",
"In terms of energy content, the maximum value of Bior3.5 was calculated in the analysis of different groups of sample data. ",
"In the calculation of Shannon entropy, Bior3.5, Coif4 and Rbio3.5 respectively obtained a primary minimum value. ",
"As for the E--S ratio, the maximum value of Bior3.5 is obtained. ",
"It can also be seen that the maximum E--S ratio standard can help us select the most suitable wavelet. ",
"Bior3.5 was selected for wavelet packet decomposition in this research.",
"\n\n3.2. ",
"Fault Feature Extraction {#sec3dot2-sensors-19-04069}\n-----------------------------\n\nThe signal collected from the spindle box of the machine tool is analyzed. ",
"The spindle box is known to have gear misalignment errors among the shafts. ",
"Sampling signal acquisition in shaft II bearing end cover. ",
"The tooth transmission relation between machine each axis: shaft I/II is 28/47, shaft II/III is 30/43. ",
"The motor speed measured during sampling was 999.6 rpm and the sampling frequency was 8192 Hz. ",
"After calculation, the rotation frequency of the test shaft is 9.93 Hz, and the engagement frequency of the gears installed on the shaft is 466.48 Hz and 297.75 Hz. ",
"The original time domain waveform of the acquired signal is shown in [Figure 8](#sensors-19-04069-f008){ref-type=\"fig\"}. ",
"The period of some faults cannot be directly identified from the information in the figure, and the waveform peak is disordered. ",
"Bior3.5 wavelet is used for five-layer wavelet packet decomposition, so the signal is decomposed into 32 frequency bands, and a five-layer binary tree structure is obtained. ",
"The last layer is \\[5 0\\], \\[5 1\\], ..., \\[5 31\\], a total of 32 nodes, and the corresponding frequency band is 0--128 Hz, 128--256 Hz, ..., 3968--4096 Hz. ",
"The characteristic frequency associated with the mismeshing fault of the gear is the first, second and third meshing frequency, and the corresponding frequency of conversion modulation, that is, the meshing frequency plus or minus 9.93 Hz.",
"\n\nThe frequency points concerned are 297.75 Hz, 595.5 Hz, 893.25 Hz, 466.48 Hz, 932.96 Hz, 1399.44 Hz, and the modulation frequencies around them. ",
"These frequency points are located in frequency bands corresponding to nodes \\[5 2\\], \\[5 3\\], \\[5 4\\], \\[5 6\\], \\[5 7\\], \\[5 9\\]. ",
"These nodes were reconstructed and Hilbert transformation was performed for spectral analysis, and the waveform obtained was shown in [Figure 9](#sensors-19-04069-f009){ref-type=\"fig\"}. [",
"Figure 9](#sensors-19-04069-f009){ref-type=\"fig\"}a,c,e deal with pairs of nodes \\[5 2\\], \\[5 4\\] and \\[5 6\\]. ",
"Gear meshing frequency between shaft II and shaft III is 297.75 Hz, and the second and third order has a larger peak near the frequency doubling, and they all have a modulation of about 9 Hz. ",
"From the size of its peak value, it can be seen that the energy at the second and third order frequencies does not decay, and is even greater than the intensity of the meshing fundamental frequency, which is consistent with the fault characteristics. [",
"Figure 9](#sensors-19-04069-f009){ref-type=\"fig\"}b,d,f processing the node \\[5 3\\], \\[5 7\\], \\[5 9\\] represent axial meshing frequency between shaft I and shaft II, the second order and third order meshing frequencies and the corresponding amplitude modulation frequencies. ",
"Through the characteristics shown in these processing results, it is proved that this processing method can be used to obtain significant changes related to specific faults in the frequency--power spectrum, and this ability to capture abnormal features can be used to extract features under different faults.",
"\n\nBy analyzing the amplitude of signal at certain frequency in signal power spectrum, the fault type can be judged. ",
"Therefore, it is very meaningful to obtain the energy values at these characteristic frequencies for fault analysis. ",
"Because there are only certain kinds of assembly faults of the spindle box of the heavy horizontal lathe, and the detection of some faults needs to be based on the amplitude analysis of some rotating frequencies. ",
"These can be used as the basis of the characteristics of the frequency mainly includes the following: (1) shaft frequency and its harmonic frequency, shaft rotation frequency of 1/2 frequency, 1/3 frequency; (2) meshing frequency and doubling frequency of the gear mounted on the shaft; (3) the gear mesh frequency plus or minus the frequency of the shaft rotation frequency value. ",
"With the amplitudes at these frequency points, they can be arranged into a set of eigenvectors to represent the operation of the axis. ",
"The feature vectors extracted from known faults can be used to train the intelligent classifier, while the features extracted from unknown fault signals can be used to judge the fault type of the axis.",
"\n\nConsidering the fault types to be diagnosed comprehensively, the feature frequency points that should be extracted are summarized in [Table 3](#sensors-19-04069-t003){ref-type=\"table\"}. ",
"Considering that there are two gears meshing on some axes, the information of their frequency points should be extracted. ",
"For the side frequency modulation of the meshing frequency, the fault can be diagnosed only by knowing whether the unilateral modulation phenomenon exists. ",
"Therefore, a representative value of the two side frequencies can be extracted. ",
"For each signal, a 23-dimensional vector is extracted to represent its characteristics.",
"\n\nIn order to extract the power spectrum value at the above characteristic points, the variables should be clearly defined, including the number of teeth of each relevant shaft of the machine tool spindle box, motor speed during testing, gear position during sampling, sampling frequency, and which axis the sensor is arranged in during sampling. ",
"After calculating the characteristic frequency, the wavelet packet decomposition node corresponding to the frequency band is reconstructed and spectral analyzed, and then the value of the characteristic frequency point is obtained. ",
"At this time, in the algorithm, the maximum value in the range of --2 Hz to +2 Hz can be extracted in order to cope with the rotation speed change in the above analysis. ",
"As for the meshing edge frequency band information, only one value needs to be extracted, so the value with a large side frequency is selected to extract it. ",
"Since the extracted values are likely to be large, they should be normalized to facilitate subsequent analysis and calculation. ",
"$E_{ij}$ represents the amplitude of a certain frequency point, and it is the energy at a discrete point *i* representing the characteristic frequency after reconstruction of the node *j*. ",
"The total energy $E = \\sum_{j}\\sum_{i}E_{ij}$. For the extracted 23-dimensional eigenvector *T*= \\[$T_{1}$, $T_{2}$, ..., $T_{23}$\\], each value corresponds to the value of a certain $E_{ij}$. After normalization, a new vector $T’$ can be obtained, $T’$= \\[$T_{1}$/*E*, $T_{2}/E$, ..., $T_{23}/E$\\], which will be used for the subsequent intelligent fault classification.",
"\n\n4. ",
"Fault Diagnosis Based on Support Vector Machine {#sec4-sensors-19-04069}\n==================================================\n\nTo realize intelligent and automatic fault diagnosis, a corresponding intelligent algorithm is needed. ",
"The acquisition of signals, decomposition and transformation of signals and extraction of fault features mentioned above are all preparations for the realization of fault diagnosis. ",
"After obtaining the characteristic vectors representing various types of faults, these samples can be used to study the intelligent classification algorithm. ",
"The problem of small sample training classification can be solved by means of an SVM method in this research.",
"\n\nThe set of spindle box fault classification can be divided by using a linear function, that is, there is a hyperplane to distinguish samples, as shown in Equation ([6](#FD6-sensors-19-04069){ref-type=\"disp-formula\"}) \\[[@B17-sensors-19-04069]\\]:$$\\left\\{ \\begin{array}{l}\n{\\omega^{T}x_{i} + b = 0,} \\\\\n{\\omega^{T}x_{i} + b \\geq 1,y_{i} = 1,} \\\\\n{\\omega^{T}x_{i} + b \\leq 1,y_{i} = - 1,} \\\\\n\\end{array} \\right.$$ where the nearest distance between ($x_{i}$,$y_{i}$) and the classification hyperplane is $\\frac{1}{\\parallel \\omega \\parallel}$, and the hyperplane with the largest distance value $\\frac{1}{\\parallel \\omega \\parallel}$ is the optimal hyperplane.",
"\n\nLinear SVM is to find the weights and biases, so that, while Equation ([6](#FD6-sensors-19-04069){ref-type=\"disp-formula\"}) can be satisfied, the maximum $\\frac{1}{\\parallel \\omega \\parallel}$ can be obtained. ",
"Finding the maximum value of $\\frac{1}{\\parallel \\omega \\parallel}$ is the same as finding the minimum value of $\\frac{1}{2}{\\parallel \\omega \\parallel}^{2}$, which is the expression of Equation ([7](#FD7-sensors-19-04069){ref-type=\"disp-formula\"}):$$\\left\\{ \\begin{array}{l}\n{\\min\\frac{1}{2}{\\parallel \\omega \\parallel}^{2},} \\\\\n{s.t.y_{i}\\left( {\\omega x_{i} + b} \\right) \\geq 1,i = 1,2,\\ldots,l.} \\\\\n\\end{array} \\right.$$\n\nThe Lagrange multiplier is introduced to solve the optimization problem:$$L\\left( {\\omega,b,\\alpha} \\right) = \\frac{1}{2}{\\parallel \\omega \\parallel}^{2} - \\sum_{i = 1}^{l}{\\alpha_{i}\\left\\lbrack {y_{i}\\left( {\\omega x_{i} + b} \\right) - 1} \\right\\rbrack},$$ where $\\alpha_{i}$, *i* = 1, 2, ..., *l* is a Lagrange coefficient. ",
"Equation ([8](#FD8-sensors-19-04069){ref-type=\"disp-formula\"}) can be simplified to Equation ([9](#FD9-sensors-19-04069){ref-type=\"disp-formula\"}):$$\\left\\{ \\begin{array}{l}\n{\\max Q\\left( \\alpha \\right) = \\sum_{i = 1}^{l}{\\alpha_{i} - \\frac{1}{2}}\\sum_{i = 1}^{l}{\\sum_{j = 1}^{l}{\\alpha_{i}\\alpha_{j}y_{i}y_{j}\\left( {x_{i}x_{j}} \\right)}},} \\\\\n{s.t.\\sum_{i = 1}^{l}\\alpha_{i}y_{i} = 0,\\alpha_{i} \\geq 0.} \\\\",
"\n\\end{array} \\right.$$\n\nSamples corresponding to $\\alpha_{i}$≠0 are support vectors. ",
"The optimal solution can be obtained:$$\\left\\{ \\begin{array}{l}\n{\\omega^{*} = \\sum_{i = 1}^{l}{{\\alpha_{i}}^{*}x_{i}y_{i}},} \\\\\n{b^{*} = - \\frac{1}{2}\\omega^{*}\\left( {x_{r} + x_{s}} \\right),} \\\\\n\\end{array} \\right.$$ where $x_{r}$ and $x_{s}$ are any support vector in their respective categories, and the discriminant function of the classifier can be obtained as in Equation ([11](#FD11-sensors-19-04069){ref-type=\"disp-formula\"}):$$f\\left( x \\right) = \\sum_{i = 1}^{l}{\\alpha_{i}}^{*}y_{i}{x_{i}}^{T}x + b^{*}.$$\n\nWhen SVM \\[[@B18-sensors-19-04069],[@B19-sensors-19-04069],[@B20-sensors-19-04069]\\] classification is used, complex problems to be classified are transformed into simple quadratic programming problems, which can guarantee the correct and optimal classification in principle \\[[@B21-sensors-19-04069]\\]. ",
"When the SVM method is applied in practice, the training samples needed are the eigenvectors of corresponding faults that have been identified. ",
"Considering that in practice the test shaft may have one fault or several faults at the same time, it is therefore necessary to train SVM with the classification ability for various faults. ",
"In principle, there are two ways: one is to train SVM that can distinguish multiple categories of samples, and the other is to train multiple SVM that can only make dichotomies. ",
"The first method requires a corresponding sample of each fault combination mode for training, which is difficult to do in practice because the fault types are randomly combined and the sample number is very limited. ",
"Considering the second approach, it is possible to train an SVM classifier for each fault to be identified. ",
"When features need to be classified and recognized, each SVM is used to identify them, so as to judge whether the signals to be diagnosed have faults corresponding to each SVM in turn. ",
"In this way, various single faults and combinations of faults can be identified.",
"\n\nWhen multiple SVM are used for recognition, features need to be divided because information on a particular dimension in a feature represents a particular failure. ",
"The faults to be identified are misalignment of two connecting shafts, looseness of bearing seat installation, misalignment of gears and excessive side clearance of meshing gear teeth. ",
"The combination of frequency points that can display these faults is shown in [Table 4](#sensors-19-04069-t004){ref-type=\"table\"}, where *f* is the rotation frequency of the corresponding shaft and $F_{1}$ and $F_{2}$ are the gear engagement frequencies.",
"\n\nAfter feature extraction of acquired data by the wavelet packet decomposition method, feature vectors are divided according to the requirements of each SVM, and 48 class A and B samples and 75 class C and D samples are finally obtained. ",
"The samples in each category are randomly assigned. ",
"One part is used to train the respective SVM, and the other part is used to verify the generalization property of the trained SVM.",
"\n\nAs for the selection of kernel function, there is no uniform standard and method, but radial basis function (RBF) has been applied successfully in various cases. ",
"When training SVM, in order to achieve better learning and classification effects, it is necessary to select the optimal punishment factor c and RBF kernel parameter *g*, and apply the cross validation method in the optimization process. ",
"The idea of this method is to divide the original data into *k* groups according to the number of cross validation *k*. ",
"After grouping, the $k^{- 1}$ group is used for training, while the remaining group is used for validation. ",
"Different groups were selected as verification groups, and the calculation process was carried out *k* times. ",
"The optimization range of *c* and *g* is set as $2^{- 10}$ to $2^{10}$, the iteration step length is 0.5, the number of cross validation is 5, and the optimization target is the cross validation accuracy of classification. ",
"After iteration and optimization, the three-dimensional diagram of the relationship between *c* and *g* and classification accuracy obtained is shown in [Figure 10](#sensors-19-04069-f010){ref-type=\"fig\"}, and the optimization results of each classifier are shown in [Table 5](#sensors-19-04069-t005){ref-type=\"table\"}.",
"\n\nAccording to the analysis results, the optimal c and g obtained in SVM training for different types of samples are not the same. ",
"Therefore, it is necessary to optimize c and g by cross-validation for each SVM. ",
"The optimized SVM model is used to classify the test samples and obtain the accuracy of fault type judgment. ",
"The summary is shown in [Table 6](#sensors-19-04069-t006){ref-type=\"table\"}.",
"\n\nIt can be seen from the table that, after training, the classification accuracy of class B, C and D classifiers on test samples reaches over 90%, with high classification accuracy. ",
"Meanwhile, the spindle box fault diagnosis method based on WPT and SVM is verified. ",
"The classification accuracy of class A is relatively low because the number of samples marked as faults is relatively small. ",
"However, its classification accuracy can also reach 80%. ",
"External validation has been executed to evaluate the developed prediction model. ",
"For the four classifiers, 80% of the data are used to train the model and the remaining 20% are for validation. ",
"From the validation results, the accuracy of fault classification results was 81.58%, 94.73%, 95%, and 96.67%, respectively. ",
"Test results of different studies by using SVM are shown in [Table 7](#sensors-19-04069-t007){ref-type=\"table\"}. ",
"This shows again that the model proposed in this paper has high precision. ",
"In the actual classification of signal testing, signal features can be extracted separately, and each SVM is used to diagnose whether the shaft has a specific fault. ",
"For shafts with two pairs of meshing gears, it is necessary to use classes C and D to diagnose the two sets of gears twice, respectively, to determine whether gears have faults or not.",
"\n\n5. ",
"Conclusions {#sec5-sensors-19-04069}\n==============\n\nThe fault types of gear box assembly of a heavy-duty horizontal lathe and corresponding frequency spectrum characteristics are determined. ",
"The signal acquisition system based on an accelerometer is set up and the vibration signal of the spindle box is collected. ",
"Based on Shannon entropy theory, the characteristics of different wavelet transform methods and wavelet basis functions are analyzed. ",
"The signal scheme of wavelet packet transformation using Bior3.5 wavelet is determined. ",
"A method of fault feature extraction is proposed and the correctness of this method is verified by analyzing the signals of known faults. ",
"For the faults to be analyzed, the characteristic vector with a dimension of 23 is determined, and the fault samples for fault diagnosis are obtained. ",
"A fault classifier based on support vector machine (SVM) is designed and its relevant parameters are optimized. ",
"Fault diagnosis is made by using a classifier. ",
"The test results show that, among the four types of faults, the SVM can identify three types of faults with an accuracy rate of over 90%, which is suitable for assembly fault diagnosis.",
"\n\nThe authors would like to thank the anonymous reviewers for their careful, constructive and insightful comments to improve the quality of the present contribution.",
"\n\nConceptualization, H.J.; Computer simulation, Y.L.; Formal analysis, A.T. and Y.L.; Data curation, Y.L.; Investigation, Y.W. and Y.L.; Writing---Original draft, A.T. and H.J.; Writing---Review and editing, Z.H.\n\nThis work is supported by the National Natural Science Foundation of China (51805116).",
"\n\nThe authors declare no conflict of interest.",
"\n\n{#sensors-19-04069-f001}\n\n{#sensors-19-04069-f002}\n\n{#sensors-19-04069-f003}\n\n{#sensors-19-04069-f004}\n\n{#sensors-19-04069-f005}\n\n{#sensors-19-04069-f006}\n\n{#sensors-19-04069-f007}\n\n{#sensors-19-04069-f008}\n\n![",
"Data processing results on shaft II. (**",
"a**) Spectral analysis after node reconstruction \\[5 2\\]. (**",
"b**) Spectral analysis after node reconstruction \\[5 3\\]. (**",
"c**) Spectral analysis after node reconstruction \\[5 4\\]. (**",
"d**) Spectral analysis after node reconstruction \\[5 7\\]. (**",
"e**) Spectral analysis after node reconstruction \\[5 6\\]. (**",
"f**) Spectral analysis after node reconstruction \\[5 9\\].](sensors-19-04069-g009){#sensors-19-04069-f009}\n\n{#sensors-19-04069-f010}\n\nsensors-19-04069-t001_Table 1\n\n###### \n\nComparison of different transformation modes under the basis function of $coif$1.",
"\n\n No. ",
" Discrete $\\mathbf{\\mathbf{E}_{\\mathbf{energy}}}$ (J) Wavelet $\\mathbf{\\mathbf{E}_{\\mathbf{entropy}}}$ Transform E--S Ratio Wavelet $\\mathbf{\\mathbf{E}_{\\mathbf{energy}}}$ (J) Packet $\\mathbf{\\mathbf{E}_{\\mathbf{entropy}}}$ Transform E--S Ratio\n ----- ------------------------------------------------------ -------------------------------------------------- ---------------------- ----------------------------------------------------- ------------------------------------------------- ----------------------\n 1 464.077 15.145 30.642 464.099 15.126 30.683\n 2 230.455 15.181 15.181 230.496 15.157 15.207\n 3 3562.405 17.064 208.772 3562.534 16.925 210.485\n 4 2801.081 16.863 166.109 2081.192 16.783 166.910\n 5 3155.939 17.006 185.583 3156.027 16.960 186.088\n 6 3841.223 17.038 225.451 3841.373 16.882 227.540\n\nsensors-19-04069-t002_Table 2\n\n###### \n\nThe results of wavelet packet decomposition based on different wavelet basis functions.",
"\n\n Wavelet Basis $\\mathbf{E}_{\\mathbf{energy}}$ (J) $\\mathbf{E}_{\\mathbf{entropy}}$ E--S Ratio Wavelet Basis $\\mathbf{E}_{\\mathbf{energy}}$ (J) $\\mathbf{E}_{\\mathbf{entropy}}$ E--S Ratio\n --------------- ------------------------------------ --------------------------------- ------------ --------------- ------------------------------------ --------------------------------- ------------\n Db8 465.160 15.042 30.923 Sym6 464.450 15.037 30.886\n Db20 467.251 15.013 31.124 Bior3.5 1436.305 13.892 103.391\n Coif2 464.570 15.068 30.831 Bior6.8 481.790 15.003 32.114\n Coif4 465.656 15.028 30.986 Rbio2.8 627.791 14.864 42.236\n Sym3 464.253 15.080 30.787 Rbio5.5 601.328 14.785 40.670\n\nsensors-19-04069-t003_Table 3\n\n###### \n\nSummary of characteristic frequencies.",
"\n\n Characteristic Frequencies\n -------------------------------------------- ----------------------------------------------------------------------------------------------------------\n Shaft rotation frequency *f* 1/3*f*, 1/2*f*, *f*, 2*f*, 3*f*, 4*f*, 5*f*\n Gear engagement frequency $F_{1}$, $F_{2}$ 1/3$F_{1}$, 1/2$F_{1}$, $F_{1}$, 2$F_{1}$, 3$F_{1}$, 1/3$F_{2}$, 1/2$F_{2}$, $F_{2}$, 2$F_{2}$, 3$F_{2}$\n Side frequency of mesh frequency $F_{1} + f$, 2$F_{1} + f$, 3$F_{1} + f$, $F_{2} + f$, 2$F_{2} + f$, 3$F_{2} + f$\n\nsensors-19-04069-t004_Table 4\n\n###### \n\nCombinations of different fault characteristic frequencies.",
"\n\n The Fault Types Combination of Characteristic Frequencies\n ----------------------------------------- ----------------------------------------------------------------------------------------\n A-Misalignment of axes *f* *f*, 2*f*, 3*f*\n B-Loose mounting of bearing seat 1/3*f*, 1/2*f*, *f*, 2*f*, 3*f*, 4*f*, 5*f*\n C-Misalignment of gears $F_{1(2)}$, $F_{1(2)} + f$, $2F_{1(2)}$, 2$F_{1(2)} + f$, 3$F_{1(2)}$, 3$F_{1(2)} + f$\n D-Large side clearance of meshing teeth 1/3$F_{1(2)}$, 1/2$F_{1(2)}$, $F_{1(2)}$, 2$F_{1(2)}$, 3$F_{1(2)}$\n\nsensors-19-04069-t005_Table 5\n\n###### \n\nThe results of parameter optimization under different SVM classifiers.",
"\n\n SVM Code Optimal Parameter *c* Optimal Parameter *g* Cross Validation Accuracy\n ---------- ----------------------- ----------------------- ---------------------------\n A 16 0.1768 96.16%\n B 11.3137 0.0313 96.15%\n C 0.3536 0.2500 97.78%\n D 1024 0.0098 91.11%\n\nsensors-19-04069-t006_Table 6\n\n###### \n\nSummary of classification results of test samples.",
"\n\n SVM Code Test Sample Size Correct Number Test Accuracy\n ---------- ------------------ ---------------- ---------------\n A 22 18 81.81%\n B 22 21 95.45%\n C 30 28 93.33%\n D 30 28 93.35%\n\nsensors-19-04069-t007_Table 7\n\n###### \n\nTest results of different research works using SVM.",
"\n\n Test Sample Size Correct Number Test Accuracy\n ---------------------------------------------------- ------------------ ---------------- ---------------\n This research (SVM-D) 60 58 96.67%\n Wu's research (SVM) \\[[@B22-sensors-19-04069]\\] 60 57 95%\n Jiang's research (SVM) \\[[@B23-sensors-19-04069]\\] 60 56 93.33%\n"
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"\n“Is this a kissing book?” ",
"moans the young boy who’s being read a bedtime story by his granddad, a look of disgust on his face. ",
"But by the end of the film that young...\n\nAuthor: Rupert Giles (though we suspect that he may have had some help from Nancy Holder)\nPublisher: Titan Books\nPublished: OUT NOW\nSo, 12 years after the show finished, how can publishers still find new ideas for books about Buffy The Vampire Slayer? ",
"More to the point how can Nancy Holder find anything new to say about Buffy The Vampire Slayer? ",
"She’s not only written an entire library of Buffy and Angel fiction, she’s also authored...\n\nWith this year’s Strictly Come Dancing just about to end, we thought we’d collect together some of sci-fi and fantasy’s most notorious and/or bizarre dance scenes to see what the judges would make of them…\nGamer\nPerformer: Ken Castle (Michael C Hall)\nMusic: “I’ve Got You Under My Skin”\nBruno Tonioli: You jerk and twitch like a puppet having its strings cut one by one. ",
"It was brave. ",
"It was modern. ",
"It was very strange. ",
"I...\n\nFans went wild as Felicia Day emerged from the wings, beaming at the enthusiastic turn-out. ",
"She waved and waved before curtsying to the right, then bowing to the left, and finishing with a little pirouette. “",
"Thank you very much,” she said, “I’m so happy to be here. ",
"It’s been fantastic to meet so many people with accents, very exotic, even, from Wales! ",
"Those who are Welsh, the voice, it’s lovely. ",
"Very nice.”",
"\nNot...\n\nComic con or cruise? ",
"If this decision has you all at sea, why not do both! ",
"Launching for the first time this year, ComicCon at Sea offers traditional shore excursions and various cruise offerings during the day, with the evenings dedicated to convention specific programming.",
"\nEmbarking from Barcelona on 14 June 2015, this seven-day luxury European cruise takes place aboard Royal Caribbean’s Allure of the Seas. ",
"Joining around 500 other fans, guests...\n\nTo coincide with the Priority Entry Tickets for the MCM London Comic Con in May going on sale, MCM have announced their first special guest for the show – Felicia Day.",
"\nHaving appeared in TV shows such as Buffy The Vampire Slayer, Dollhouse, Supernatural and Eureka, she also wrote, produced and starred in the web series The Guild and Dragon Age: Redemption. ",
"Felicia Day is also the founder of the hugely successful YouTube channel Geek & Sundry....\n\nHave you ever wondered what Buffy The Vampire Slayer would look like if the series had continued but as a cartoon?",
"\nWell thanks to the talented artist/animator Stephen Byrne, you can now get an idea of what the opening titles of such a show would look like (Click play below to watch the video).",
"\nThe video also features a new version of the show’s iconic theme by composer Jeff Ball and too many cool references for us too list. ",
"The video even appears...\n\nEvery year fans flock to San Diego Comic Con for a chance to see the star studded panels, meet their heroes and check out the awesome cosplay. ",
"They also have the opportunity to pick up some of the incredible exclusive items that the dealers have on offer at the event.",
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"\nHere are just a few highlights that we discovered while looking through the programme:\nLife and death...\n\nHaving observed certain power shifts within the television industry throughout the 80s and 90s, self-proclaimed “die-hard TV geeks” Des Doyle, John Wallace and Ryan Patrick McGuffey hit upon the idea of creating a film that would document the rise of the TV showrunner. ",
"Widespread interest in the topic enabled them to fund the project through Kickstarter, and at this May’s MCM London Comic Con, the creators of the now completed..."
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"1. ",
"Field of the Invention\nThe present invention relates to a storage system capable of generational management, and a back-up method for a storage system.",
"\n2. ",
"Description of the Related Art\nFor example, in various organizations, such as businesses, self-governing bodies, schools, or the like, large volumes of data of various different types are managed. ",
"These large volumes of data are managed by a storage system formed separately from a host computer. ",
"The storage system is constituted by comprising at least one or more storage device, such as a disk array device, for example.",
"\nA disk array device is constituted by arranging storage devices, such as hard disk drives, or semiconductor memory devices, for example, in an array configuration. ",
"The disk array device provides a storage region based on a RAID (Redundant Array of Independent Inexpensive Disks).",
"\nIn the storage system, by managing the storage contents of a main volume, separately from the main volume, provision is made for cases in which data in the main volume is lost, or the like. ",
"For example, as one such method, a technique is known whereby the contents of the volume at a prescribed point in time are saved, by copying the storage contents of the main volume, exactly, to a secondary volume (Japanese Patent Laid-open No. (",
"Hei)11-259348). ",
"In another such method, a technique is also known whereby an image (snapshot), which takes a static impression of the main volume at a prescribed point in time, is acquired, in such a manner that the data of the main volume can be managed in a plurality of generations (Japanese Patent Laid-open No. ",
"2002-278819).",
"\nIf the storage contents of the main volume are copied exactly to another volume, then a back-up volume of the same volume of the main volume is required, in order to store back-up data for a single generation. ",
"Consequently, if the back-up data is managed by means of a plurality of generations, then a large number of volumes are required and hence the back-up costs increase. ",
"Moreover, each time a new-generation volume is created, then it is necessary to execute an initial copy for transferring the whole volume, and hence the system load increases.",
"\nIf the storage contents at a particular point in time are taken as a reference, and a differential snapshot is used to manage the differential only, then the required storage contents are relatively small compared to the aforementioned method, and back-up data can be managed in the form of a plurality of generations. ",
"However, the actual data used for managing generations on the basis of differential snapshots is not backed up, and there only exists one version of this data. ",
"Furthermore, the data for generational management cannot serve the purpose of data back-up on its own. ",
"The user is able to restore the data of a desired generation by referring to both the main volume forming the reference, and the data used for generational management.",
"\nTherefore, if a fault occurs in the main volume, it is not possible to restore data simply by means of the data for generational management in the differential snapshot. ",
"Furthermore, in the event that the data for generational management has been lost, it becomes impossible to revert to the storage contents of the main volume at a prescribed point of time in the past."
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"It's back to the drawing board to see what is possible.\"",
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"Fort Bliss soldiers who went AWOL in December found, Army says\n\nSAN ANTONIO — Two soldiers who went missing in December from their post at Fort Bliss were apprehended and returned to duty in late January and February, the Army said Wednesday, ending a bizarre double AWOL incident.",
"\n\nArmy Pfcs. ",
"Melvin Jones and Jake Obad-Mathis, both 20 years old and assigned to the 1st Armored Division at Fort Bliss in El Paso, Texas, were apprehended by law enforcement agencies on Jan. 27 and Feb. 26, respectively, a 1st Armored Division spokesman said.",
"\n\nThe soldiers were last seen together Dec. 19 in a car owned by Jones, Stars and Stripes reported in January. ",
"Obad-Mathis was reported AWOL when he failed to show up for duty the next day, while Jones, who was on leave, was not considered AWOL until Jan. 3.",
"\n\nArmy officials cooperated with law enforcement agencies to find and apprehend the solders, the official said, adding that the incident is under investigation.",
"\n\nArmy officials did not immediately respond to questions regarding possible disciplinary actions taken against the two soldiers for the violation.",
"\n\nObad-Mathis was AWOL for more than 30 days, which is when an AWOL soldier is dropped from the duty roster and declared a deserter, an Army personnel spokeswoman told Stars and Stripes in January.",
"\n\nIt is unclear why the soldiers left their post together or why they were gone for weeks. ",
"Carin Obad, the mother of Obad-Mathis, told Stars and Stripes on Wednesday that the Army instructed family members not to discuss details about the incident with news reporters.",
"\n\n“They wanted to go on an adventure and it got away from them a little bit,” Obad said, adding she was happy about the safe return of her son.",
"\n\nShe declined to give specifics about where the two soldiers had been, citing the Army’s request to offer few details, though she did say they were not in Mexico. ",
"One bridge crossing into Juarez, Mexico, is about four miles from the Fort Bliss visitors’ center.",
"\n\nThe disappearance of the two soldiers ignited criticism of the often confusing ways commanders and police officials work together to find AWOL troops. ",
"There is no military equivalent to a missing person report, which triggers investigative resources such as phone tracking after 24 or 48 hours of a reported disappearance, and the jurisdictions between military and civilian law enforcement do not allow for close collaboration, Obad and other critics have said.",
"\n\n“We couldn’t access [Obad-Mathis’] bank account or track his last steps. ",
"A lot of things that could happen couldn’t happen,” Obad said, describing the limitations of what she could do with police without a missing person report.",
"\n\nThe Army Criminal Investigation Command, commonly known as CID, launched an investigation shortly after they were informed of the incident nearly two weeks after it began, Stars and Stripes reported.",
"\n\n“Your leadership in his company is horrendous and we would like someone to step up and find our son,” Obad said in a Facebook posting Dec. 28, apparently criticizing her son’s commanders at Fort Bliss.",
"\n\nObad said she did not see Army officials or the police take the issue seriously until reports on their disappearance and subsequent lack of investigation surfaced in the media.",
"\n\nThere were 88 cases of AWOL soldiers at Fort Bliss last year, with 47 classified as deserters out of a total population of 34,000 troops, said Gilbert Telles, a Fort Bliss spokesman.",
"\n\n\"The efforts we take to look for soldiers consume considerable resources. ",
"Our Army is smaller now than it's been since before WWII, and we are busier than we've ever been,” said Col. ",
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"\n\nMaggie Haswell, a former Air Force security forces specialist who volunteers to search for missing troops and veterans, said Wednesday that she was relieved to hear the soldiers were safe and alive, but said the incident illustrated broader issues of how the services look for missing troops.",
"\n\n“By declaring someone AWOL, the chain of command is saying we’re not putting forth the resources,” Haswell said. “",
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"/*\n * Copyright (c) 2005, 2006, Oracle and/or its affiliates. ",
"All rights reserved.",
"\n * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.",
"\n *\n * This code is free software; you can redistribute it and/or modify it\n * under the terms of the GNU General Public License version 2 only, as\n * published by the Free Software Foundation. ",
" Oracle designates this\n * particular file as subject to the \"Classpath\" exception as provided\n * by Oracle in the LICENSE file that accompanied this code.",
"\n *\n * This code is distributed in the hope that it will be useful, but WITHOUT\n * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or\n * FITNESS FOR A PARTICULAR PURPOSE. ",
" See the GNU General Public License\n * version 2 for more details (a copy is included in the LICENSE file that\n * accompanied this code).",
"\n *\n * You should have received a copy of the GNU General Public License version\n * 2 along with this work; if not, write to the Free Software Foundation,\n * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.",
"\n *\n * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA\n * or visit www.oracle.com if you need additional information or have any\n * questions.",
"\n */\n\npackage javax.lang.model;\n\nimport java.util.",
"Collections;\nimport java.util.",
"Set;\nimport java.util.",
"HashSet;\n\n/**\n * Source versions of the Java™ programming language.",
"\n *\n * See the appropriate edition of\n * <cite>The Java™ Language Specification</cite>\n * for information about a particular source version.",
"\n *\n * <p>Note that additional source version constants will be added to\n * model future releases of the language.",
"\n *\n * @author Joseph D. Darcy\n * @author Scott Seligman\n * @author Peter von der Ahé\n * @since 1.6\n */\npublic enum SourceVersion {\n /*\n * Summary of language evoluation\n * 1.1: nested classes\n * 1.2: strictfp\n * 1.3: no changes\n * 1.4: assert\n * 1.5: annotations, generics, autoboxing, var-args...\n * 1.6: no changes\n */\n\n /**\n * The original version.",
"\n *\n * The language described in\n * <cite>The Java™ Language Specification, First Edition</cite>.",
"\n */\n RELEASE_0,\n\n /**\n * The version recognized by the Java Platform 1.1.",
"\n *\n * The language is {@code RELEASE_0} augmented with nested classes as described in the 1.1 update to\n * <cite>The Java™ Language Specification, First Edition</cite>.",
"\n */\n RELEASE_1,\n\n /**\n * The version recognized by the Java 2 Platform, Standard Edition,\n * v 1.2.",
"\n *\n * The language described in\n * <cite>The Java™ Language Specification,\n * Second Edition</cite>, which includes the {@code\n * strictfp} modifier.",
"\n */\n RELEASE_2,\n\n /**\n * The version recognized by the Java 2 Platform, Standard Edition,\n * v 1.3.",
"\n *\n * No major changes from {@code RELEASE_2}.",
"\n */\n RELEASE_3,\n\n /**\n * The version recognized by the Java 2 Platform, Standard Edition,\n * v 1.4.",
"\n *\n * Added a simple assertion facility.",
"\n */\n RELEASE_4,\n\n /**\n * The version recognized by the Java 2 Platform, Standard\n * Edition 5.0.",
"\n *\n * The language described in\n * <cite>The Java™ Language Specification,\n * Third Edition</cite>. ",
" First release to support\n * generics, annotations, autoboxing, var-args, enhanced {@code\n * for} loop, and hexadecimal floating-point literals.",
"\n */\n RELEASE_5,\n\n /**\n * The version recognized by the Java Platform, Standard Edition\n * 6.",
"\n *\n * No major changes from {@code RELEASE_5}.",
"\n */\n RELEASE_6,\n\n /**\n * The version recognized by the Java Platform, Standard Edition\n * 7.",
"\n *\n * @since 1.7\n */\n RELEASE_7;\n\n // Note that when adding constants for newer releases, the\n // behavior of latest() and latestSupported() must be updated too.",
"\n\n /**\n * Returns the latest source version that can be modeled.",
"\n *\n * @return the latest source version that can be modeled\n */\n public static SourceVersion latest() {\n return RELEASE_7;\n }\n\n private static final SourceVersion latestSupported = getLatestSupported();\n\n private static SourceVersion getLatestSupported() {\n try {\n String specVersion = System.getProperty(\"java.specification.version\");\n if (\"1.7\".equals(specVersion))\n return RELEASE_7;\n else if (\"1.6\".equals(specVersion))\n return RELEASE_6;\n } catch (SecurityException se) {}\n\n return RELEASE_5;\n }\n\n /**\n * Returns the latest source version fully supported by the\n * current execution environment. ",
" {@code RELEASE_5} or later must\n * be returned.",
"\n *\n * @return the latest source version that is fully supported\n */\n public static SourceVersion latestSupported() {\n return latestSupported;\n }\n\n /**\n * Returns whether or not {@code name} is a syntactically valid\n * identifier (simple name) or keyword in the latest source\n * version. ",
" The method returns {@code true} if the name consists\n * of an initial character for which {@link\n * Character#isJavaIdentifierStart(int)} returns {@code true},\n * followed only by characters for which {@link\n * Character#isJavaIdentifierPart(int)} returns {@code true}.",
"\n * This pattern matches regular identifiers, keywords, and the\n * literals {@code \"true\"}, {@code \"false\"}, and {@code \"null\"}.",
"\n * The method returns {@code false} for all other strings.",
"\n *\n * @param name the string to check\n * @return {@code true} if this string is a\n * syntactically valid identifier or keyword, {@code false}\n * otherwise.",
"\n */\n public static boolean isIdentifier(CharSequence name) {\n String id = name.toString();\n\n if (id.length() == 0) {\n return false;\n }\n int cp = id.codePointAt(0);\n if (!",
"Character.isJavaIdentifierStart(cp)) {\n return false;\n }\n for (int i = Character.charCount(cp);\n i < id.length();\n i += Character.charCount(cp)) {\n cp = id.codePointAt(i);\n if (!",
"Character.isJavaIdentifierPart(cp)) {\n return false;\n }\n }\n return true;\n }\n\n /**\n * Returns whether or not {@code name} is a syntactically valid\n * qualified name in the latest source version. ",
" Unlike {@link\n * #isIdentifier isIdentifier}, this method returns {@code false}\n * for keywords and literals.",
"\n *\n * @param name the string to check\n * @return {@code true} if this string is a\n * syntactically valid name, {@code false} otherwise.",
"\n * @jls 6.2 Names and Identifiers\n */\n public static boolean isName(CharSequence name) {\n String id = name.toString();\n\n for(String s : id.split(\"\\\\.\", -1)) {\n if (!",
"isIdentifier(s) || isKeyword(s))\n return false;\n }\n return true;\n }\n\n private final static Set<String> keywords;\n static {\n Set<String> s = new HashSet<String>();\n String [] kws = {\n \"abstract\", \"continue\", \"for\", \"new\", \"switch\",\n \"assert\", \"default\", \"if\", \"package\", \"synchronized\",\n \"boolean\", \"do\", \"goto\", \"private\", \"this\",\n \"break\", \"double\", \"implements\", \"protected\", \"throw\",\n \"byte\", \"else\", \"import\", \"public\", \"throws\",\n \"case\", \"enum\", \"instanceof\", \"return\", \"transient\",\n \"catch\", \"extends\", \"int\", \"short\", \"try\",\n \"char\", \"final\", \"interface\", \"static\", \"void\",\n \"class\", \"finally\", \"long\", \"strictfp\", \"volatile\",\n \"const\", \"float\", \"native\", \"super\", \"while\",\n // literals\n \"null\", \"true\", \"false\"\n };\n for(String kw : kws)\n s.add(kw);\n keywords = Collections.unmodifiableSet(s);\n }\n\n /**\n * Returns whether or not {@code s} is a keyword or literal in the\n * latest source version.",
"\n *\n * @param s the string to check\n * @return {@code true} if {@code s} is a keyword or literal, {@code false} otherwise.",
"\n */\n public static boolean isKeyword(CharSequence s) {\n String keywordOrLiteral = s.toString();\n return keywords.contains(keywordOrLiteral);\n }\n}\n"
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"\n\nWomen's Environment and Development Organization - WEDO is an international advocacy network actively working\nto transform society to achieve a healthy and peaceful planet\nwith social, political, economic and environmental justice for\nall through the empowerment of women, in all their diversity,\nand their equal participation with men in decision-making from\ngrassroots to global arenas."
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"VIDEO ABOVE\n\nSusie Polston said she called 911 and SCDNR responded. ",
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"The Oligopeptide Synthesis Facility is a shared resource that provides rapid and efficient peptide synthesis, protein sequencing and amino acid analysis services to Cancer Center members and other faculty in the Division of the Biological Sciences. ",
"The synthetic peptides can be used to investigate the functional domain of known proteins; to study the interaction of proteins with receptors, membranes or DNA; to separate gene products based on heterogeneity of small domains of the proteins and for the production of domain-specific antibodies. ",
"Sequence information for particular proteins is necessary for preparing oligonucleotide probes for the purpose of cloning cDNA; in study of posttranslational processing of both native and recombinant proteins and in many active site and chemical modification studies designed to define the structure-function properties of enzymes and other proteins. ",
"The programs and projects supported by this Facility represent a diverse, multi-disciplinary approach to obtaining an improved understanding of cancer biology, genetics, immunology, chromosomal abnormalities and therapeutic modalities. ",
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"Introduction {#s1}\n============\n\nThe endoplasmic reticulum (ER) is the site where proteins destined for the cell surface and endomembrane system enter the secretory pathway. ",
"Newly synthesized secretory and membrane-associated proteins undergo disulfide-bond formation and isomerization in the ER to yield correctly folded and assembled proteins. ",
"Under physiological condition, ER-protein load and protein-folding capacity achieves an equilibrium state. ",
"Changes in ER homeostasis due to increased protein synthesis, accumulation of misfolded proteins, or alterations in the calcium or redox balance of ER lead to a condition called ER stress [@pone.0086753-Lai1], [@pone.0086753-Hetz1]. ",
"To cope with this stress, the cells have developed an adaptive signaling pathway called the unfolded protein response (UPR) or ER stress response. ",
"If homeostasis is not restored, the UPR is chronically activated and leads to cell death (apoptosis) [@pone.0086753-Tabas1], [@pone.0086753-Szegezdi1].",
"\n\nAccumulating evidence indicates that pathological conditions that interfere with ER homeostasis will give rise to chronic activation of UPR, which contributes to the pathogenesis of many diseases, including neurodegenerative disorders, type 2 diabetes, atherosclerosis, liver disease, and cancer [@pone.0086753-Yoshida1]--[@pone.0086753-Hosoi1]. ",
"A more specific example of one such disorder is Alzheimer's disease (AD). ",
"AD is a progressive neurological disorder characterized by a decline in cognitive processes, eventually leading to dementia [@pone.0086753-Ozcan1], [@pone.0086753-Huang1], [@pone.0086753-Salminen1]. ",
"The hallmarks of this disease include the accumulation of extracellular amyloid-β (Aβ) peptides and intracellular aggregates of phosphorylated tau proteins, along with the perturbation of calcium homeostasis and neuronal death [@pone.0086753-Ittner1]. ",
"Recent reports have indicated that UPR is activated in the brain of patients with AD. ",
"There is also increased expression of the ER chaperone Grp78 (which is indicative of UPR activation) in the brains of AD patients [@pone.0086753-Hoozemans1]. ",
"Additionally, autopsy studies have revealed increased phosphorylated (activated) PERK, eIF2α, and IRE1 in the brains of patients with AD, compared to specimens from subjects without the disease. ",
"UPR-positive staining is localized to the neurons, and not glial cells, which is consistent with a role for ER stress in AD pathogenesis [@pone.0086753-Unterberger1].",
"\n\nDHCR24 (also known as hDiminuto/Seladin-1) is an enzyme that acts as a 3β-hydroxysteroid-Δ24 reductase, and its level has been found to decrease in the brain of AD patients. ",
"DHCR24 catalyzes the final step of cholesterol biosynthesis, which is the conversion of desmosterol to cholesterol [@pone.0086753-Waterham1]. ",
"In addition to its cholesterol-synthesizing activity, several biologically important activities of DHCR24 have also been reported. ",
"Overexpression of DHCR24 protects neuronal cells from apoptosis induced by hydrogen peroxide or Aβ [@pone.0086753-Greeve1]. ",
"Furthermore, DHCR24 also interacts with and induces the accumulation of p53 [@pone.0086753-Wu1]. ",
"It is thought that DHCR24 acts as an anti-apoptotic protein because reduced expression of the DHCR24 gene is associated with increased apoptosis of adrenocortical cells. ",
"We have previously cloned the DHCR24 gene, which is abundantly expressed in cortisol-producing adrenocortical adenomas [@pone.0086753-Sarkar1]. ",
"Using mouse embryonic fibroblast cells (MEFs) obtained from *DHCR24* ^−/−^ mice, we demonstrated that DHCR24 can directly scavenge H~2~O~2~, protecting these cells from oxidative-stress-induced apoptosis [@pone.0086753-Lu1]. ",
"Therefore, DHCR24 is a multifunctional protein, possessing both cholesterol-synthesizing and anti-apoptotic activities. ",
"However, the molecular mechanisms underlying its anti-apoptotic activity are not fully understood.",
"\n\nUtilizing the database obtained from Allen institute for brain science, we found that DHCR24 is mainly expressed in the anterior orbital gyrus and hippocampus area, suggesting that it has an important role in neuron function associated with learning and memory. ",
"A recombinant adenovirus driving the overexpression of DHCR24 (Ad-DHCR24) used in a previous study has shown that overexpression of DHCR24 protects MEFs from apoptosis initiated by the ER stress inducer, Tunicamycin [@pone.0086753-Lu1]. ",
"To elucidate the roles of DHCR24 in neuronal cells, the current study investigated the neuroprotective function of DHCR24 under ER stress, and found that DHCR24 protected neuronal cells from ER stress-induced apoptosis by attenuating ER stress signaling, possibly through scavenging intracellular reactive oxygen species and elevating cholesterol levels.",
"\n\nMaterials and Methods {#s2}\n=====================\n\nCell Culture {#s2a}\n------------\n\nN2A cells (American Type Culture Collection, Manassas, VA) were cultured in DMEM/F12 (high glucose) supplemented with 10% fetal bovine serum. ",
"In some experiments, cells were treated with 1 µM Akt inhibitor IV (Calbiochem, San Diego, CA). ",
"Cell images were obtained with a phase-contrast microscope (IMT-2, Olympus, Tokyo, Japan) equipped with a digital camera (PDMC II, Polaroid, Waltham, MA).",
"\n\nAnalysis of Adherent Cell Numbers and Apoptosis Assay {#s2b}\n-----------------------------------------------------\n\nDetached cells were removed from the culture plate by washing the plate once with medium. ",
"The adherent cells were trypsinized and collected by centrifugation. ",
"They were then stained with Trypan Blue using Trypan Blue-Staining Cell Viability Assay Kit (Biyuntian, Shanghai, China). ",
"The numbers of cells from different treatments were counted and compared.",
"\n\nDNA fragmentation was analyzed by the terminal deoxynucleotidyl transferase-mediated deoxyuridine triphosphate (dUTP)-biotin nick end labeling (TUNEL) method (Takara, Otsu, Japan) according to the manufacturer's protocol.",
"\n\nCellular apoptosis was assessed by *in situ* DNA fragmentation and immunocytochemistry-based caspase-3 assay. ",
"The procedure for immunocytochemical analysis was described previously [@pone.0086753-Lu1], [@pone.0086753-Lu2]. ",
"Briefly, after fixation and blocking, the cells were incubated with rabbit antibody directed against active caspase-3 (Sigma-Aldrich, St. Louis, Missouri, USA) followed by incubation with anti-rabbit IgG antibody conjugated to Alexa Fluor 568 (Molecular Probes, Eugene, OR). ",
"For the detection of Alexa fluor-568 fluorescence, the main beam splitter for excitation, the secondary beam splitter for emission, and barrier filter were set to 568 nm, 570 nm, and 585 nm long pass, respectively. ",
"Several images were captured with the same set of optical parameters. ",
"Densitometric analysis was performed using Multi Gauge software in LAS-1000 (Fuji Film).",
"\n\nDetermination of Intracellular Cholesterol {#s2c}\n------------------------------------------\n\nLipid was extracted by the method of Bligh and Dyer [@pone.0086753-BLIGH1]. ",
"Total sterol content in the lipid was determined by measuring the content of 3β-hydroxysterols using an enzymatic cholesterol assay kit (Roche Diagnostics, Mannheim, Germany).",
"\n\nWestern Blot Analysis {#s2d}\n---------------------\n\nWhole cell lysate for Western blot analysis was prepared as previously described [@pone.0086753-Lu1], [@pone.0086753-Lu2]. ",
"For Western blot analysis, whole cell lysate (50 µg protein/lane) was first subjected to SDS-PAGE using 10% gel. ",
"After that, the proteins in the gel were transferred onto polyvinylidene difluoride membrane (Amersham Pharmacia, Piscataway, NJ). ",
"The blot was probed with the appropriate primary antibody as described below, followed by incubation with horseradish peroxidase-conjugated anti-rabbit or anti-mouse IgG antibodies. ",
"Primary antibodies: Rabbit anti-phospho-p38 MAPK (T180/Y182), anti-caspase12, anti-cleaved caspase-3, anti-p-PERK, anti-p-Akt, anti-Akt and anti-phospho-JNK (T183/Y185) antibodies were purchased from Cell Signaling (Beverly, MA). ",
"Mouse monoclonal anti-Bip/GRP78 antibody was from BD Biosciences (Bedford, MA). ",
"Rabbit anti-CHOP (GADD153) and mouse anti-myc antibodies were from Santa Cruz Biotechnology (CA, USA). ",
"Rabbit anti-actin, anti-caspase-12, and anti-cleaved caspase-3 antibodies were from Sigma-Aldrich. ",
"Positive protein bands in the blot were visualized using enhanced chemiluminescence reagents (Pierce, Rockford, IL). ",
"The images of the blotted membranes were obtained by an LAS-1000 lumino-image analyzer (Fuji Film, Tokyo, Japan). ",
"Densitometric analysis was performed with the same instrument.",
"\n\nImmunocytochemical Analysis {#s2e}\n---------------------------\n\nConfocal laser scanning microscopy was performed as previously described [@pone.0086753-Lu3]. ",
"Briefly, after the fixation and blocking steps, the cells were incubated with a mixture of rabbit anti-caveolin-1 polyclonal antibodies (Santa Cruz Biotechnology, Santa Cruz, CA) and mouse anti-IGF-I β monoclonal antibody (BD Biosciences, Bedford, MA) for 60 minutes, and then with a mixture of anti-mouse IgG antibody conjugated to Alexa fluor-488 (Molecular Probes, Eugene, OR) and anti-rabbit IgG antibody conjugated to Alexa fluor-568 (Molecular Probes). ",
"In other experiments, the cells were incubated with antibody against cleaved caspase-3, full-length caspase-12, or CHOP followed by secondary antibody. ",
"Images were obtained using a confocal laser microscope (LSM510, Carl Zeiss, Jena, Germany). ",
"Several images were captured with the same set of optical parameters, and merged using Adobe photoshop software (ver. ",
"7.0.1, Adobe Systems, San Jose, CA). ",
"All images were taken with a 40/1.4 numerical aperture (NA) oil differential interference contrast Plan-Neofluar (Zeiss) using a 488-nm Ar/Kr laser line or a 543- HeNe laser line. ",
"Scanning speed and laser intensity were adjusted to avoid photobleaching of the fluorophores and damage to the cells. ",
"The microscope was equipped with a microenvironmental chamber to maintain a physiological condition. ",
"For quantification of fluorescence intensity, nonsaturated images were taken with a full-open pinhole. ",
"For multichannel imaging, each fluorescent dye was imaged sequentially in the frame-interlace mode to eliminate cross-talk between the channels. ",
"Three independent experiments were performed for Ad-lacZ- and Ad-DHCR24-infected cells, and 40 cells were randomly captured from a single group with the same set of optical parameters and their fluorescence intensity was measured and calculated to obtain an average value per cell.",
"\n\nAll image processing was performed using the Zeiss LSM 510 image examiner software.",
"\n\nMeasurement of ROS Production {#s2f}\n-----------------------------\n\nIntracellular ROS production was measured by a fluorescent-dye technique [@pone.0086753-Lu3]. ",
"N2A cells were cultured on glass cover-slips and then treated for 30 min with 20 µM 2′, 7′-dichlorofluorescin diacetate (DCFHDA, Molecular Probes) in phosphate buffered saline (PBS). ",
"The cover slips were fixed and mounted on a glass slide. ",
"To detect the fluorescence of 2′, 7′-dichlorofluorescein, the main beam splitter for excitation, the secondary beam splitter for emission, and barrier filter were set at 488 nm, 570 nm, and 505 nm long pass, respectively. ",
"Several images were captured with the same set of optical parameters. ",
"Densitometric analysis was performed using Multi Gauge software in LAS-1000 (Fuji Film). ",
"Three independent experiments were performed for Ad-lacZ and Ad-DHCR24 infected cells, and 40 cells were randomly captured from a single group with the same set of optical parameters and their fluorescence intensity was measured and calculated to average value per cell.",
"\n\nStatistical Analysis {#s2g}\n--------------------\n\nStatistical analysis was performed with ANOVA followed by Bonferroni's multiple t-tests, and statistical significances were considered at the *P*\\<0.05 level.",
"\n\nResults {#s3}\n=======\n\nAdenovirus-delivered DHCR24 Protects N2A Cells from Apoptosis Induced by ER Stress {#s3a}\n----------------------------------------------------------------------------------\n\nWe have previously constructed a recombinant adenovirus to drive the expression of DHCR24 tagged with a myc epitope at the C-terminus (Ad-DHCR24), and demonstrated that DHCR24 expressed from Ad-DHCR24 can protect mouse embryonic fibroblasts from oxidative stress-induced apoptosis [@pone.0086753-Lu1]. ",
"We first confirmed the transduction efficiency of Ad-DHCR24 in N2A cells by immunocytochemical analysis using an antibody against the myc tag. ",
"More than 85% transduction efficiency was achieved as revealed by the presence of DHCR24-myc in the cytoplasm of the infected cells ([Fig. ",
"1A and B](#pone-0086753-g001){ref-type=\"fig\"}). ",
"Adenovirus-driven expression of β-galactosidase (Ad-LacZ) was used as a negative control.",
"\n\n{#pone-0086753-g001}\n\nTo induce ER stress, we used tunicamycin (TM), an inhibitor of N-linked glycosylation, which is widely used as an ER stress inducer. ",
"We found that N2A cells overexpressing Ad-DHCR24 were resistant to TM. ",
"After 48 hours of TM exposure, these cells still survived, with only insignificant loss of cells ([Fig. ",
"1B and C](#pone-0086753-g001){ref-type=\"fig\"}). ",
"In contrast, Ad-LacZ-infected cells were progressively lost. ",
"TUNEL assay showed that 24 hours after treatment with TM, roughly 40% of N2A cells infected with Ad-LacZ underwent apoptosis in response to TM, whereas virtually all N2A cells infected with Ad-DHCR24 were resistant to apoptosis ([Fig. ",
"1 D and E](#pone-0086753-g001){ref-type=\"fig\"}). ",
"These results demonstrated that overexpression of DHCR24 induced by adenovirus could protect neural cells from ER-stress-induced apoptosis.",
"\n\nOverexpression of DHCR24 Inhibits Apoptotic Cell Signaling {#s3b}\n----------------------------------------------------------\n\nTo confirm and identify the relevant apoptotic signaling pathway affected by the overexpression of DHCR24, both Ad-DHCR24-infected and Ad-LacZ-infected N2A cells cultured on cover slips were subjected to immunocytochemical analysis using an antibody against the activated (cleaved) form of caspase-3. ",
"The detection of active caspase-3 is widely used to detect the presence of apoptosis in many cell lines. ",
"As shown in [Fig. ",
"2 A and B](#pone-0086753-g002){ref-type=\"fig\"}, the number of Ad-LacZ-infected cells displaying intense fluorescence signals was significantly higher compared to Ad-DHCR24-infected cells. ",
"This result suggested that overexpression of DHCR24 inhibited the activation of caspase-3, thus blocking the progression and onset of apoptosis.",
"\n\n![",
"DHCR24 suppresses caspase-mediated apoptotic signaling pathway.\\\nPanel A, N2A cells were infected with the indicated adenoviral construct for 48 hours and then exposed to TM for 48 hours, followed by immunocytochemical analysis. ",
"The cells were probed with mouse antibodies against the cleaved form of caspase-3, followed by anti-mouse IgG antibody conjugated to Alexa fluor-568. ",
"The nuclei were stained with Hoechst. ",
"Images were obtained with a confocal laser microscope. ",
"Scale bar, 50 µm. ",
"Panel B, The relative fluorescence signal intensities of Panel A were represented as percentage of Ad-LacZ. Values are expressed as mean ± S.D. (n = 3) \\*p\\<0.05 versus Ad-LacZ. Similar results were obtained from three separate experiments. ",
"Panel C, N2A cells were infected with the indicated adenoviral construct for 48 hours and then exposed to TM for 0 \\[TM(−)\\] or 48 hours \\[TM(+)\\], followed by immunocytochemical analysis. ",
"The cells were probed with mouse anti-caspase-12 antibody, followed by an anti-mouse IgG antibody conjugated to Alexa fluor-488. ",
"Images were obtained with a confocal laser microscopy. ",
"Scale bar, 100 µm. ",
"Panel D, The ratios of fluorescence intensity outside the nuclei to that inside the nuclei in Panel C images are represented. ",
"Values are expressed as mean ± S.D. (n = 3) \\*p\\<0.05 vs. Ad-LacZ. Similar results were obtained from three separate experiments. ",
"Panel E, N2A cells were infected with the indicated adenoviruses for 48 hours and then exposed to TM for 0 or 48 hours. ",
"Whole cell lysates (50 µg protein/lane) were subjected to Western blot analysis using anti-cleaved caspase-3, anti-caspase-12 (both full-length and cleaved form), and anti-actin antibodies.](pone.0086753.g002){#pone-0086753-g002}\n\nIt has been reported that caspase-12 is an ER stress-specific caspase that mediates ER stress-induced apoptosis, and is activated during the disruption of ER calcium homeostasis. ",
"Caspase-12 is co-localized to the ER and mediates cytotoxicity induced by Aβ [@pone.0086753-Nakagawa1], [@pone.0086753-Fujita1]. ",
"Thus, we investigated whether caspase-12 was activated under TM-induced ER stress. ",
"We found that the enzyme was mainly localized to the ER of both Ad-LacZ- and Ad-DHCR24-infected cells when these cells were cultured in normal growth medium ([Fig. ",
"2 C](#pone-0086753-g002){ref-type=\"fig\"} TM (−)) (under non-ER-stress conditions), consistent with previous reports. ",
"However, when these cells were exposed to TM, the fluorescence signal of caspase-12 was more diffuse in Ad-LacZ-infected cells (indicative of translocation from the ER to the cytoplasm) than in Ad-DHCR24-infected cells, where translocation was inhibited. ",
"This suggested that overexpression of DHCR24 probably inhibited the caspase-12-mediated apoptotic signaling pathway that was strongly activated by TM in the control cells ([Fig. ",
"2 C](#pone-0086753-g002){ref-type=\"fig\"} TM (+)).",
"\n\nThe difference in apoptotic pathway signaling exhibited by cells that did overexpress DHCR24 and cells that did not was confirmed by western blot using antibodies against both cleaved caspase-3 and full-length caspase-12. ",
"As shown in [Fig. ",
"2 E](#pone-0086753-g002){ref-type=\"fig\"}, cleaved caspase-3 was more abundant in Ad-LacZ infected cells than in Ad-DHCR24-infected cells 48 hours after TM treatment. ",
"Similar results were observed for the activation of caspase-12. ",
"Taken together, these results demonstrated that TM-induced ER stress led to the activation of caspase-mediated apoptosis, but this process could be blocked by overexpression of DHCR24.",
"\n\nOverexpression of DHCR24 Reduces/delays Activation of ER Stress-related Cell Signaling during UPR in N2A Cells {#s3c}\n--------------------------------------------------------------------------------------------------------------\n\nIt has been reported that Bip (immunoglobulin heavy-chain binding protein, Grp78) and CHOP (C/EBP homologous protein, CHOP) are upregulated and activated in response to ER stress [@pone.0086753-Hoozemans1], [@pone.0086753-Oyadomari1]. ",
"We next investigated whether there was any difference in the expression levels of Bip and CHOP between Ad-DHCR24-infected cells and control cells during TM-induced ER stress. ",
"Western blot analysis indicated that the levels of Bip and CHOP were upregulated in both Ad-DHCR24-infected and control cells after exposure to TM for 24 and 48 hours ([Figs. ",
"3A and 3B](#pone-0086753-g003){ref-type=\"fig\"}), confirming the presence of TM-induced ER stress. ",
"However, the upregulation of both Bip and CHOP was much weaker in Ad-DHCR24-infected cells. ",
"Even 48 hours after TM treatment, the level of DHCR24 in the cells was still strong, probably sufficient to protect the cells from TM-induced apoptosis. ",
"Immunocytochemical analysis showed that CHOP was almost un-detectable in both Ad-DHCR24- and Ad-LacZ-infected cells before TM treatments ([Fig. ",
"3C and D](#pone-0086753-g003){ref-type=\"fig\"}). ",
"Strong fluorescence signals were observed in the nuclei of most Ad-LacZ-infected cells in response to TM. ",
"In the case of Ad-DHCR24-infected cells, fewer cell nuclei exhibited similarly strong fluorescence signals after TM treatment. ",
"This suggested that overexpression of DHCR24 could suppress TM-induced translocation and activation of CHOP ([Fig. ",
"3 C, D and E](#pone-0086753-g003){ref-type=\"fig\"}).",
"\n\n{#pone-0086753-g003}\n\nProlonged activation of stress-related signaling can induce apoptosis [@pone.0086753-Tabas1], [@pone.0086753-Schroder1], [@pone.0086753-Chakrabarti1]. ",
"To further confirm the effect of DHCR24 overexpression on ER stress-related cell signaling, we investigated the activation of PERK, p-38 and JNK by western blot analysis. ",
"In the case of Ad-LacZ-infected cells, PERK was strongly activated as early as 3 hours after TM treatment ([Fig. ",
"4](#pone-0086753-g004){ref-type=\"fig\"}). ",
"However, the activation of PERK was delayed to 6 hours and was somewhat weaker in the case of Ad-DHCR24-infected cells. ",
"Similarly, the activation of p38 and JNK was significantly reduced in Ad-DHCR24-infected cells compared to the controls. ",
"Taken together, these results demonstrated that adenovirus-driven overexpression of DHCR24 protected neuronal cells from ER stress-induced apoptosis by delaying and reducing the activation of ER stress-related signaling at the early stage of ER stress.",
"\n\n{#pone-0086753-g004}\n\nDHCR24 Scavenges Intracellular ROS during ER Stress {#s3d}\n---------------------------------------------------\n\nSeveral studies have demonstrated that oxidative stress and the onset of ER stress are connected and that some antioxidants are able to reduce ER stress [@pone.0086753-Tabas1], [@pone.0086753-Ding1], [@pone.0086753-Malhotra1]. ",
"It has also been reported that loss of PERK can cause defects in the sensitivity to cell death under pathological conditions linked to ROS-mediated ER stress [@pone.0086753-Tabas1], [@pone.0086753-Verfaillie1]. ",
"We have previously reported that DHCR24 can protect mouse embryonic fibroblast from H~2~O~2~-induced apoptosis by scavenging intracellular ROS [@pone.0086753-Lu1]. ",
"Thus, we investigated whether DHCR24 could scavenge ROS precipitated by ER stress in neuronal cells. ",
"Intracellular ROS was detected by DCFHDA, which was used as a fluorescent probe. ",
"As shown in [Fig. ",
"5](#pone-0086753-g005){ref-type=\"fig\"}, the fluorescence signal was stronger when the cells were exposed to TM for 3 hours compared to zero time treatment in the case of Ad-LacZ-infected cells, suggesting that the excess intracellular ROS was generated by ER stress. ",
"However, compared to Ad-LacZ-infected cells, Ad-DHCR24-infected cells showed a significantly weaker signal when the cells were exposed to TM for 3 hours, indicating the presence of a weaker oxidative environment. ",
"These data suggested that overexpression of DHCR24 suppressed the excess generation of intracellular ROS induced by ER stress, and this is consistent with our previous study [@pone.0086753-Lu1].",
"\n\n{#pone-0086753-g005}\n\nElevated Cholesterol Levels Induced by Ad-DHCR24 May Contribute to the Neuroprotective Function of DHCR24 {#s3e}\n---------------------------------------------------------------------------------------------------------\n\nAs DHCR24 is involved in the conversion of desmosterol to cholesterol, we next measured the intracellular cholesterol levels of both Ad-LacZ and Ad-DHCR24-infected cells. ",
"The cholesterol level of Ad-DHCR24-infected cells was significantly elevated compared to that of Ad-LacZ-infected cells ([Fig. ",
"6A](#pone-0086753-g006){ref-type=\"fig\"}). ",
"Cholesterol in neuronal cell membranes plays an important role in maintaining the structure and function of caveolae/lipid rafts, the sites where many neuronal growth factors receptors (e.g., insulin-like growth factor (IGF-1) and neuron growth factor (NGF)) are located [@pone.0086753-Huo1], [@pone.0086753-Smart1]. ",
"We previously reported that cholesterol loss induced by double deletion of the *DHCRC24* gene can destroy the cellular localization of insulin receptor and impair the Akt-Bad cell survival signaling pathway in mouse embryonic fibroblasts [@pone.0086753-Lu2], [@pone.0086753-Lu4]. ",
"Thus the effect of DHCR24 overexpression on the structure of caveolae and location of IGF-1 receptor (IGF-1R) in N2A cells was examined by immunocytochemical analysis. ",
"As shown in [Fig. ",
"6B and C](#pone-0086753-g006){ref-type=\"fig\"}, the merging of green fluorescence (which represents caveolin 1(the molecular marker of caveolae)) and red fluorescence (which represents IGF-1R) yielded yellow fluorescence in Ad-LacZ-infected cells, confirming the localization of IGF-1R in the caveolae. ",
"The spots-like green and red signals were much stronger and more concentrated in Ad-DHCR24-infected cells compared to Ad-LacZ-infected cells. ",
"Western blot analysis was performed to evaluate the activation of Akt induced by overexpression of DHCR24. ",
"As shown in [Figs. ",
"6D and 6E](#pone-0086753-g006){ref-type=\"fig\"}, the phosphorylation of Akt was slightly stronger in Ad-DHCR24-infected cells, suggesting that overexpression of DHCR24 slightly induced the activation of Akt. ",
"Consistent with this, treatment of Ad-DHCR24-infected cells with the Akt inhibitor IV following TM exposure significantly reversed the neuron-protective effect provided by the overexpression of DHCR24 ([Fig. ",
"6 E](#pone-0086753-g006){ref-type=\"fig\"}). ",
"Taken together, these data suggested that overexpression of DHCR24 improved the structural organization of caveolae, as well as the localization of IGF-1R, and thereby enhanced the activation of IGF-1-Akt survival signaling.",
"\n\n{#pone-0086753-g006}\n\nOverexpression of DHCR24 Protects Neuronal Cells from Aβ-induced Apoptosis {#s3f}\n--------------------------------------------------------------------------\n\nFinally, we investigated the effects of DHCR24 overexpression on rescuing neuronal cells from Aβ-induced apoptosis. ",
"For both Ad-LacZ- and Ad-DHCR24-infected cells, the level of Bip was elevated at 24 and 48 hours following treatment with Aβ. ",
"This indicated that the ER stress occurred in response to Aβ. ",
"However, the induction of Bip in response to Aβ was more robust in Ad-LacZ-infected cells than in Ad-DHCR24-infected cells, both 24 and 48 hours after Aβ treatment ([Fig. ",
"7A and D](#pone-0086753-g007){ref-type=\"fig\"}). ",
"These results suggested that Aβ could induce ER stress in neuronal cells and overexpression of DHCR24 could suppress the induction of Bip induced by Aβ ([Fig. ",
"7 C and D](#pone-0086753-g007){ref-type=\"fig\"}).",
"\n\n{#pone-0086753-g007}\n\nDiscussion {#s4}\n==========\n\nThe present study demonstrated that overexpression of DHCR24 in N2A cells (a neuronal cell line) could rescue the cells from ER stress-induced apoptosis ([Fig. ",
"1](#pone-0086753-g001){ref-type=\"fig\"}). ",
"The anti-apoptotic action of DHCR24 has been suggested by previous reports, including our own work [@pone.0086753-Greeve1], [@pone.0086753-Sarkar1]. ",
"Greeve and *et al.* [",
"@pone.0086753-Greeve1] reported decreased expression of DHCR24 in the brain area affected by Alzheimer's disease. ",
"These investigators further showed that overexpression of DHCR24 protects neuronal cells from apoptosis induced by hydrogen peroxide and Aβ, strongly suggesting that DHCR24 may play a role in the scavenging of ROS produced by oxidative stress [@pone.0086753-Greeve1]. ",
"Recently, we demonstrated that DHCR24 protects MEFs from hydrogen peroxide-induced apoptosis through scavenging ROS in both *in vivo* and *in vitro* systems. ",
"We further showed that overexpression of DHCR24 can also protect MEFs from ER stress-induced apoptosis by scavenging the ROS generated by ER stress [@pone.0086753-Lu1]. ",
"The result of the current study showed that DHCR24 also protected neuronal cells from ER stress-induced apoptosis.",
"\n\nDHCR24 suppressed ER stress-induced apoptosis through suppressing the activations of caspase-3 and caspase-12 ([Fig. ",
"2](#pone-0086753-g002){ref-type=\"fig\"}), suggesting that the ER stress-specific apoptotic signaling pathway was attenuated by DHCR24. ",
"Furthermore, induction of both Bip and CHOP, the two common ER stress-related molecular markers, was also reduced by the action of DHCR24 ([Fig. ",
"3](#pone-0086753-g003){ref-type=\"fig\"}), with the consequence of less severe stress levels in Ad-DHCR24-infected cells compared to control cells. ",
"This phenomenon was supported by the data obtained from western blot analysis using antibody against p-PERK, p-p38 or p-JNK ([Fig. ",
"4](#pone-0086753-g004){ref-type=\"fig\"}). ",
"This effect of DHCR24 on stress-related signaling pathway may be due in part to its ability to scavenge ROS, as well as its role in cholesterol biosynthesis. ",
"ROS generation in Ad-DHCR24-infected N2A cells was much weaker than in control N2A cells 3 hours and 24 hours after exposure to TM ([Fig. ",
"5](#pone-0086753-g005){ref-type=\"fig\"}), consistent with our previous study showing that DHCR24 can scavenge the ROS generated in MEFs under ER stress [@pone.0086753-Lu1]. ",
"It is well accepted that oxidative stress and ER stress are the common pathological mechanisms in AD [@pone.0086753-Bhandary1]. ",
"Aβ oligomers have recently been shown to accumulate in induced pluripotent stem cells (iPSC)-derived neurons and astrocytes from patients with a familial amyloid precursor protein (APP)-E693Δ mutation and sporadic AD, leading to ER and oxidative stress [@pone.0086753-Kondo1]. ",
"Here, we also demonstrated that overexpression of DHCR24 could protect N2A cells from hydrogen peroxide-induced apoptosis (data not shown). ",
"Together with the findings reported by other investigators, we believe that DHCR24 could protect neuronal cells from both oxidative stress and ER stress-induced apoptosis. ",
"This conclusion is further supported by the data presented in the present study ([Fig. ",
"7](#pone-0086753-g007){ref-type=\"fig\"}), which showed that Aβ oligomers could induce apoptosis and the expression of Bip in control N2A cells, whereas both events were suppressed in Ad-DHCR24-infected cells.",
"\n\nRecently, a new theory on AD has emerged, which considers AD as a third type of diabetes. ",
"This opinion is supported by a number of studies, especially since insulin-related cell survival signaling was shown to be downregulated in the affected areas of the brains of AD patients [@pone.0086753-Steen1]--[@pone.0086753-Gasparini1]. ",
"Additionally, many studies have reported abnormal insulin-signaling or insulin-like growth factor I (IGF-1) signaling pathways in AD-associated case studies or cellular experiments. ",
"Further support for this comes from clinical study revealing the effectiveness of long-trial intranasal insulin therapy for patients with amnestic mild cognitive impairment and patients with AD [@pone.0086753-Craft1], [@pone.0086753-Dandona1]. ",
"DHCR24 protects MEFs from serum withdrawal-induced apoptosis through its function in cholesterol biosynthesis and maintenance of the structure of caveolae and insulin-Akt-Bad signaling [@pone.0086753-Lu2]. ",
"We have also reported that IGF-1R localized in the caveolae in neuronal cells and cholesterol are necessary for the neuron-protective function of IGF-1 [@pone.0086753-Lu4]. ",
"Therefore, we believe that DHCR24 could also protect neurons from apoptosis through its involvement in cholesterol biosynthesis and by facilitating the structural formation and function of caveolae. ",
"This belief was supported by the fact that the cholesterol level of N2As that overexpressed DHCR24 was significantly elevated compared to the cholesterol level of the controls ([Fig. ",
"6 A](#pone-0086753-g006){ref-type=\"fig\"}). ",
"In addition, overexpression of DHCR24 also facilitated the structure and the colocalization of IGF-1Rwith caveolae ([Fig. ",
"6 B and C](#pone-0086753-g006){ref-type=\"fig\"}). ",
"Taken together, our results strongly supported the function of DHCR24 in maintaining the structure of caveolae and facilitating the IGF-1-mediated cell survival signaling in neuronal cells.",
"\n\nIn conclusion, the present study demonstrated that DHCR24 protected neuronal cells from ER stress-induced apoptosis, possibly through scavenging ROS and improving the function of membrane cholesterol. ",
"The neuron-protective function of DHCR24 is generally accepted, although its role as an AD indicator has recently been challenged. ",
"Through data obtained from microarray and RT-PCR experiments, Sharpe LJ and *at el.* [",
"@pone.0086753-Sharpe1] have pointed out that the expression of DHCR24 is not decreased in the affected brain area of AD patients. ",
"This is conflicting with an earlier report by Greeve and *at el*. [",
"@pone.0086753-Greeve1]. ",
"These authors also suggested that the relationship between AD and DHCRC24 might be linked to its cholesterol biosynthesis function. ",
"We believe that regardless of the expression level of DHCR24 in AD patients, its neuroprotective function is apparent, possibly through multiple mechanisms, including cholesterol biosynthesis. ",
"The expression of DHCR24 is thought to decrease with ages. ",
"In the APPSLxPS1 mutant mouse model of AD, the levels of desmosterol in the cerebellum of nine-months-old animals were found to increase compared to control. ",
"The level of desmosterol in the hippocampus of 21-months-old mutant can increase by as much as 600% compared to the level of desmosterol in the control. ",
"By this age, there would be abundant deposits of amyloid, accompanied by decreased expression of DHCR24 [@pone.0086753-Vanmierlo1]. ",
"Interestingly, in that particular study, some of the AD patients who carry certain DHCR24 polymorphism (rs718265 GG) have lower levels of Aβ~42~, indicating the DHCR24 gene may be associated with AD risk [@pone.0086753-Lamsa1]. ",
"We believe that the reduced neuroprotective function of DHCR24 would arise as a result of either decreased expression of DHCR24 or polymorphism in the DHCR24 gene, either of which can be an important factor in the pathogenesis of AD.",
"\n\n[^1]: **Competing Interests:**The authors have declared that no competing interests exist.",
"\n\n[^2]: Conceived and designed the experiments: XL BG. ",
"Performed the experiments: YL DJ WW. ",
"Analyzed the data: SC DS. ",
"Contributed reagents/materials/analysis tools: TL XQ. ",
"Wrote the paper: XL. ",
"Corrected the revised manuscript and rebuttal letter: AKC.",
"\n"
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"Leave a Testimonial\n\nThank you for allowing us to provide you with service. ",
"Please complete this form and your comment will be added to our testimonial page. ",
"If for any reason you do not wish to be on our testimonial page please let us know.",
"\n\nSpecial Discounts\n\nContact Us Now\n\nFree Estimate\n\nLawson Family Plumbing can beat almost any\ncompetitors estimate by up to 10% or more in\nsome cases. ",
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"Buddy Holly Center\n\nThe Buddy Holly Center is a performance and visual arts center in Lubbock, Texas, dedicated to Buddy Holly as well as the music of Lubbock and West Texas more broadly. ",
"The building in which it is located opened as the city's Fort Worth and Denver South Plains Railway depot in 1928. ",
"In 1996, the City of Lubbock obtained a sizable collection of Holly-related artifacts from his estate, and the next year it purchased the former depot. ",
"In 1999, the new Buddy Holly Center opened as the home of the newly acquired Buddy Holly collection as well as a replacement for the city's Fine Arts Center, which had been established in 1984.",
"\n\nThe Center features a permanent gallery for the Buddy Holly collection, which showcases artifacts and documents from Holly's childhood as well as his professional career, as well as the Texas Musician Hall of Fame, the Lubbock Fine Arts Gallery, and three additional visual arts galleries that host traveling exhibits. ",
"In 2002, an oversized sculpture of Holly's distinctive horn-rimmed glasses was installed outside the Center's main entrance, and in 2013 Crickets drummer Jerry Allison's restored house was relocated to the site and opened to the public.",
"\n\nHistory\n\nBuilding \n\nThe building in which the Buddy Holly Center is located opened as Lubbock's Fort Worth and Denver South Plains Railway depot in 1928. ",
"It was designed by Fort Worth-based architect Wyatt C. Hedrick in the Spanish Renaissance Revival style. ",
"The depot, which provided the city with both passenger and freight service, was the largest built on the Lubbock–Estelline branch of the Fort Worth and Denver, itself a subsidiary of the Burlington Route. ",
"It operated as a railroad station until 1953, when it was closed and subsequently used for salvaging and warehousing. ",
"In 1976, the building was reopened as the Depot Restaurant, which is regarded by the City of Lubbock as \"one of the first successful examples of adaptive use in the city\". ",
"In 1979, it became the first structure to be designated a Lubbock Historic Landmark by the Lubbock City Council, and it was added to the National Register of Historic Places in 1990. ",
"The restaurant closed in 1997, and later that year the city purchased the former depot.",
"\n\nCenter \n\nIn 1984, the City of Lubbock established a Fine Arts Center; although it developed successful artistic and popular programs, it was housed in \"an aging structure that was never intended for museum or gallery functions\". ",
"In 1996, the city obtained a large number of Buddy Holly-related artifacts from his estate. ",
"The next year the city then purchased the recently closed Depot Restaurant, and began restoring, renovating, and expanding it into a facility that could both replace the Fine Arts Center and serve as a home for its newly acquired Buddy Holly collection. ",
"The original depot building was restored to its original specifications, and while the addition was largely built to match it, it also features a frieze with steel forms shaped like Holly's iconic Fender Stratocaster electric guitar.",
"\n\nThe new Buddy Holly Center, envisioned as a municipal performance and visual arts center dedicated to Holly as well as the music of Lubbock and West Texas more broadly, opened on September 3, 1999, four days before Holly's birthday. ",
"The guitar-shaped permanent gallery of the Buddy Holly collection is featured alongside the Center's other attractions: the Texas Musician Hall of Fame (which often features temporary exhibits dedicated to West Texas musicians), the Lubbock Fine Arts Gallery, and three additional visual arts galleries that host traveling exhibits.",
"\n\nIn May 2002, an oversized sculpture of Holly's distinctive horn-rimmed glasses created by Steve Teeters was installed next to the Center's main entrance. ",
"In 2010, the western wing of the Center underwent a major renovation that resulted in the replacement of doors and windows, the repointing of exterior bricks and stone, new flooring, lighting, and paint for the interior gallery space, the removal of freestanding exhibit walls, the addition of black-out shades to windows to protect artifacts from exposure to sunlight, and a new permanent education space.",
"\n\nCrickets drummer Jerry Allison's house, which had previously been restored and then relocated to the site of the Buddy Holly Center, opened to the public for the first time on September 7, 2013. ",
"The structure, which was where Allison and Holly wrote many of their songs, is available for tours.",
"\n\nCollection \n\nThe Buddy Holly Center's Buddy Holly collection, which the City of Lubbock acquired in 1996, is headlined by the Fender Stratocaster that Holly played during his final concert and the pair of glasses that he was wearing at the time of his death. ",
"It also includes numerous other artifacts from his career as a professional musician, including a recording microphone, performing outfits, a guitar strap customized by Holly himself, and numerous albums. ",
"Furthermore, the collection includes numerous photographs (both promotional and personal in nature) as well as documents, such as postcards, fan mail, original tour itineraries, and business cards. ",
"A handwritten letter from Holly to A.V. Bamford and correspondence with Decca Records are among the numerous letters in the collection. ",
"Various artifacts and documents from Holly's childhood, including homework assignments, a slingshot, a leather-crafting kit, and his personal collection of 45-rpm records are also on public display.",
"\n\nThe Center also has Holly's 1958 Ariel Cyclone motorcycle on long-term loan from Lubbockite George McMahan; after Holly's death, the motorcycle was owned by country musicians Waylon Jennings and Jessi Colter before being sold at auction.",
"\n\nExhibits and events \n\nTemporary exhibits hosted by the Buddy Holly Center have included visual arts exhibitions showcasing subjects ranging from underwater photography to watercolors from children's literature.",
"\n\nIn October 2014, Paul McCartney played a concert at the Center, during which he recounted Holly's musical influence on the Beatles. ",
"It also hosts an annual Summer Showcase Concert Series, which offers free performances in its Meadows Courtyard from May to August featuring bands of various musical genres, including rock, rhythm and blues, funk, soul, country, metal, and mariachi. ",
"In August 2015, Latin roots rock musician Patricia Vonne played a concert in the Meadows Courtyard.",
"\n\nThe Buddy Holly Center offers free admission, as well as free trolley tours of various Holly-related sites in Lubbock, on February 3, \"The Day the Music Died\" (the day Holly was killed in an airplane crash along with Ritchie Valens and J.P. \"The Big Bopper\" Richardson). ",
"The Center has also waived admission fees for other occasions, including for its participation in the City of Lubbock's First Friday Art Trail (on multiple occasions, including in August 2011, May 2015, and August 2015), as well as on the occasion of its 15th anniversary in September 2014.",
"\n\nThe Center has additionally hosted various other community events, including workshops on wire drawings, a Music, Art and Drama Camp open to children between the ages of 8 and 12, and various Day of the Dead events, including an art gallery, a family art workshop, and a concert by a Tejano band.",
"\n\nSee also\n List of music museums\n\nReferences\n\nExternal links \n \n \n\nCenter\nCategory:Music museums in the United States\nCategory:Art museums in Texas\nCategory:Museums in Lubbock, Texas\nCategory:Former railway stations in the United States\nCategory:1999 establishments in Texas"
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"\"You're taking eighth-graders and trying to mold their minds with material that is highly confrontational and controversial,\" parent Angela McMillan said at Wednesday's meeting of the Edison Park Local School Council. ",
"View Full Caption DNAinfo/Alex Nitkin\n\nEDISON PARK — Parents and teachers at Edison Park Elementary School, 6200 N. Olcott Ave., ",
"grappled Wednesday night with how to teach lessons on police misconduct to classes full of students whose parents patrol the streets every day.",
"\n\nLast month, CPS leaders announced that eighth-graders and high school sophomores in public schools across the city will learn this year about disgraced former Chicago Police Cmdr. ",
"Jon Burge, who led a group of rogue officers known as the \"midnight crew\" in torturing suspects to force confessions from 1972-1991.",
"\n\nThe new curriculum was developed as part of a sweeping reparations package approved by city leaders in 2015, when Mayor Rahm Emanuel and the City Council voted to set aside $5.5 million for the more than 100 Burge torture victims.",
"\n\nRELATED: A Police Torture Lesson Plan: Here's What CPS Kids Will Learn About Burge\n\nBut some parents on the city's Far Northwest Side, where thousands of officers hang up their hats when their shifts are over, fear their kids may be subjected to a wholesale lashing of the Chicago Police Department.",
"\n\nAngela McMillan, whose daughter just started eighth grade at Edison Park Elementary, said she was \"infuriated, appalled and disgusted\" over a curriculum she called \"out of place and out of context.\"",
"\n\n\"You're taking eighth-graders and trying to mold their minds with material that is highly confrontational and controversial,\" McMillan said during Wednesday's meeting of the Edison Park Local School Council. \"",
"It's contradictory to how they live their personal lives with their families, where they eat dinner every night and celebrate Christmas ... I think it's deplorable.\"",
"\n\nMcMillan asked the LSC for a way to opt her daughter out of the program, but the Burge material is embedded in the district's core social studies curriculum, Edison Park Principal Jeffrey Finelli told her.",
"\n\n\"It would be a little like saying 'I don't like quadratic equations, so I'm going to opt out of algebra,'\" Finelli said.",
"\n\nRELATED: Police Union: CPS Must Tell 'Full Truth' About Burge Torture Era\n\nSocial studies teacher Emily Skowronek walked parents through the 116-page curriculum, which joins lessons on Japanese internment and Jim Crow laws as part of a larger \"civil rights\" unit taught to eighth-graders every year.",
"\n\nSkowronek will spark discussions among students about how to improve police-community relations going forward, but she'll leave the Burge episode squarely in the past, she said.",
"\n\n\"There are a lot of bad apples in every professions, and we'll try to portray that to the kids,\" Skowronek said. \"",
"Just because there were a few people who did bad things, that doesn't mean the whole department is bad.\"",
"\n\nParents like Leticia Kaner, a 19-year police veteran, questioned why students would learn such an ugly chapter from the department's history without including lessons on the sweeping Police Department reforms enacted in the last year.",
"\n\n\"The whole curriculum is really one-sided, especially when our general orders can change in the blink of an eye,\" Kaner said. \"",
"Civilians don't get how many changes we're going through as police officers right now.\"",
"\n\nThe eighth-grade civil rights unit won't be taught until spring, and in the meantime administrators at Edison Park will work with other Far Northwest Side schools to iron out a common method of teaching the sensitive subject, they said.",
"\n\n\"We know our students and know how we want to approach this, but what that looks like right now is unanswered,\" Finelli told parents Wednesday. \"",
"We know what the curriculum says, but as for how it’s carried out, we’ll be better informed as the year goes on with more discussion.\""
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"[Stereum hirsutum (Wild) Pers. ",
"action in dye degradation].",
"\nStereum hirsutum, a white rot fungus, has a good growth in solid state fermentation. ",
"This was carried on with wheat bran, soy bran and a mixture of both. ",
"Mycelia grown on soy bran showed the highest decolorization activity on Ponceau 2R (xylidine), indigo carmine and malachite green. ",
"Optimal relationship between decolorization and detoxification of malachite green was 30 g of fresh weight (mycelia plus substrate) in 500 ml malachite green solution, 42 U/l of laccase was measured in this solution. ",
"Decolorization was carried on without the addition either of nutrients or mediators. ",
"Conditions for maximal decolorization did not agree with those for maximal ligninolytic enzyme production, but effectiveness of laccase activity on decolorization was evidenced by electrophoretic analysis, that allowed laccase identification and its decolorization activity in gels stained with indigo carmine and malachite green, with ABTS as mediator. ",
"Detoxification was assayed using the sensible fungus Phanerochaete chrysosporium."
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"It appears that there are several wills in the late 19th century for Adams County, Illinois, that were microfilmed twice.",
"\n\nWe’re using that will for the next issue of Casefile Clues. ",
"I need to make certain that I am precise in my citation so that I cite the correct filming of these records as both filmings have been digitized by Ancestry.com."
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", 19164100, 23955126?",
"\n4791026*z - 4\nWhat is the s'th term of 27801, 55308, 82815?",
"\n27507*s + 294\nWhat is the f'th term of -347, -4335, -15689, -38093, -75231, -130787, -208445, -311889?",
"\n-614*f**3 + f**2 + 307*f - 41\nWhat is the x'th term of 1065, 2165, 3957, 6435, 9593, 13425?",
"\n-x**3 + 352*x**2 + 51*x + 663\nWhat is the i'th term of -2304, -2309, -2374, -2529, -2804, -3229?",
"\n-5*i**3 + 30*i - 2329\nWhat is the p'th term of -22615, -45240, -67863, -90484, -113103, -135720, -158335?",
"\np**2 - 22628*p + 12\nWhat is the m'th term of -3033798, -6067568, -9101352, -12135156, -15168986, -18202848, -21236748?",
"\n-m**3 - m**2 - 3033760*m - 36\nWhat is the m'th term of 261993859, 261993861, 261993863?",
"\n2*m + 261993857\nWhat is the j'th term of 17258534, 17258539, 17258544, 17258549, 17258554?",
"\n5*j + 17258529\nWhat is the o'th term of 6332122, 25328494, 56989116, 101313988?",
"\n6332125*o**2 - 3*o\nWhat is the m'th term of 3641697, 7283387, 10925077, 14566767, 18208457, 21850147?",
"\n3641690*m + 7\nWhat is the p'th term of -4407, -18443, -41845, -74619, -116771, -168307, -229233?",
"\n-p**3 - 4677*p**2 + 2*p + 269\nWhat is the g'th term of -19289363, -19289347, -19289331, -19289315, -19289299?",
"\n16*g - 19289379\nWhat is the c'th term of 988, 4035, 7082, 10129?",
"\n3047*c - 2059\nWhat is the s'th term of -14444, -15360, -16892, -19046, -21828?",
"\n-s**3 - 302*s**2 - 3*s - 14138\nWhat is the q'th term of -40, 425, 1688, 4145, 8192, 14225, 22640?",
"\n66*q**3 + 3*q**2 - 6*q - 103\nWhat is the f'th term of 17985, 18013, 18041?",
"\n28*f + 17957\nWhat is the n'th term of 5954966, 11909932, 17864906, 23819894, 29774902, 35729936?",
"\nn**3 - 2*n**2 + 5954965*n + 2\nWhat is the y'th term of -674595, -674599, -674603?",
"\n-4*y - 674591\nWhat is the u'th term of -59506396, -59506398, -59506400?",
"\n-2*u - 59506394\nWhat is the q'th term of -834, -972, -1148, -1362, -1614?",
"\n-19*q**2 - 81*q - 734\nWhat is the o'th term of -6882319, -6882330, -6882341, -6882352, -6882363, -6882374?",
"\n-11*o - 6882308\nWhat is the n'th term of 78572, 155903, 233234, 310565?",
"\n77331*n + 1241\nWhat is the z'th term of -79, -1072, -2577, -4468, -6619, -8904, -11197, -13372?",
"\n21*z**3 - 382*z**2 + 6*z + 276\nWhat is the c'th term of -313898, -2511972, -8478382, -20097296, -39252882, -67829308, -107710742, -160781352?",
"\n-314028*c**3 + 122*c + 8\nWhat is the m'th term of -105601, -105502, -105403, -105304, -105205?",
"\n99*m - 105700\nWhat is the c'th term of -3870340, -7740691, -11611038, -15481381?",
"\n2*c**2 - 3870357*c + 15\nWhat is the w'th term of 10176, 10108, 9974, 9774, 9508, 9176, 8778?",
"\n-33*w**2 + 31*w + 10178\nWhat is the k'th term of -2007298, -4014772, -6022246, -8029720, -10037194, -12044668?",
"\n-2007474*k + 176\nWhat is the m'th term of -4748003, -4748010, -4748021, -4748036?",
"\n-2*m**2 - m - 4748000\nWhat is the y'th term of 2029, 1618, 629, -1226, -4235?",
"\n-48*y**3 - y**2 - 72*y + 2150\nWhat is the r'th term of 382, 2541, 7982, 18211, 34734, 59057, 92686?",
"\n251*r**3 + 135*r**2 - 3*r - 1\nWhat is the m'th term of -3109574, -6219135, -9328694, -12438251, -15547806?",
"\nm**2 - 3109564*m - 11\nWhat is the h'th term of -2379, -10260, -23377, -41730?",
"\n-2618*h**2 - 27*h + 266\nWhat is the y'th term of -19299574, -38599148, -57898722, -77198296, -96497870?",
"\n-19299574*y\nWhat is the i'th term of 204674, 204704, 204774, 204902, 205106, 205404?",
"\n3*i**3 + 2*i**2 + 3*i + 204666\nWhat is the s'th term of 8061, 15203, 21467, 26853, 31361, 34991?",
"\n-439*s**2 + 8459*s + 41\nWhat is the t'th term of 1348, 110, -1124, -2354, -3580, -4802, -6020?",
"\n2*t**2 - 1244*t + 2590\nWhat is the a'th term of -2227, -5910, -9603, -13312, -17043, -20802, -24595, -28428?",
"\n-a**3 + a**2 - 3679*a + 1452\nWhat is the j'th term of -118943, -475740, -1070397, -1902908, -2973267, -4281468, -5827505?",
"\nj**3 - 118936*j**2 + 4*j - 12\nWhat is the u'th term of -9396827, -9396824, -9396821, -9396818, -9396815?",
"\n3*u - 9396830\nWhat is the z'th term of -23659517, -47319040, -70978563?",
"\n-23659523*z + 6\nWhat is the n'th term of -1910, -3833, -5692, -7463, -9122?",
"\n4*n**3 + 8*n**2 - 1975*n + 53\nWhat is the c'th term of -195998, -784023, -1764074, -3136151, -4900254, -7056383, -9604538?",
"\n-196013*c**2 + 14*c + 1\nWhat is the v'th term of -3168, -6303, -9452, -12615, -15792, -18983?",
"\n-7*v**2 - 3114*v - 47\nWhat is the i'th term of 414, 3125, 10558, 25101, 49142, 85069, 135270, 202133?",
"\n398*i**3 - 27*i**2 + 6*i + 37\nWhat is the v'th term of 8376654, 8376650, 8376646, 8376642, 8376638?",
"\n-4*v + 8376658\nWhat is the b'th term of 12682739, 12682737, 12682735?",
"\n-2*b + 12682741\nWhat is the f'th term of 126996, 126963, 126932, 126903?",
"\nf**2 - 36*f + 127031\nWhat is the y'th term of -4229, -10524, -16817, -23108, -29397?",
"\ny**2 - 6298*y + 2068\nWhat is the t'th term of -1644, -8307, -26286, -61239, -118824, -204699, -324522?",
"\n-943*t**3 - 62*t - 639\nWhat is the c'th term of -7371, -6616, -5357, -3594, -1327?",
"\n252*c**2 - c - 7622\nWhat is the j'th term of -794013, -794219, -794419, -794607, -794777, -794923, -795039?",
"\nj**3 - 3*j**2 - 204*j - 793807\nWhat is the q'th term of -1016, -1776, -2024, -1502, 48?",
"\n43*q**3 - 2*q**2 - 1055*q - 2\nWhat is the d'th term of 34135, 36264, 38407, 40570, 42759, 44980?",
"\nd**3 + d**2 + 2119*d + 32014\nWhat is the n'th term of -6565, -6592, -6637, -6700, -6781?",
"\n-9*n**2 - 6556\nWhat is the g'th term of -32784, -32929, -33074, -33219, -33364?",
"\n-145*g - 32639\nWhat is the t'th term of -130062, -124467, -118872?",
"\n5595*t - 135657\nWhat is the v'th term of -1372, -1120, -880, -652?",
"\n-6*v**2 + 270*v - 1636\nWhat is the f'th term of -726460, -726484, -726508, -726532, -726556, -726580?",
"\n-24*f - 726436\nWhat is the p'th term of -830, -1355, -1880?",
"\n-525*p - 305\nWhat is the m'th term of -54610, -54689, -54770, -54853, -54938?",
"\n-m**2 - 76*m - 54533\nWhat is the r'th term of -539, -2106, -3659, -5192, -6699?",
"\nr**3 + r**2 - 1577*r + 1036\nWhat is the v'th term of 25957, 103895, 233813, 415711, 649589, 935447?",
"\n25990*v**2 - 32*v - 1\nWhat is the x'th term of -352039, -352054, -352065, -352072, -352075?",
"\n2*x**2 - 21*x - 352020\nWhat is the g'th term of -8055661, -16111262, -24166863, -32222464, -40278065?",
"\n-8055601*g - 60\nWhat is the n'th term of 4179036, 4178967, 4178894, 4178817, 4178736?",
"\n-2*n**2 - 63*n + 4179101\nWhat is the w'th term of 65127, 130221, 195279, 260301, 325287?",
"\n-18*w**2 + 65148*w - 3\nWhat is the m'th term of -1488746, -1488743, -1488740, -1488737?",
"\n3*m - 1488749\nWhat is the u'th term of -1442, -666, 1478, 5686, 12654?",
"\n116*u**3 - 12*u**2 - 1546\nWhat is the l'th term of -22334, -22827, -23656, -24827, -26346, -28219, -30452?",
"\n-l**3 - 162*l**2 - 22171\nWhat is the n'th term of -16318, -129928, -438298, -1038808, -2028838, -3505768, -5566978, -8309848?",
"\n-16230*n**3 - 88\nWhat is the a'th term of 11266, 8511, 5756, 3001, 246?",
"\n-2755*a + 14021\nWhat is the d'th term of 285657, 571236, 856777, 1142286, 1427769, 1713232?",
"\nd**3 - 25*d**2 + 285647*d + 34\nWhat is the j'th term of -13224397, -26448792, -39673187?",
"\n-13224395*j - 2\nWhat is the n'th term of -185177, -183098, -181019, -178940, -176861?",
"\n2079*n - 187256\nWhat is the w'th term of -113674, -227623, -341572?",
"\n-113949*w + 275\nWhat is the v'th term of 48292689, 48292686, 48292683, 48292680?",
"\n-3*v + 48292692\nWhat is the v'th term of 6148, 4789, 2526, -641, -4712, -9687?",
"\n-452*v**2 - 3*v + 6603\nWhat is the i'th term of 262294, 262671, 263048, 263425?",
"\n377*i + 261917\nWhat is the k'th term of -129569590, -129569591, -129569592?",
"\n-k - 129569589\nWhat is the a'th term of -2192991, -4385984, -6578977, -8771970?",
"\n-2192993*a + 2\nWhat is the q'th term of -542, -19775, -39010, -58247, -77486, -96727?",
"\n-q**2 - 19230*q + 18689\nWhat is the f'th term of 66, 482, 1342, 2742, 4778?",
"\n16*f**3 + 126*f**2 - 74*f - 2\nWhat is the z'th term of -15112, -60280, -135560, -240952, -376456, -542072?",
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"\n2*a**3 - 45*a**2 + 690*a + 22\nWhat is the o'th term of 74287, 74268, 74229, 74164, 74067, 73932, 73753, 73524?",
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"Q:\n\nCan you extract types from template parameter function signature\n\nIs there a way that I can extract the types from a function signature in the form foo(bar) and get access to just foo or bar. ",
"So if I have the template:\ntemplate<typename signiture>\nclass type{\n};\n\nwhere signiture is foo(bar) and then have a function in the class that reads\nfoo function(bar b){\n //do stuff\n}\n\nI am interfacing with std::function and have found it more convenient to use the foo(bar) syntax instead of using multiple template params like so:\ntemplate<typename return_t,param_tps... args>\nclass type{\n return_t function(param_ps args...){\n return something;\n }\n};\n\nLet me know if i can clarify please?",
"\nThanks in Advance.",
"\nEDIT:\nfor clarification i am interested in a function with N number of parameters to be determined by whatever the specific class instance specifies.",
"\nEDIT 2:\nThe code that this question is based off of is as follows:\nusing std::function;\ntemplate <typename signiture>\nclass Observer;\n\ntemplate <typename return_t, typename ...args_t>\nclass Observer<return_t(args_t...)> {\nprotected:\n using signature = return_t(args_t...);\n typedef function<signature> func_t;\n ~Observer(){}\n func_t what_to_do;\npublic:\n Observer(Subject<signature>& subject,func_t what):what_to_do(what){\n subject.",
"Attach(what_to_do);\n }\n return_t operator()(args_t... args){\n what_to_do(args...);\n }\n};\n\nusing std::function;\nusing std::vector;\ntemplate <typename signature>\nclass Subject;\n\ntemplate <typename return_t,typename...param_tps>\nclass Subject<return_t(param_tps...)> {\n using signature=return_t(param_tps...);\npublic:\n void Attach(std::function<signature> o){\n obs.push_back(o);\n }\n void operator()(param_tps... params){\n for (typename vector<std::function<signature>>::const_iterator i=obs.begin(); i!=obs.end(); ++i) {\n (*i)(params...);\n }\n }\nprotected:\n ~Subject(){}\n vector<std::function<signature>> obs;\n\n};\n\nIt is an implementation of the Observer pattern that is non-virtual using std::function to glue things together between the two. ",
"I wanted to use the foo(bar) syntax because everything it is more conducive to the ease of use of the classes. ",
"The issue was converting the function type signature into the return type and the parameter types in order to specify the proper operator() in the subject class so that it can notify the observers with the correct data. ",
"\nThe changes that were made were based upon the example given below as follows:\ntemplate<typename t>struct type;\ntemplate<typename R,typename... args_t>\nstruct type<R(args_t...)>{\n//use R and args_t as needed\n}\n\nThanks to all who helped.",
"\n\nA:\n\nHere is a very basic solution that works for functions accepting one parameter( it seems you are placing this constraint in the question, but a generalized solution is quite easy to provide, as shown in the following):\ntemplate<typename S>\nstruct type; // You can leave this undefined, because the template is\n // supposed to be instantiated with a function type, and\n // that is matched by the specialization below.",
"\n\ntemplate<typename R, typename Arg>\nstruct type<R(Arg)>\n{\n // Just use R and Args as you wish here..\n};\n\nHere is a possible example (live demo on Coliru):\n#include <type_traits>\n\ntemplate<typename S>\nstruct signature;\n\ntemplate<typename R, typename Arg>\nstruct signature<R(Arg)>\n{\n using return_type = R;\n using argument_type = Arg;\n};\n\nint main()\n{\n using ret = signature<void(int)>::return_type;\n using arg = signature<void(int)>::argument_type;\n\n static_assert(std::is_same<ret, void>{}, \"!\");",
"\n static_assert(std::is_same<arg, int>{}, \"!\");",
"\n}\n\nIn case you are interested in a more general solution for the variadic case, this is probably what you're looking for:\n#include <tuple>\n\nstruct type; // You can leave this undefined, because the template is\n // supposed to be instantiated with a function type, and\n // that is matched by the specialization below.",
"\n\ntemplate<typename R, typename... Args>\nstruct type<R(Args...)>\n{\n // Just use R and Args... as you with here..\n};\n\nAnd here is a possible usage example (live demo on Coliru):\n#include <tuple>\n#include <type_traits>\n\ntemplate<typename S>\nstruct signature;\n\ntemplate<typename R, typename... Args>\nstruct signature<R(Args...)>\n{\n using return_type = R;\n using argument_type = std::tuple<Args...>;\n};\n\nint main()\n{\n using ret = signature<void(int, double)>::return_type;\n using arg1 = std::tuple_element_t<0, signature<void(int, double)>::argument_type>;\n using arg2 = std::tuple_element_t<1, signature<void(int, double)>::argument_type>;\n\n static_assert(std::is_same<ret, void>{}, \"!\");",
"\n static_assert(std::is_same<arg1, int>{}, \"!\");",
"\n static_assert(std::is_same<arg2, double>{}, \"!\");",
"\n}\n\n"
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"The commune is the most northerly of the cercle. ",
"It contains 11 villages in an area of 737 square kilometers and at the time of 2009 census had a population of 9,435. ",
"The village of Daban is 80 km northwest of Kati, the chef-lieu of the cercle.",
"\n\nReferences\n\nExternal links\n.",
"\n\nCategory:Populated places in Koulikoro Region\nCategory:Communes of Mali"
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"Helsingin Vuosaaressa sijaitsevassa Citymarketissa tapahtui torstaina erikoinen episodi.",
"\n\nVälikohtaus sattui Vuosaaren Citymarketissa. ",
"Kuvassa Malmin marketti. ",
"JOHN PALMéN\n\nIsä oli 4-vuotiaan ja kouluikäisen lapsensa kanssa kaupan leluosastolla torstaina puolen päivän aikaan, kun pienempi lapsista sai raivokohtauksen.",
"\n\n- Kyse oli 80 euroa maksavasta lelusta, jota hän ei saanut, perheen äiti kertoo Iltalehdelle.",
"\n\nÄiti kertoo, että 4-vuotiaan tytön mielenosoitus saattoi kuulua pitkälle, mutta hän ihmettelee kaupan toimintaa tilanteessa. ",
"Kyseessä oli kuitenkin lapsille suunnattu osasto.",
"\n\n- Paikalle tuli myyjä, joka kertoi muiden asiakkaiden kyllästyneen lapsen kiukutteluun ja käski perhettä poistumaan kaupasta, jos lasta ei saada rauhoittumaan, äiti kuvailee.",
"\n\nLapsen kiukuttelua oli kestänyt 10-15 minuuttia ennen kuin kehotus poistumiseen tuli. ",
"Perheelle ei selvinnyt, olivatko muut asiakkaat todella valittaneet lapsen kiukuttelusta.",
"\n\n- Onko sallittua, että myyjä voi lapsen kiukutteluun vedoten poistaa asiakkaan kaupasta? ",
"Kiukuttelu kohdistui ainoastaan vanhempaan eikä myyjään tai muihin asiakkaisiin, äiti kysyy.",
"\n\n”Rauhallisempi asiointikokemus”\n\nKauppa myöntää tapauksen ja sen, että toiset asiakkaat olivat valittaneet huutaneesta lapsesta.",
"\n\n- Myyjä oli mennyt paikalle selvittääkseen ja rauhoittaakseen tilanteen. ",
"Toimintaohjeidemme mukaan myyjän tulee tarjota asiakkaalle apua tilanteen rauhoittamiseksi. ",
"Tässä tapauksessa myyjämme oli pyytänyt asiakasta rauhoittamaan lapsen tai vaihtoehtoisesti poistumaan, tavaratalon päällikkö Petri Siitonen vastaa Iltalehdelle.",
"\n\nHän pahoittelee tapahtunutta ja kertoo keskustelleensa tilanteesta sekä toimintaohjeista henkilökunnan kanssa. ",
"Kauppa uskoo silti toimineensa oikein muiden asiakkaiden kannalta.",
"\n\n- Pahoittelemme tapahtunutta, tarkoituksemme ei ollut loukata asiakastamme, vaan myötävaikuttaa kaikkien asiakkaidemme mahdollisuuteen saada rauhallinen asiointikokemus, hän kertoo.",
"\n\nOletko joutunut ikäviin tilanteisiin lapsen raivokohtauksen vuoksi? ",
"Kerro kokemuksistasi!",
"\n\nKommenttini:\n\nNimimerkkini:"
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"Revolution in real time. ",
"Physician practice management in the 21st century.",
"\nChange in the health care system is stimulating the interest of physicians policymakers, and third-party payers in practice management activities. ",
"New analytic techniques and management concepts enable interested parties to obtain more and better information about the overall results of practice. ",
"These new techniques and concepts offer physicians an opportunity to exert more influence on the direction and productivity of practice than in the past. ",
"This report reviews the development of three emerging fields and projects their impact on practice management. ",
"Outcomes management technology involves adjusting for the differences in patient biology between practices and describes outcomes of patient interventions in standard ways. ",
"Medical informatics involves the conversion of data into information that allows the physician manager to direct outcomes, quality, and costs. ",
"Quality management involves empowering employees and physicians to participate in improving the overall performance of the practice. ",
"The use of techniques for analysis and intervention is changing the definition of practice management beyond its conventional preoccupation with billing and accounts receivable. ",
"In fact, a new science of practice management is emerging that describes the interactions between the health care organization and the outcomes of services to patients. ",
"These tools empower physicians to direct the performance of the practice, to match services to patient needs, and to manage the costs of practice operations. ",
"Thus, the new methods and technologies of practice management have the potential to bring substantial benefits to patients, office staff, and physicians."
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"heres whats new.",
"\n*added a new fancy ass orange ramp to the room on last.",
"\n*added a fancy ass wire going from the tower to the junction so players will know that the tower controls the junction.",
"\n*added some more walkways to last.",
"\n*added a fence on last to stop red from standing in there spawn and still being able to defend.",
"\n*added teleports so you dont get stuck in spawnrooms. (",
"blu's seems to not work tho :/ )\n\nheres whats changed\n*made it so the doors to the tower open earlier.",
"\n*the healthpack near the snipernest at 3 to last has been moved onto the sniper nest. (",
"did any snipers use the nest thing tho?)",
"\n*i changed the length of the timer and how much time you get from capping a point.",
"\n\n*fixed the hole in the upper tunnel near the badwater like area.",
"\n*blocked a sightline\n*added a bit of clipping near the ramp to the sniper battlement near last, so you step up instead of just stoping.",
"\n*added signs near the first point so ppl know where the control point is.",
"\n*made it so you cant stand on the spinning signs near first and second.",
"\n*the red wall/cover holding up a pathway between first and second got moved closer to first, to block a sightline.",
"\n*added spinning lights to last.",
"\n*added a bit more health to last.(maby i increased the ammo? ",
"i dont remember tho.)",
"\n*i blocked some annoying sightlines.",
"\n*fixed an issue where you could stand ontop the slopped roofs.",
"\n*made you capture the tower quicker.",
"\n*tweaked the big blue wall thingy near second.",
"\n*blocked some sightlines near first.",
"\n*fixed blu not teleporting from the first blu spawn to the second blu spawn.",
"\n*blocked a sightline near red spawn with a tiny wooden fence.",
"\n*i made the hole on last deeper.",
"\n*i added lights to the pathway leading from behind the red baracade near the hill near second, to second.",
"\n*did i tell you that i blocked some sightlines?"
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"\nSource table\nID | Name | Funct | Phone1 | Phone2 | Phone3\n1 | John | boss | 112233 | 114455 | 117788\n2 | Jane | manager | NULL | NULL | 221111\n3 | Tony | merchant | 441100 | 442222 | NULL\n\nWanted result\nID | Name | Funct | Phone | Ord\n1 | John | boss | 112233 | 1\n1 | John | boss | 114455 | 2\n1 | John | boss | 117788 | 3\n2 | Jane | manager | 221111 | 3\n3 | Tony | merchant | 441100 | 1\n3 | Tony | merchant | 442222 | 2\n\nOrd is a column where is the order number (Phone1...Phone3) of the original column\nEDITED:\nOK, UNION would be fine when phone numbers are in separed columns, but what if the source is following (all numbers in one column)?:",
"\nID | Name | Funct | Phones\n1 | John | boss | 112233,114455,117788\n2 | Jane | manager | 221111\n3 | Tony | merchant | 441100,442222\n\nHere I understand, that column Ord is a non-sense (so ignore it in this case), but how to split numbers to separed rows?",
"\n\nA:\n\nThe easiest way is to use union all:\nselect id, name, funct, phone1 as phone, 1 as ord\nfrom source\nwhere phone1 is not null\nunion all\nselect id, name, funct, phone2 as phone, 2 as ord\nfrom source\nwhere phone2 is not null\nunion all\nselect id, name, funct, phone3 as phone, 3 as ord\nfrom source\nwhere phone3 is not null;\n\nYou can write this with a cross apply as:\nselect so.*",
"\nfrom source s cross apply\n (select s.id, s.name, s.funct, s.phone1 as phone, 1 as ord union all\n select s.id, s.name, s.funct, s.phone2 as phone, 2 as ord union all\n select s.id, s.name, s.funct, s.phone3 as phone, 3 as ord\n ) so\nwhere phone is not null;\n\nThere are also methods using unpivot and cross join/case.",
"\n\n"
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"Background\n==========\n\nLiver transplantation (LT) is widely accepted as a definitive therapy for end-stage liver disease (ESLD). ",
"However, due to the donor liver graft shortage, patients suffering from ESLD are not transplanted in a timely manner \\[[@b1-anntransplant-23-507]\\]. ",
"Various types of LT, such as non-heart beating donor LT, split LT, and living donor liver transplantation (LDLT), have been performed to expand the donor pool, but the shortage of liver grafts persists \\[[@b2-anntransplant-23-507],[@b3-anntransplant-23-507]\\]. ",
"Reaching beyond the ABO blood type barrier is considered less feasible for organ transplantation due to the strong immunological antibody-antigen reaction \\[[@b4-anntransplant-23-507]\\]. ",
"In addition, incompatible living liver donors have been emergently applied to rescue patients who present with rapid hepatic decompensation but remain on the waiting list; however, long-term outcomes are not good after ABO-incompatible (ABOi) LT \\[[@b5-anntransplant-23-507]\\]. ",
"The introduction of the anti-CD20 monoclonal antibody (rituximab) made ABOi LT a clinical reality. ",
"Thereafter, patient and graft outcomes improved after ABOi LDLT \\[[@b6-anntransplant-23-507]\\].",
"\n\nPartial liver grafts are required to regenerate rapidly due to the recipient demands of LDLT. ",
"A study by Kawasaki et al. ",
"\\[[@b7-anntransplant-23-507]\\] reported that transplanted grafts with insufficient mass against recipient body size did not undergo functional failure due to vigorous graft regeneration after LDLT. ",
"Previous studies have reported associations between liver graft regeneration and multiple factors, including recipient clinical status, graft ischemia-reperfusion injury, hepatic vascular hemodynamics, and donor condition in LDLT \\[[@b8-anntransplant-23-507]--[@b11-anntransplant-23-507]\\]. ",
"Immunosuppressive drugs also have negative or positive effects on graft regeneration after LDLT \\[[@b12-anntransplant-23-507]--[@b14-anntransplant-23-507]\\]. ",
"A complex immunoreaction develops between the donor graft and the host body in patients who undergo ABOi LDLT \\[[@b4-anntransplant-23-507]\\].",
"\n\nThe present study investigated whether ABO incompatibility affects postoperative liver graft regeneration using a propensity score (PS) matching analysis in the LDLT setting. ",
"The incidence of major complications was compared between patients who underwent ABO-compatible (ABOc) and ABOi LDLTs. ",
"Finally, we analyzed the associations between perioperative factors and postoperative liver graft regeneration in patients who underwent ABOi LDLT.",
"\n\nMaterial and Methods\n====================\n\nStudy population\n----------------\n\nA total of 335 adult patients (age ≥18 years) undergoing elective ABOc or ABOi LDLT from March 2009 to February 2016 at St. Mary's Hospital (Seoul, South Korea) were analyzed. ",
"Retrospective reviews of perioperative recipient and donor data were performed using the hospital electronic medical records system. ",
"Defective or inadequate data in the recipient's or donor's records were excluded from the present study. ",
"The Institutional Review Board of Seoul St. Mary's Hospital Ethics Committee granted approval for this study (KC17RIS0001). ",
"Informed consent was waived due to the retrospective study design.",
"\n\nSurgery and anesthesia\n----------------------\n\nThe surgical procedure and anesthetic care of recipients undergoing LDLT were explained previously in detail \\[[@b15-anntransplant-23-507],[@b16-anntransplant-23-507]\\]. ",
"Briefly, the piggyback method was applied using the right hepatic lobe of the donor, and by reconstructing the middle hepatic vein so that the segmental hepatic veins (from liver segments V and VIII) of the recipient anastomosed to the middle hepatic vein of the donor using a Dacron conduit with an inner diameter of 10 mm (Gelweave; Vascutek, Inchinnan, UK). ",
"Hepatic vessels, including the hepatic vein, portal vein, and hepatic artery, were serially anastomosed followed by biliary ductal reconstruction. ",
"After the anastomoses with the hepatic vessels, patency of hepatic blood flow was evaluated using spectral Doppler ultrasonography.",
"\n\nThe patients were intraoperatively managed with balanced anesthesia. ",
"Hemodynamic homeostasis of the patients was adjusted with appropriate circulatory supplementation and a vasopressor was administered under invasive hemodynamic monitoring.",
"\n\nGrouping for the propensity score matching analysis\n---------------------------------------------------\n\nThe study population was classified into the ABOc and ABOi LDLT groups. ",
"The perioperative recipient factors and donor-graft factors were applied in the PS matching analysis between the ABOc and ABOi groups. ",
"Preoperative recipient factors included age, sex, body mass index (BMI), comorbidities, model for end-stage liver disease (MELD) score, hepatic decompensation complications, and inflammatory markers. ",
"Intraoperative recipient factors included duration of surgery, use of a strong vasopressor (epinephrine or norepinephrine), severe post-reperfusion syndrome (PRS), average vital signs, total amount of blood products transfused, hourly fluid infusion, hourly urine output, average lactate, and the neutrophil-to-lymphocyte ratio (NLR). ",
"Donor-graft factors were age, sex, BMI, graft fatty percentage and type, total ischemic time, preoperative liver graft/standard liver volume (SLV) ratio, and average hepatic vascular hemodynamics on postoperative days (PODs) 1, 3, and 5.",
"\n\nDesensitization and immunosuppression protocol\n----------------------------------------------\n\nAll recipients scheduled for ABOi LDLT were treated with a single dose of intravenous rituximab (375 mg/m^2^ body surface area), which attenuates B cell immunity \\[[@b17-anntransplant-23-507]\\], 2 weeks before the surgery. ",
"Recipients' blood samples were collected to measure the anti-ABO isohemagglutinin titers (anti-A IgM, anti-B IgM, anti-A IgG, and anti-B IgG) at hospital admission, at each point of plasmapheresis, and during the follow-up period. ",
"Fresh frozen plasma (FFP) in AB^+^ blood type was used for plasmapheresis. ",
"Plasmapheresis was continued to achieve the desired isohemagglutinin titer (≤1: 8) before surgery, and postoperative isoagglutinin titers were monitored and treated if the titers increased \\>1: 8 to 16.",
"\n\nThe immunosuppression regimen (tacrolimus, methylprednisolone, and mycophenolate mofetil \\[MMF\\]) was administered postoperatively. ",
"The trough level of tacrolimus was preserved between 7 and 10 ng/mL for the first month after surgery and tapered between 5 and 7 ng/mL thereafter. ",
"Methylprednisolone was administered immediately before graft reperfusion and then gradually tapered. ",
"MMF was withdrawn 3--6 months after surgery. ",
"Basiliximab (interleukin-2 receptor blocker) was administered on the day of LDLT prior to the surgery and on POD 4.",
"\n\nPerioperative cluster of differentiation marker measurement\n-----------------------------------------------------------\n\nThe cluster of differentiation (CD) marker levels (%), including CD 4+, 8+, 19+, 20+, and 25+ were estimated 1 day before surgery and on POD 14 in patients undergoing ABOi LDLT. ",
"Blood samples were obtained in test tubes (BD Vacutainer, K2 EDTA; Becton, Dickinson and Company, Franklin Lakes, NJ, USA) and sent to the laboratory for analysis of the CD markers using Navios flow cytometry (Beckman Coulter, Indianapolis, IN, USA).",
"\n\nPostoperative graft volume measurement\n--------------------------------------\n\nLiver graft volumes were measured using abdominal computed tomography (CT) images in the recipients and donors. ",
"Total liver volume and right lobe volume of donors were determined on preoperative volumetric CT images, and liver graft volumes of recipients were identified by postoperative volumetric CT images on PODs 7 and 21. ",
"Experienced radiologists measured absolute liver volume (ALV; mL) using volumetry software (AW VolumeShare 4; General Electric Healthcare, Chicago, IL, USA). ",
"Relative liver volume (RLV;%) was calculated as the ratio of the liver graft volume to the SLV. ",
"SLV was defined as follows: 1072.8×body surface area (BSA)−345.7 (BSA=weight \\[kg\\]^0.425^×height \\[cm\\]^0.725^×0.007184) \\[[@b18-anntransplant-23-507]\\].",
"\n\nClinical outcomes\n-----------------\n\nPostoperative clinical outcomes were liver graft volumes on PODs 7 and 21, hospital and intensive care unit (ICU) stays, early allograft dysfunction (EAD) \\[[@b19-anntransplant-23-507]\\], re-operation, infection, duration of mechanical ventilation, re-intubation, and major graft complications, including acute cellular rejection defined as clinically suspected findings and a Rejection Activity Index ≥3 \\[[@b20-anntransplant-23-507]\\], and acute antibody-mediated rejection (AMR) based on histopathological findings such as monocytic, eosinophilic, or neutrophilic microvascular/capillary inflammation \\[[@b21-anntransplant-23-507]\\], biliary complications, hepatic vessel thrombosis, and *de novo* cancer occurrence. ",
"Overall patient survival was assessed on the last out-patient visit during the follow-up period.",
"\n\nStatistical analysis\n--------------------\n\nComparisons of perioperative recipient factors and donor-graft factors were evaluated using the Mann-Whitney *U*-test, and χ^2^ test or Fisher's exact test, as appropriate, before PS matching. ",
"The normality of the continuous data was tested using the Shapiro--Wilk test. ",
"A PS matching analysis was used to minimize the influence of potential confounding factors on intergroup differences based on ABO compatibility. ",
"The PS scores were elicited from matched patients at a 1: 1 ratio using greedy matching algorithms without replacement. ",
"Standardized mean differences \\<0.25 were considered to indicate well-balanced matching between the groups \\[[@b22-anntransplant-23-507],[@b23-anntransplant-23-507]\\]. ",
"The Wilcoxon signed-rank sum test and McNemar's test were applied to the pair-matched data. ",
"Continuous data are expressed as median and interquartile range (IQR), and categorical data are shown as numbers and proportions. ",
"The perioperative determinants that affected liver graft volume on POD 21 were analyzed using univariate and multivariate linear regression in patients undergoing ABOi LDLT. ",
"Potentially significant factors (*p*\\<0.1) in the univariate analysis were included in a multivariate analysis. ",
"When various perioperative factors were correlated with each other, the most clinically relevant factors were selected. ",
"The values in univariate and multivariate linear analyses are presented as β (regression coefficient) and 95% confidence intervals (CIs). ",
"All tests were two-sided, and a *p*-value \\<0.05 was considered significant. ",
"Overall survival of patients was evaluated using the Kaplan-Meier analysis and compared between the groups using the log-rank test. ",
"We used stratified Cox regression to account for the PS matching. ",
"The stratified Cox regression analysis was applied to evaluate the hazard ratio (HR) and 95% CI of ABO compatibility on overall survival of patients. ",
"All statistical analyses were performed using the SAS 9.4 software package (SAS Institute, Cary, NC, USA).",
"\n\nResults\n=======\n\nStudy population\n----------------\n\nNinety-five patients were excluded due to deficient or missing donor-related variables (10 patients), radiological images of liver grafts (11 patients), intraoperative hemodynamic variables of recipients (10 patients), and emergency LDLT (64 patients). ",
"Eventually, 335 patients were enrolled in our study. ",
"The study populations were featured as follows: median age was 54.0 (IQR: 49.0--59.0) years; mainly male sex (70.7%); median BMI was 23.8 (IQR: 22.0--26.4) kg/m^2^. The median follow-up period was 3.3 (IQR: 1.3--5.4) years after LDLT. ",
"Hepatocellular carcinoma was diagnosed in 177 (52.8%) patients, and the most common indication for LDLT was hepatitis B virus (60.3%), followed by alcohol abuse (20.9%), hepatitis C virus (7.5%), toxin and drug-related hepatitis (3.8%), autoimmune hepatitis (2.7%), and cryptogenic hepatitis (4.8%). ",
"The median MELD score was 13.7 (IQR: 9.2--23.4) points, and hepatic decompensation, including severe encephalopathy (West-Haven criteria III or IV), variceal hemorrhage, and ascites (\\>1 L), was seen in 77 (22.9%), 89 (26.5%), and 143 (42.6%) patients, respectively.",
"\n\nA total of 300 (89.6%) patients underwent ABOc LDLT. ",
"Identical LDLTs were performed in 201 (60.0%) patients, and compatible LDLTs were conducted in 99 (29.5%) patients. ",
"Thirty-five (10.4%) patients underwent ABOi LDLT. ",
"The preoperative frequency of plasmapheresis was 5.0 (IQR: 4.0--6.0) to reduce the isohemagglutinin titer to \\<1: 8 in patients who underwent ABOi LDLT.",
"\n\nComparison of perioperative factors before and after propensity score matching\n------------------------------------------------------------------------------\n\nBefore PS matching, preoperative NLR and sodium levels were higher in the ABOi group than in the ABOc group, but AST and ALT levels were lower in the ABOi group than in the ABOc group. ",
"The mean intraoperative NLR was higher in the ABOi group than in the ABOc group, and more bicarbonate was administered to the ABOi group than to the ABOc LDLT group. ",
"Donor-graft findings, such as age, sex, BMI, preoperative liver graft volume/SLV ratio, graft fatty percentage and type, total ischemic time, and hepatic vascular hemodynamic, were not different between the ABOc and ABOi groups ([Table 1](#t1-anntransplant-23-507){ref-type=\"table\"}). ",
"After PS matching, no significant differences in perioperative factors were found in the recipients or donors between the ABOc and ABOi groups ([Table 2](#t2-anntransplant-23-507){ref-type=\"table\"}).",
"\n\nLiver graft volumes according to ABO incompatibility before and after propensity score matching\n-----------------------------------------------------------------------------------------------\n\nALV and RLV on POD 21 were smaller in the ABOi group than in the ABOc LDLT before PS matching. ",
"ALV in the ABOi group was significantly smaller on POD 21 than that in the ABOc group after PS matching, but the difference in RLV on POD 21 was marginally comparable between the groups ([Table 3](#t3-anntransplant-23-507){ref-type=\"table\"}).",
"\n\nClinical outcomes according to ABO incompatibility in the propensity score-matched patients\n-------------------------------------------------------------------------------------------\n\nMajor events in the hospital (hospital and ICU stays, EAD development, re-operation, infection, duration of mechanical ventilation, and re-intubation) and major graft complications (acute cellular rejection, acute AMR, biliary complications, hepatic vessel thrombosis, and *de novo* cancer occurrence) were comparable between the groups during the follow-up period ([Table 4](#t4-anntransplant-23-507){ref-type=\"table\"}).",
"\n\nNo difference in overall patient survival was observed between the groups during the follow-up period ([Figure 1](#f1-anntransplant-23-507){ref-type=\"fig\"}). ",
"The association between overall patient survival and ABOi LDLT was not significant based on the ABOc LDLT using a stratified Cox regression analysis (HR: 1.33; 95% CI: 0.30--5.96; *p*=0.706).",
"\n\nAssociation between perioperative factors and liver graft volume on postoperative day 21 in patients who underwent ABO-incompatible living donor liver transplantation\n----------------------------------------------------------------------------------------------------------------------------------------------------------------------\n\nIn patients undergoing ABOi LDLT ([Table 5](#t5-anntransplant-23-507){ref-type=\"table\"}), a univariate linear analysis related to the ALV on POD 21 showed that potentially significant predictors were involved in preoperative recipient factors (MELD score, hematocrit, NLR, total bilirubin, and INR), intraoperative recipient factors (central venous pressure \\[CVP\\], mean pulmonary arterial pressure, and furosemide administration), a donor graft factor (preoperative graft volume/SLV ratio), and specific ABOi LDLT factors (the preoperative serum cellular levels \\[%\\] of CD19+, which is a protein expressed on B lymphocytes, and CD4+, which is a protein expressed on T-lymphocytes; and CD4+ cell on POD 14). ",
"A multivariate linear analysis identified that preoperative graft volume/SLV ratio and CD4+ cellular level on POD 14 were independently associated with the ALV on POD 21 (R^2^=0.689; *p*=0.009). ",
"In RLV on POD 21 ([Table 6](#t6-anntransplant-23-507){ref-type=\"table\"}), a univariate linear analysis showed that potentially valid predictors were included with preoperative recipient factors (age, male sex, BMI, hematocrit, platelet, and INR), and intraoperative recipient factors (severe PRS, CVP, FFP transfusion, and furosemide infusion), a donor graft factor (preoperative graft volume/SLV ratio), and specific ABOi LDLT factors (preoperative plasmapheresis frequency and CD4+ cellular level, and CD4+ cellular level on POD 14). ",
"A multivariate linear analysis showed that preoperative graft volume/SLV ratio and CD4+ cellular level on POD 14 were independently related to the RLV on POD 21 (R^2^=0.779; *p*\\<0.001).",
"\n\nDiscussion\n==========\n\nThe main findings in the present study were that graft volume on POD 21 was significantly smaller in recipients undergoing ABOi LDLT than in those undergoing ABOc LDLT. ",
"Larger preoperative graft volume/SLV ratio and a lower CD4+ cellular level on POD 14 were independently associated with ALV and RLV on POD 21 in patients undergoing ABOi LDLT, respectively. ",
"In the PS-matched patients, the difference in major patient and graft complications was comparable between the groups, and there was no association between ABOi LDLT and overall patient mortality against the ABOc LDLT.",
"\n\nMany studies have suggested that various independent factors, including ischemia-reperfusion injury \\[[@b24-anntransplant-23-507],[@b25-anntransplant-23-507]\\], liver graft volume \\[[@b10-anntransplant-23-507],[@b26-anntransplant-23-507]\\], steatosis \\[[@b27-anntransplant-23-507]--[@b29-anntransplant-23-507]\\], donor age \\[[@b30-anntransplant-23-507],[@b31-anntransplant-23-507]\\], and portal circulation \\[[@b8-anntransplant-23-507],[@b32-anntransplant-23-507]\\], are associated with liver graft regeneration in experimental and clinical LT settings. ",
"The regeneration rate of dual-graft LDLT using both ABOi and ABOc grafts was comparable but postoperative complications (antibody-mediated rejection and biliary problems) were not presented. ",
"Dual-LDLT using ABOi and ABOc grafts is considered a feasible treatment for small-for-size syndrome \\[[@b33-anntransplant-23-507]\\]. ",
"In the present study, the perioperative factors related to graft regeneration were matched between the ABOc and ABOi groups using a PS-matching analysis. ",
"Our study and the study by Song et al. ",
"shared favorable ABOi LDLT findings that the ABOi LDLT was acceptable for patients with ESLD because postoperative morbidity and mortality were comparable compared with ABOc LDLT. ",
"However, in the PS-matched patients of our study, the ABO incompatibility between liver grafts and recipients affected graft regeneration capability, and probably tended to suppress growth of the graft mass even though the difference in regeneration was not large enough to result in clinical significance. ",
"Because insufficient graft regeneration is closely connected to graft dysfunction \\[[@b34-anntransplant-23-507]\\], patients with poor donor-graft quality could have undergone inappropriate graft regeneration in the ABOi LDLT group, resulting in delayed recovery of graft function. ",
"Poor graft regeneration was considered as a cause of graft dysfunction in patients who underwent ABOi LDLT. ",
"To assess graft regeneration, it is necessary to discriminate true graft regeneration from graft edema \\[[@b35-anntransplant-23-507]\\] or a portal hyperperfusion-related graft size increase \\[[@b8-anntransplant-23-507]\\], which is related to poor postoperative outcome. ",
"Therefore, it was helpful to investigate graft regeneration using serial volumetric assessments in patients with poor donor-graft quality and quantity, as well as inappropriate graft circulation.",
"\n\nThe high serum level of CD4+ T-lymphocytes on POD 14 had a negative effect on increasing liver graft size on POD 21 in ABOi LDLT. ",
"Until now, little has been reported on the direct relationship between CD4+ T-lymphocyte level and graft regeneration in the LT setting. ",
"In experimental studies, T-lymphocytes play an important role in neutrophil-mediated inflammation in liver grafts after ischemia-reperfusion. ",
"Particularly, CD4+ T-lymphocytes trigger recruitment of neutrophils into the liver graft \\[[@b36-anntransplant-23-507]\\]. ",
"Warm ischemia-reperfusion injury in the hepatic microcirculation caused CD4+ T-lymphocytes to migrate into liver grafts. ",
"The CD4+ T-lymphocytes interact with platelets and sinusoidal endothelial cells and disrupt the microcirculation \\[[@b37-anntransplant-23-507]\\]. ",
"Aggressive CD4+ T-lymphocyte-induced inflammation in the graft can be associated with reduced regeneration ability of allograft cells under the ABOi condition.",
"\n\nThe preoperative graft volume/SLV ratio was positively associated with liver graft volume on POD 21. ",
"A previous study suggested that small-for-size grafts after major hepatectomy or LT regenerate vigorously and attain a graft size similar to the initial native liver \\[[@b38-anntransplant-23-507]\\]. ",
"A small-for-size graft with minimal ischemic injury regenerates robustly after partial liver graft transplantation \\[[@b39-anntransplant-23-507]\\]. ",
"One study reported that the partial graft sizes affect graft regeneration duration, with a smaller starting graft volume delaying the regeneration progress in LDLT \\[[@b11-anntransplant-23-507]\\]. ",
"Partial grafts have impaired ability to recover and maintain appropriate graft function in LDLT of patients in poor clinical condition, representing a high MELD score \\[[@b40-anntransplant-23-507]\\]. ",
"The present study suggests that sufficient allograft volume at the transplanted point may ensure postoperative vigorous regeneration of grafts against ABO incompatibility.",
"\n\nDuring ABOi transplantation, acute AMR has been reported frequently in heart, kidney, and pancreatic allografts, manifesting in organ dysfunction and a pathological microvasculature \\[[@b41-anntransplant-23-507],[@b42-anntransplant-23-507]\\]. ",
"During LT, AMR development is rare due to the ability of the liver to modulate immunological reactions and thus compensate for injury. ",
"In particular, by decreasing the titer to \\<1: 16 in plasma exchange therapy (plasmapheresis) and administering rituximab, ABOi AMR can be prevented \\[[@b43-anntransplant-23-507],[@b44-anntransplant-23-507]\\]. ",
"Acute AMR usually develops during the first several weeks after surgery, leading to rapid liver graft failure in highly sensitized patients \\[[@b45-anntransplant-23-507]\\]. ",
"Liver grafts with AMR present with specific histopathological features, including monocytic, eosinophilic, or neutrophilic microvasculitis accompanied by dilatation, disruption, and edema (of the portal veins, capillaries, and inlet venules, respectively) \\[[@b21-anntransplant-23-507]\\]. ",
"In the present study, there was no difference in the incidence of AMR between the 2 groups, and the development of AMR was not statistically associated with the liver graft volume on POD 21. ",
"However, based on previous studies \\[[@b21-anntransplant-23-507],[@b43-anntransplant-23-507]\\], AMR onset may be related to poor postoperative outcomes, including impaired graft regeneration in patients scheduled for ABOi LDLT. ",
"Therefore, meticulous monitoring of titers is necessary to identify whether additional treatment is needed. ",
"Further investigation of the association between partial liver graft regeneration and AMR development is also required because ABOi partial grafts are now widely used \\[[@b44-anntransplant-23-507]\\].",
"\n\nThe present study has several limitations. ",
"First, although we tried to balance the confounding factors between the ABOc and ABOi groups using a PS matching analysis, hidden biases were not totally eliminated as unknown factors may have remained. ",
"Second, because of the study design, the complex effect of immunosuppressants on graft regeneration was not investigated. ",
"Patients undergoing ABOi LDLT required more careful immunosuppression therapy than those undergoing ABOc LDLT owing to their immunologic reaction related to the ABO blood barrier \\[[@b46-anntransplant-23-507],[@b47-anntransplant-23-507]\\]. ",
"Further study would be helpful to guide the immunosuppressant levels between the graft regeneration and the ABOi-specified complications. ",
"Third, the difference in graft regeneration between ABOc and ABOi LDLT was not large enough to have clinical significance in our study; therefore, the ability to apply the study results in selecting ABOi LDLT donors is limited. ",
"Fourth, we could not assess the change in postoperative graft volumes over the long term. ",
"Future study is required to investigate the association between ABO incompatibility and long-term graft regeneration. ",
"Finally, because of the small sample size in this study, the results pertaining to patient survival, acute graft rejection, and biliary complications should be interpreted cautiously.",
"\n\nConclusions\n===========\n\nIn conclusion, because ABO incompatibility is associated with reduced graft regeneration within 1 month after LDLT in volume, the liver grafts for ABOi LDLT require more meticulous monitoring than those for ABOc LDLT. ",
"Graft regeneration must be investigated using a volumetric assessment in patients with poor donor-graft quality and quantity, as well as inappropriate graft circulation in those who underwent ABOi LDLT, to distinguish true graft regeneration from graft edema or portal hyperperfusion-related graft size. ",
"Preoperative sufficient graft volume is one of the important factors required to improve graft regeneration after ABOi LDLT. ",
"Because an aggressive increase in CD4+ cells is associated with poorer graft regeneration, careful management of inflammation, including the CD4+ cellular level, is required under the complex immunosuppression regimen in the ABOi LDLT group, although the pathological process of graft regeneration was uncertain under CD4+ T-lymphocyte-induced inflammation. ",
"The ABOi LDLT is a feasible option for patients with ESLD. ",
"However, when a graft with extended criteria is utilized for ABOi LDLT, close and serial graft volumetric monitoring is required regarding postoperative graft regeneration status.",
"\n\n**Conflicts of interest**\n\nNone.",
"\n\n**Source of support:** The statistical consultation was supported by a grant of the Korea Health Technology R&D Project through the Korea Health Industry Development Institute (KHIDI), funded by the Ministry of Health & Welfare, Republic of Korea (grant number: HI14C1062)\n\nABOi\n\n: ABO-incompatible\n\nABOc\n\n: ABO-compatible\n\nLDLT\n\n: living donor liver transplantation\n\nMELD\n\n: model for end-stage liver disease\n\nALV\n\n: absolute liver volume\n\nRLV\n\n: relative liver volume\n\n{#f1-anntransplant-23-507}\n\n###### \n\nPerioperative characteristics between the ABO-Compatible and ABO-Incompatible living donor liver transplantation groups before propensity score-matching.",
"\n\n Characteristics Total subject (n=335) ABOc (n=300) ABOi (n=35) *p* \n --------------------------------------------------------------------------- ----------------------- ------------------ ------------- ------------------ -------- ------------------ -------\n **Preoperative recipient findings** \n Age (years) 54.0 (49.0--59.0) 53.0 (49.0--59.0) 55.0 (50.0--58.0) 0.566\n Gender (Male) 237 (70.7%) 210 (70.0%) 27 (77.1%) 0.379\n Body mass index (kg/m^2^) 23.8 (22.0--26.4) 23.9 (21.8--26.4) 23.4 (22.2--25.2) 0.546\n Diagnosis 0.949\n Alcohol 70 (20.9%) 64 (21.3%) 6 (17.1%) \n Hepatitis B 202 (60.3%) 179 (59.7%) 23 (65.7%) \n Hepatitis C 25 (7.5%) 22 (7.3%) 3 (8.6%) \n Autoimmune 9 (2.7%) 9 (3.0%) 0 (0.0%) \n Toxin and drung 13 (3.8%) 12 (4.0%) 1 (2.9%) \n Cryptogenic 16 (4.8%) 14 (4.7%) 2 (5.7%) \n Comorbidity \n Diabetes mellitus 87 (25.9%) 78 (26.0%) 9 (25.7%) 0.971\n Hypertension (18.2%) 53 (17.7%) 8 (22.9%) 0.451\n MELD score (points) 13.7 (9.2--23.4) 14.1 (9.4--23.6) 13.4 (8.4--20.2) 0.385\n Hepatic decompensation \n Severe encephalopathy 77 (22.9%) 68 (22.7%) 9 (25.7%) 0.685\n Variceal hemorrahage 89 (26.5%) 82 (27.3%) 7 (20.0%) 0.353\n Ascites (\\>1L) 143 (42.6%) 131 (43.7%) 12 (34.3%) 0.288\n Laboratory parameters \n Hematocrit (%) 29.7 (25.4--35.8) 29.8 (25.3--35.9) 29.6 (26.4--35.1) 0.960\n NLR 2.2 (1.4--4.0) 2.10 (1.3--3.8) 3.2 (2.2--4.8) 0.005\n PLR 64.3 (28.1--95.8) 65.8 (31.0--95.7) 62.5 (3.0--104.7) 0.149\n Platelet count (x10^3^/μL) 62.0 (46.0--101.0) 62.0 (46.0--101.7) 59.0 (38.0--99.0) 0.547\n Total bilirubin (mg/dL) 1.8 (0.8--6.8) 1.9 (0.9--6.8) 1.7 (0.6--3.7) 0.289\n Creatinine (mg/dL) 0.8 (0.6--1.0) 0.7 (0.6--1.1) 0.8 (0.6--0.9) 0.707\n International normalized ratio 1.4 (1.2--1.8) 1.4 (1.2--1.8) 1.4 (1.2--1.7) 0.798\n AST (unit/L) 46.0 (31.0--74.0) 47.0 (33.0--77.0) 34.0 (26.0--57.0) 0.009\n ALT (unit/L) 31.0 (21.0--49.0) 32.0 (22.0--52.7) 22.0 (18.0--37.0) 0.003\n Sodium (mEq/L) 140.0 (136.0--142.0) 139.0 (136.0--141.0) 141.0 (139.0--143.0) 0.027\n **Intraoperative recipient findings** \n Duration of surgery (min) 515.0 (465.0--585.0) 520.0 (465.0--589.0) 505.0 (455.0--530.0) 0.182\n Strong vasopressor 51 (15.2%) 47 (15.7%) 4 (11.4%) 0.509\n Severe PRS 49 (14.6%) 46 (15.3%) 3 (8.6%) 0.284\n Average of vital signs \n CVP (mmHg) 9.6 (8.0--11.5) 9.6 (8.0--11.5) 10.3 (7.7--11.5) 0.918\n MPAP (mmHg) 18.8 (16.3--20.8) 18.6 (16.1--20.8) 18.9 (17.6--20.8) 0.303\n Stroke volume variation (%) 6.6 (5.0--8.6) 6.6 (5.0--8.5) 6.5 (5.0--10.3) 0.921\n Cardiac index (L/min/m^2^) 4.1 (3.6--4.9) 4.1 (3.6--4.8) 4.0 (3.2--5.0) 0.984\n SVRI (dynes-sec/cm^5^/m^2^) 1266.1 (1011.5--1581.1) 1284.4 (1011.5--1591.5) 1231.0 (1007.1--1321.4) 0.631\n Mean blood pressure (mmHg) 77.0 (70.9--83.5) 77.0 (70.9--83.6) 76.7 (70.9--82.0) 0.629\n Heart rate (beats/min) 86.4 (76.8--97.0) 86.4 (76.6--97.0) 85.4 (80.2--95.0) 0.940\n Blood product transfusion (unit) \n Packed red blood cell 7.0 (4.0--12.0) 7.5 (3.0--12.7) 6.0 (5.0--12.0) 0.818\n Fresh frozen plasma 7.0 (4.0--10.0) 7.0 (4.0--10.0) 6.0 (4.0--9.0) 0.115\n Platelet concentrates 4.0 (0.0--10.0) 4.0 (0.0--10.0) 2.0 (0.0--10.0) 0.565\n Cryoprecipitate 0.0 (0.0--0.0) 0.0 (0.0--0.0) 0.0 (0.0--0.0) 0.452\n Fluid infusion (mL/kg/h) 10.0 (7.6--12.9) 10.0 (7.6--12.9) 10.6 (7.4--14.8) 0.523\n Urine output (mL/kg/h) 1.4 (0.7--2.2) 1.4 (0.7--2.2) 1.4 (0.7--2.1) 0.799\n Drug administraion \n Bicarbonate (mEq) 0.0 (0.0--60.0) 0.0 (0.0--57.5) 40.0 (0.0--100.0) 0.019\n Insulin (unit) 10.0 (2.0--25.0) 10.0 (3.0--25.0) 6.0 (0.0--25.0) 0.142\n Frusemide (mg) 10.0 (0.0--20.0) 10.00 (0.0--20.0) 10.0 (0.0--20.0) 0.281\n Mean lactate (mmol/L) 4.1 (3.3--5.4) 4.0 (3.3--5.4) 4.5 (3.3--5.4) 0.806\n Mean NLR 12.7 (8.1--19.2) 12.1 (7.9--18.8) 16.0 (12.8--23.2) 0.002\n **Donor graft findings** \n Age (years) 32.0 (25.0--42.0) 32.0 (25.0--42.0) 29.0 (24.0--36.0) 0.519\n Gender (Male) 128 (38.2%) 114 (38.0%) 14 (40.0%) 0.818\n Body mass index (kg/m^2^) 23.3 (21.1--25.5) 23.4 (21.1--25.5) 21.5 (19.9--26.1) 0.125\n Graft volume/SLV ratio\\* 55.9 (47.6--68.7) 56.1 (48.0--68.2) 53.9 (43.2--70.2) 0.372\n Graft fatty percentage (%) 3.0 (0.0--5.0) 4.0 (0.0--5.0) 3.0 (0.0--5.0) 0.989\n Graft fatty type 0.572\n No fat 93 (27.8%) 82 (27.3%) 11 (31.4%) \n Microvesicular 11 (3.3%) 10 (3.3%) 1 (2.9%) \n Macrovesicular 216 (64.5%) 196 (65.3%) 20 (57.1%) \n Mixed 15 (4.4%) 12 (4.0%) 3 (8.6%) \n Total ischemic time (min) 92.7 (69.0--117.0) 92.7 (68.0--117.0) 92.0 (74.0--109.0) 0.746\n Average of hepatic vascular hemodynamic on postoperative days 1, 3, and 5 \n Hepatic arterial resistive index 0.6 (0.6--0.6) 0.6 (0.5--0.6) 0.6 (0.6--0.7) 0.122\n Portal venous flow (mL/min) 2057.0 (1495.0--2613.6) 1884.8 (1461.7--2722.5) 2004.6 (1591.2--2840.1) 0.674\n\nMELD -- Model for end-stage liver disease; NLR -- neutrophil to lymphocyte ratio; PLR -- platelet to lymphocyte ratio; AST -- asparate aminotransferase; ALT -- alanine aminotransferase; PRS -- postreperfusion syndrome; CVP -- central venous pressure; MPAP -- mean pulmonary artery pressure; SVRI -- systemic vascular resistive index; SLV -- standard liver volume. ",
"Values are expressed as number (proportion), mean ±SD or median (IQR).",
"\n\n###### \n\nPerioperative characteristics between the ABO-Compatible and ABO-Incompatible living donor liver transplantation after propensity score-matching.",
"\n\n Characteristics Total subject (n=64) ABOc (n=32) ABOi (n=32) *p* SD \n --------------------------------------------------------------------------- ---------------------- ------------------ ------------- ------------------ -------- ------------------ --------- ------\n **Preoperative recipient factors** \n Age (years) 54.0 (49.0--57.0) 54.0 (49.0--57.0) 54.0 (49.5--57.0) 0.954 5.0\n Gender (Male) 48 (75.0%) 24 (75.0%) 24 (75.0%) \\>0.999 0.0\n Body mass index (kg/m^2^) 24.1 (22.2--26.3) 24.4 (21.7--28.6) 23.6 (22.4--25.3) 0.419 16.1\n Comorbidity \n Diabetes mellitus 14 (21.9%) 7 (21.9%) 7 (21.9%) \\>0.999 0.0\n Hypertension 11 (17.2%) 5 (15.6%) 6 (18.8%) 0.763 8.3\n MELD score (points) 12.0 (8.0--20.0) 11.5 (8.0--20.5) 12.0 (8.0--20.0) 0.565 11.7\n Hepatic decompensated complication \n No 64 (100.0%) 32 (100.0%) 32 (100.0%) \\- \\-\n Inflammatory marker \n NLR 2.7 (1.7--4.1) 1.9 (1.6--3.1) 3.1 (2.2--4.8) 0.230 21.6\n Platelet to lymphocyte ratio 62.0 (47.5--108.5) 71.0 (52.5--115.5) 60.5 (38.5--100.5) 0.378 4.7\n **Intraoperative recipient factors** \n Duration of surgery (min) 487.5 (455.0--547.5) 477.5 (445.0--570.0) 497.5 (460.0--525.0) 0.886 8.5\n Strong vasopressor 8 (12.5%) 4 (12.5%) 4 (12.5%) \\>0.999 0.0\n Severe PRS 6 (9.4%) 4 (12.5%) 2 (6.3%) 0.317 21.6\n Average of vital signs \n CVP (mmHg) 10.2 (8.4--11.5) 9.7 (8.7--11.3) 10.6 (8.3--12.0) 0.891 3.7\n MPAP (mmHg) 19.0 (16.8--21.0) 18.9 (16.3--21.0) 19.0 (17.9--21.0) 0.564 4.6\n Mean blood pressure (mmHg) 76.9 (71.0--82.2) 77.3 (71.8--84.6) 76.4 (70.7--81.2) 0.715 16.5\n Heart rate (beats/min) 84.6 (80.4--97.0) 82.4 (77.6--98.7) 87.6 (81.6--95.6) 0.648 14.7\n Blood product transfusion (unit) \n Packed red blood cell 6.5 (3.0--11.0) 7.5 (3.0--10.5) 6.0 (5.0--11.5) 0.920 5.4\n Fresh frozen plasma 6.0 (4.0--9.5) 5.0 (4.0--10.0) 6.0 (4.0--8.5) 0.744 0.0\n Platelet concentrates 0.0 (0.0--0.0) 0.0 (0.0--1.5) 0.0 (0.0--0.0) 0.889 3.5\n Cryoprecipitate 0.0 (0.0--0.0) 0.0 (0.0--0.0) 0.0 (0.0--0.0) \\- \\-\n Hourly fluid infusion (mL/kg/h) 9.6 (7.7--12.7) 9.3 (7.7--11.7) 10.6 (7.8--13.8) 0.358 17.7\n Hourly urine output (mL/kg/h) 830.0 (497.5--1140.0) 830.0 (585.0--1115.0) 830.0 (425.0--1175.0) 0.442 20.7\n Mean lactate (mmol/L) 4.2 (3.3--5.3) 4.0 (3.3--5.1) 4.5 (3.5--5.5) 0.709 10.0\n Mean NLR 86.5 (81.9--89.7) 85.3 (80.9--88.9) 87.5 (84.2--90.6) 0.344 17.2\n **Donor graft factors** \n Age (years) 28.5 (23.5--35.0) 28.0 (23.0--35.0) 28.5 (23.5--35.5) 0.641 16.4\n Gender (Male) 43 (67.2%) 23 (71.9%) 20 (62.5%) 0.467 20.1\n Body mass index (kg/m^2^) 22.1 (20.6--24.9) 22.4 (21.1--24.1) 21.7 (20.2--26.3) 0.898 0.7\n Graft fatty percentage (%) 3.0 (0.0--5.0) 5.0 (0.5--5.0) 3.0 (0.0--5.0) 0.815 17.7\n Graft fatty type \n No fatty change 18 (28.1) 8 (25.0) 10 (31.3) 0.997 14.0\n Microvesicular 2 (3.1) 1 (3.1) 1 (3.1) 0.0\n Macrovesicular 39 (60.9) 21 (65.6) 18 (56.3) 19.1\n Mixed 5 (7.8) 2 (6.3) 3 (9.4) 11.5\n Total ischemic time (min) 92.7 (72.5--115.0) 94.5 (70.5--120.0) 91.5 (73.0--108.5) 0.654 11.4\n Preop liver graft/SLV ratio (%) 53.4 (44.1--70.3) 51.2 (44.1--67.9) 54.1 (44.6--70.3) \\>0.999 1.3\n Average of hepatic vascular hemodynamic on postoperative days 1, 3, and 5 \n Hepatic arterial resistive index 0.7 (0.6--0.7) 0.7 (0.6--0.7) 0.7 (0.6--0.7) 0.204 16.5\n Portal venous flow (mL/min) 1942.5 (1584.0--2663.2) 1855.6 (1559.8--2648.8) 2032.0 (1631.0--2743.0) 0.884 3.9\n\nMELD -- model for end-stage liver disease; PRS -- postreperfusion syndrome; CVP -- central venous pressure; MPAP -- mean pulmonary arterial pressure; NLR -- neutrophil to lymphocyte ratio; SLV -- standard liver volume. ",
"Values are numbers (percentages) for categorical variables and mean ±SD, median (IQR) others.",
"\n\n###### \n\nComparison of liver graft regeneration between the ABO-Compatible and ABO-Incompatible living donor liver transplantation before and after propensity score matching.",
"\n\n Total subject (n=335) Before propensity score matching After propensity score matching \n -------------------------------------- ------------------------- ---------------------------------- --------------------------------- ------- ------------------------- ------------------------- ---------\n **Absolute liver graft volume (mL)** \n Preoperative day 840.0 (736.6--1008.2) 840.0 (736.7--1010.9) 846.3 (706.8--1000.7) 0.761 823.4 (730.9--981.7) 857.5 (733.4--1016.2) 0.798\n Postoperative day 7 1143.4 (1018.8--1257.5) 1151.2 (1028.4--1259.5) 1069.32 (925.4--1207.4) 0.061 1121.9 (1014.5--1253.4) 1061.9 (919.4--1183.2) 0.300\n Postoperative day 21 1161.9 (1022.4--1335.0) 1188.0 (1037.5--1367.7) 1092.2 (966.4--1157.3) 0.003 1202.0 (1107.8--1455.2) 1098.4 (964.0--1162.0) 0.007\n **Relative liver graft volume (%)** \n Standard liver volume (mL) 1530.4 (1375.4--1638.3) 1524.8 (1373.9--1635.0) 1559.1 (1392.8--1660.8) 0.516 1585.7 (1433.3--1705.5) 1564.6 (1395.3--1663.9) 0.701\n Preoperative day 55.9 (47.6--68.6) 56.0 (47.9--68.2) 53.8 (43.1--70.1) 0.371 51.2 (44.1--67.9) 54.1 (44.6--70.3) \\>0.999\n Postoperative day 7 74.1 (66.3--84.6) 74.7 (66.9--84.9) 68.8 (62.0--82.3) 0.022 71.2 (63.7--84.4) 66.7 (62.3--80.6) 0.846\n Postoperative day 21 78.4 (68.7--88.4) 79.0 (69.8--89.1) 68.8 (64.8--82.7) 0.001 78.9 (70.3--88.6) 68.9 (62.5--83.3) 0.023\n\nValues are expressed as number (proportion) and median (IQR).",
"\n\n###### \n\nComparison of postoperative outcomes between ABO-Compatible and ABO-Incompatible living donor liver transplantation after propensity score matching.",
"\n\n Characteristics ABOc (n=32) ABOi (n=32) *p* \n ------------------------------------- ------------- -------------- ------ -------------- ---------\n **Major events in hospital** \n Hospital stay (day) 22.0 (21.0--29.5) 25.5 (21.0--34.0) 0.438\n Intensive care unit stay (day) 7.0 (6.0--7.0) 7.0 (6.0--7.0) 0.956\n Early allograft dysfunction 5 (15.6%) 5 (15.6%) \\>0.999\n Re-operation 1 (3.1%) 3 (9.7%) 0.317\n Infection 3 (9.4%) 3 (9.7%) \\>0.999\n Mechanical ventilation (min) 0.0 (0.0--391.5) 0.0 (0.0--217.5) 0.626\n Re-intubation 3 (9.4%) 1 (3.2%) 0.317\n **Major graft complications** \n Acute graft rejection 6 (18.8%) 7 (21.9%) 0.782\n Acute cellular rejection 4 (12.5%) 5 (15.6%) \\>0.999\n Acute antibody-mediated rejection 2 (5.9%) 2 (5.9%) \\>0.999\n Biliary complication 13 (40.6%) 12 (37.5%) 0.796\n Hepatic vessel thrombosis -- -- -- \n *De novo* cancer occurrence 3 (9.4%) 5 (15.6%) 0.480\n\nValues are expressed as number (proportion) and median (IQR)*.*",
"\n\n###### \n\nAssociation between perioperative factors and absolute liver graft volume on postoperative day 21 in patients who underwent ABO-Incompatible living donor liver transplantation.",
"\n\n Characteristics Absolute liver graft volume (mL) on POD 21 \n -------------------------------------------------------------------- -------------------------------------------- ------------------ ------- ------- ----------------- -------\n **Preoperative recipient factors** \n MELD score (points) 7.31 0.31 to 14.30 0.041 \n Laboratory parameters \n Hematocrit (%) −9.47 −20.45 to 1.52 0.088 \n Neutrophil/lymphocyte ratio 18.06 0.40 to 35.72 0.045 \n Total bilirubin (mg/dL) 6.89 0.12 to 13.66 0.046 \n International normalized ratio 154.86 −0.92 to 310.65 0.051 \n **Intraoperative recipient factors** \n Average of vital signs \n CVP (mmHg) 42.01 19.71 to 64.30 0.001 \n MPAP (mmHg) 12.07 −1.88 to 26.02 0.087 \n Frusemide (mg) 2.84 −0.04 to 5.72 0.054 \n **Donor graft factors** \n Preoperative graft volume/SLV ratio\\* 7.15 2.94 to 11.34 0.002 11.65 4.44 to 18.86 0.006\n **Specific factors in ABOi LDLT** \n Cluster of difference (CD) level (%) on one day before the surgery \n CD 4+ −5.61 −10.63 to −0.58 0.031 \n CD 19+ 188.52 −42.76 to 419.81 0.098 \n CD level (%) on postoperative day 14 \n CD 4+ −7.54 −13.66 to −1.40 0.019 −6.87 −13.13 to −0.62 0.035\n\nMELD -- model for end-stage liver disease; CVP -- central venous pressure; MPAP -- mean pulmonary arterial pressure; ABOi LDLT -- ABO-incompatible living donor liver transplantation.",
"\n\n###### \n\nAssociation between perioperative factors and relative liver graft volume on postoperative day 21 in patients who underwent ABO-Incompatible living donor liver transplantation.",
"\n\n Characteristics Relative liver graft volume (%) on POD 21 \n -------------------------------------------------------------------- ------------------------------------------- ----------------- ------- ------- ----------------- -------\n **Preoperative recipient factors** \n Age (years) 0.62 −0.08 to 1.32 0.080 \n Gender (Male) 14.00 1.35 to 26.64 0.031 \n Body mass index (kg/m^2^) −2.40 −4.61 to −0.18 0.035 \n Laboratory parameters \n Hematocrit (%) −0.93 −1.81 to −0.05 0.039 \n Platelet count (x10^3^/μL) −0.07 −0.16 to 0.01 0.088 \n International normalized ratio 12.93 0.13 to 25.71 0.048 \n **Intraoperative recipient factors** \n Severe PRS 23.54 6.63 to 40.44 0.008 \n Average of vital signs \n CVP (mmHg) 1.87 −0.30 to 4.04 0.089 \n Blood product transfusion (unit) \n Fresh frozen plasma 1.65 −0.18 to 3.47 0.075 \n Frusemide (mg) 0.23 −0.00 to 0.47 0.055 \n Donor graft factors \n Preoperative graft volume/SLV ratio\\* 0.84 0.59 to 1.08 0.000 0.76 0.49 to 1.03 0.000\n **Specific factors in ABOi LDLT** \n Preoperative frequency of plasmapheresis −1.37 −2.95 to 0.21 0.087 \n Cluster of difference (CD) level (%) on one day before the surgery \n CD 4+ −0.37 −0.797 to 0.054 0.083 \n CD level (%) on postoperative day 14 \n CD 4+ −0.50 −0.95 to −0.04 0.032 −0.36 −0.62 to − 0.09 0.011\n\nPRS -- postreperfusion syndrome; CVP -- central venous pressure; ABOi LDLT -- ABO-incompatible living donor liver transplantation.",
"\n\n[^1]: Study Design\n\n[^2]: Data Collection\n\n[^3]: Statistical Analysis\n\n[^4]: Data Interpretation\n\n[^5]: Manuscript Preparation\n\n[^6]: Literature Search\n\n[^7]: Funds Collection\n"
] | {
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"Qui êtes-vous ?",
"\n\nAbout Pertin-nce:\nCreated by Quebec City producer Maxime Tanguay in 2005, Pertin-nce has been one of those nearly hidden netlabel treasures that has consistantly churned out a unique spectrum of offbeat electronica for the sheer pleasure of creation -- patiently and steadfastly holding to an ethic of playful experimentation and open collaboration amidst an ever expanding and evolving digital music scene.",
"\nDistribution:\nPertin-nce started of as a net-label & CD-R label from Québec, Canada to promote and distribute emerging sounds within the genres of: techno, house, digital dub, ambient, minimal & experimental.",
"\nPertin-nce releases free electronic music via its own web site pertin-nce.ca and also sells other music on sites like Beatport, Wasabeat & bandcamp."
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"Cardiovascular implantable electronic device (CIED) infections have been recognized as a growing problem, with reported incidence of 1% for initial placement and up to 7% for device reintervention \\[[@CIT0001]\\]. ",
"Pulse generator pocket infection, with or without associated bloodstream infection (BSI), is the most common clinical presentation, and the majority of patients present within 12 months of device placement or revision \\[[@CIT0002], [@CIT0003]\\]. ",
"Local manifestations of generator pocket infection include pain, erythema, swelling, drainage, dehiscence of the surgical incision, and generator or lead erosion \\[[@CIT0004], [@CIT0005]\\]. ",
"If undiagnosed or untreated, generator pocket infection may progress to involve intravascular or intracardiac portions of leads, which can present as systemic infection.",
"\n\nAlthough surgical site infections limited to superficial layers of skin, such as a stitch abscess can be managed with antibiotic therapy alone, the American Heart Association guidelines \\[[@CIT0004]\\] recommend complete extraction of an infected device for deep incisional or pocket infection. ",
"Complete removal of infected system in combination with antibiotic therapy, guided by identification and susceptibilities of the causative pathogen, is essential for curing infection \\[[@CIT0004]\\]. ",
"In clinical practice, providers frequently start empiric antimicrobial therapy at initial presentation, even in cases where infection is clinically limited to a generator pocket. ",
"Considering that majority of CIED infections are caused by staphylococci, and given the prevalence of oxacillin resistance among these organisms \\[[@CIT0006]\\], vancomycin is often initiated empirically until culture results are available \\[[@CIT0007]\\]. ",
"However, administration of antibiotics before device extraction frequently leads to negative intraoperative pocket and device cultures \\[[@CIT0008], [@CIT0009]\\], resulting in continuation of broad-spectrum antibiotic therapy after extraction. ",
"Unnecessary use of broad-spectrum antibiotics has been associated with emergence of antimicrobial resistance, one of the biggest medical challenges in modern times and a substantial threat to public health \\[[@CIT0010]\\].",
"\n\nAt present, there are no established criteria to differentiate patients with CIED pocket infection who may have underlying BSI, and therefore should immediately start empiric antibiotic therapy, from those in whom underlying BSI is unlikely and in whom it is therefore reasonable to withhold empiric antibiotic therapy, to optimize the yield of intraoperative cultures and avoid unnecessary antibiotic exposure. ",
"The aim of the current investigation was to identify independent predictors of BSI in patients presenting with device pocket infection.",
"\n\nMETHODS {#s1}\n=======\n\nWe retrospectively reviewed all adults (aged \\>18 years) presenting with CIED infection at our institution from 1 January 2005 through 1 January 2016. ",
"Cases of device infection were identified using our institutional CIED registry and defined according to standardized criteria \\[[@CIT0011]\\]. ",
"Cases were classified as CIED pocket infection if inflammatory changes were observed at the pocket site, including erythema, swelling, pain, warmth, drainage, purulence, erosion, and dehiscence \\[[@CIT0004]\\]. ",
"Patients with lead or pocket erosion with no inflammatory changes at the pocket site were excluded. ",
"Patients with CIED pocket infection were then subclassified as either having or not having concomitant BSI based on blood culture results.",
"\n\nBlood culture contamination was defined by criteria proposed by Bekeris et al \\[[@CIT0012]\\]. ",
"Accordingly, a blood culture was considered contaminated if ≥1 of the following was identified in only 1 bottle of a series of blood culture specimens: coagulase-negative *Staphylococcus* species (CoNS), *Propionibacterium* (now *Cutibacterium*) *acnes*, *Micrococcus* species, viridans-group streptococci, *Corynebacterium* species, or *Bacillus* species. ",
"A blood culture series was defined as ≥1 specimen collected serially within a 24-hour period to detect an episode of BSI. ",
"Patients with contaminated blood culture specimens were not included in our microbiologic analysis.",
"\n\nTo ensure accuracy and consistency of clinical and laboratory data collected from initial evaluation, we included only patients who were evaluated at our institution . ",
"To avoid false-negative blood culture data, we also excluded from analysis patients who received antibiotic therapy before blood cultures were obtained, as well as those who had abnormal white blood cell (WBC) count owing to medication effects or secondary to well-defined noninfectious comorbid conditions.",
"\n\nDemographic, clinical, laboratory, imaging and microbiologic data were extracted from each chart. ",
"Systemic inflammatory response (SIRS) was defined as the presence of ≥1 of the following clinical parameters: temperature ≥38.3°C or ≤36°C, pulse rate \\>90/min, respirations \\>20/min, and WBC count \\>12 000/μL or \\<4000/μL. Hypotension was defined as systolic blood pressure \\<90 mm Hg or a ≥40 mm Hg from baseline.",
"\n\nSignificant microbial growth from operative cultures was defined as isolation of ≥20 colony-forming units from 10 mL of sonicate fluid, colony growth reported in ≥2 quadrants of the culture plate from a single specimen, or microbial growth of the same organism reported from ≥2 operative samples \\[[@CIT0008]\\]. ",
"Device removal and associated complications were documented. ",
"The study was approved by the Mayo Clinic Institutional Review Board.",
"\n\nStatistical Analysis {#s2}\n--------------------\n\nWe used χ^2^ tests for categorical variables and Wilcoxon rank sum tests to associate continuous variables with 2-level categorical data. ",
"We assessed the predictors of BSI using univariate and multivariable logistic regression models. ",
"Odds ratios, 95% confidence intervals, and likelihood ratio *P* values were reported for the logistic models. ",
"Statistical tests were 2 sided, with differences considered statistically significant at *P* \\< .05. ",
"Analyses were performed using JMP 13.0 software (SAS Institute).",
"\n\nRESULTS {#s3}\n=======\n\nDemographic, Clinical, and Device Data {#s4}\n--------------------------------------\n\nOverall, 429 cases of CIED infection were seen during the study period ([Figure 1](#F1){ref-type=\"fig\"}). ",
"Among those, 248 of 429 (57.8%) had inflammatory changes at the pocket site, and 95 of them met study criteria. ",
"The excluded cases are outlined in [Figure 1](#F1){ref-type=\"fig\"}. ",
"Of these 95 study subjects, 68 (71.6%) were classified as non-BSI and 27 (28.4%) as BSI. ",
"Demographic data and underlying comorbid conditions for both groups are summarized with statistical analysis in [Table 1](#T1){ref-type=\"table\"}. ",
"There was no significant difference in the median age at presentation for the non-BSI and BSI groups. ",
"There was no difference in the prevalence of medical comorbid conditions or device characteristics between groups.",
"\n\n###### \n\nDemographic, Clinical, and Device Characteristics of 95 Patients With CIED Pocket Infection\n\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Variable | Patients, No. (%)",
"^a^ | | |\n+==============================================================================+======================+=====================+=====+\n| Demographics | | | |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Age, median (IQR), y | 67 (53--81) | 69 (58.8--77.7) | .77 |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Male sex | 48 (70.5) | 18 (66.6) | .71 |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Comorbid conditions | | | |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Type 2 diabetes mellitus | 15 (22.0) | 4 (14.8) | .89 |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Chronic kidney disease | 13 (19.1) | 4 (14.8) | .62 |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Chronic heart failure | 40 (58.8) | 20 (74) | .16 |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| COPD | 7 (10.2) | 2 (7.1) | .66 |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Immunosuppression | 5 (7.3) | 2 (7.4) | .99 |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Prosthetic valve | 11 (16.4) | 7 (25.9) | .27 |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Previous CIED infection | 4 (5.8) | 3 (11.1) | .38 |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Device | | | |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Initial placement | 18 (26.4) | 9 (33.3) | .50 |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Device revision | 48 (70.5) | 18 (66.6) | .71 |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| ICD | 27 (39.7) | 12 (44.4) | .89 |\n| | | | |\n| Permanent pacemaker | 26 (38.2) | 9 (33.3) | |\n| | | | |\n| CRT | 15 (22) | 6 (22.2) | |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n| Time from implantation/revision to development of symptoms, median (IQR), d | 140 (35--623) | 106.5 (22.2--756.7) | .35 |\n+------------------------------------------------------------------------------+----------------------+---------------------+-----+\n\nAbbreviations: BSI, bloodstream infection; CIED, cardiovascular implantable electronic device; COPD, chronic obstructive pulmonary disease; CRT, cardiac resynchronization therapy; ICD, implantable cardioverter defibrillator; IQR, interquartile range.",
"\n\n^a^Data represent No. (%) ",
"of patients, unless otherwise specified.",
"\n\n{#F1}\n\nSymptoms Reported at Initial Evaluation {#s5}\n---------------------------------------\n\nConcomitant lead or generator erosion was reported in 25 of 68 patients (36.7%) in the non-BSI group and 3 of 27 (11.1%) in the BSI group (*P*= .01). ",
"In the BSI group, pocket infection was complicated with CIED valvular endocarditis in 2 of 27 cases (both due to *Staphylococcus aureus*) and with CIED lead endocarditis in 5 of 27 (1 due to *S. aureus,* 2 due to *Enterococcus faecalis,* and 2 due to CoNS). ",
"There were 2 cases of CIED lead and tricuspid valve endocarditis related to *S. aureus*. ",
"A majority of patients in the non-BSI group reported symptoms at the pocket site \\>14 days in duration (60.2% vs 33.3%; *P*= .02). ",
"General malaise was reported more frequently in the BSI (92.8%) than in the non-BSI (2.9%) cohort (*P*\\< .001).",
"\n\nVital Signs at Initial Evaluation {#s6}\n---------------------------------\n\nPatients with pocket infection and BSI had a higher median maximum temperature than the non-BSI group (38.1°C \\[interquartile range (IQR)\\], 36.5°C --38.8°C\\] vs 36.6°C \\[36.5°C --36.8°C\\]; *P*\\< .001) and more frequently documented fever or hypothermia at presentation (40.7% vs 1.4%, respectively; *P*\\< .001). ",
"Patients with BSI were also more likely to present with hypotension (40.7% vs 4.4%, respectively; *P*\\< .001), pulse rate \\>90/min (37% vs 4.4%; *P*\\< .001), respirations \\>20/min (38.4% vs 16.4; *P* = .02), septic shock (21.4% vs 0%; *P*\\< .001), and organ failure (42.3% vs 1.4%; *P* \\< .001), primarily acute kidney injury (63% \\[7 of 11\\]).",
"\n\nLaboratory Data at Initial Presentation {#s7}\n---------------------------------------\n\nThe BSI group had a higher median WBC count at admission than the non-BSI group (12.4 × 10^9^/L \\[IQR, 8.7--16.1 × 10^9^/L\\] vs 7.2 × 10^9^/L \\[5.6--9.1 × 10^9^/L\\], respectively; *P* \\< .001), with leukocytosis or leukopenia present in 59.2% vs 2.9%, respectively (*P* \\< .001). ",
"The BSI cohort also had higher median serum creatinine (1.1 \\[IQR, 0.9--1.65\\] vs 1 \\[0.8--1.3\\] mg/dL; *P* = .04) and aspartate aminotransferase (29 U/L \\[25--47) vs 22 \\[20--27\\] U/L; *P*= .02) values. ",
"There were no statistically significant differences in platelet count, blood lactate level, erythrocyte sedimentation rate, or C-reactive protein level.",
"\n\nPredictors of BSI {#s8}\n-----------------\n\nWe performed univariate and multivariable analysis of clinical predictors associated with BSI in patients presenting with CIED pocket infection. ",
"In the univariate analysis ([Table 2](#T2){ref-type=\"table\"}), patients with BSI were more likely to meet SIRS criteria, have hypotension, and have ≥2 abnormal laboratory values (C-reactive protein, erythrocyte sedimentation rate, aspartate aminotransferase, and blood urea nitrogen). ",
"Symptom duration \\>14 days before presentation was correlated with a lower risk of BSI. ",
"Given the small sample size, only variables from the univariate analysis with *P*values \\<.05 were included in multivariable analysis ([Table 3](#T2){ref-type=\"table\"}). ",
"Independent variables associated with BSI in multivariable analysis included the presence of SIRS criteria and hypotension.",
"\n\n###### \n\nClinical Presentation in 95 Patients With CIED Pocket Infection\n\n Patients, No. (%)",
"^a^ \n -------------------------------------------------- ---------------------- -------------------- -----------\n Symptoms \n Reported symptoms for \\>14 d 41 (60.2) 9 (33.3) .02\n Chills 4 (5.8 24 (85.7) \\<.001^b^\n General malaise 2 (2.9) 26 (92.8) \\<.001^b^\n Altered mental status 0 4 (14.8) \\<.001^b^\n Signs at physical examination \n Erythema 46 (67.6) 18 (66.6) .93\n Tenderness 31 (45.5) 17 (62.9) .13\n Swelling 21 (30.8) 12 (44.4) .21\n Warmth 4 (5.8) 3 (11.1) .38\n Purulent drainage 6 (22.2) 18 (26.4) .67\n Fluctuance 3 (4.4) 4 (14.8) .08\n Surgical incision dehiscence 9 (13.2) 2 (7.4) .42\n Lead or generator erosion 25 (36.7) 3 (11.1) .01^b^\n Vital signs at initial evaluation \n Maximum temperature, median (IQR), °C 36.6 (36.5--36.8) 38.1 (36.52--38.8) \\<.001^b^\n Maximum temperature \\>38.3°C or \\<36°C 1 (1.4) 11 (40.7) \\<.001^b^\n Pulse rate, median (IQR), beats/min 70 (62--80) 75 (65--100) .08\n Pulse rate, \\>90 beats/min 3 (4.4) 10 (37) \\<.001^b^\n Respirations, median (IQR), respirations/min 18 (16--19) 18 (17.5--22.5) .04^b^\n Respirations \\>20/min 11 (16.4) 10 (38.4) .02^b^\n SBP, median (IQR), mm Hg 120 (108--131.5) 112 (85--125) .02^b^\n SBP \\<90 mm Hg or ≥40--mm Hg drop from baseline 3 (4.4) 11 (40.7) \\<.001^b^\n Septic shock 0 6 (21.4) \\<.001^b^\n Organ failure 1 (1.4) 11 (42.3) \\<.001^b^\n Laboratory data at initial presentation \n WBC count, median (IQR), ×10^9^/L 7.2 (5.6--9.1) 12.4 (8.7--16.1) \\<.001^b^\n WBC count \\>12 × 10^9^/L or \\<4 × 10^9^/L 2 (2.9) 16 (59.2) \\<.001^b^\n Serum creatinine, median (IQR), mg/dL 1 (0.8--1.3) 1.1 (0.9--1.7) .04^b^\n BUN, median (IQR), mg/dL 20 (14--28) 1.2 25 (17.7--37) .06\n AST, median (IQR), U/L 22 (20--27) 29 (25--47) .02^b^\n\nAbbreviations: AST, aspartate aminotransferase; BSI, bloodstream infection; CIED, cardiovascular implantable electronic device; IQR, interquartile range; SBP, systolic blood pressure; BUN, blood urea nitrogen; WBC, white blood cell.",
"\n\n^a^Data represent No. (%) ",
"of patients, unless otherwise specified.",
"\n\n^b^Significant at *P* \\< .05.",
"\n\n###### \n\nUnivariate and Multivariable Analysis of Predictors Associated With Bloodstream Infection in the 95 Patients With CIED Pocket Infection\n\n Risk Factor Univariate OR (95% CI) *P* Value Multivariable OR (95% CI) *P* Value\n -------------------------------------------------- ------------------------ ----------- --------------------------- -----------\n Age \\>65 y 1.2 (.5--3.1) .58 ... ...\n ≥2 Medical comorbid conditions^a^ 1.0 (.4--2.6) .87 ... ...\n SIRS criteria^b^ 29.5 (8.8--98.1) \\<.001^c^ 28.9 (7.6--144) \\<.001^c^\n SBP \\<90 mm Hg or \\>40--mm Hg drop from baseline 14.8 (3.7--59.7) \\<.001^c^ 12.8 (1.9--101) \\<.001^c^\n ≥2 Abnormal laboratory values^d^ 3.0 (1.1--7.7) .02^c^ 2.7 (.5--13.0) .19\n Reported symptoms for \\>14 d 0.3 (.1--.8) .02^c^ 0.29 (.06--1.1) .58\n\nAbbreviations: CI, confidence interval; CIED, cardiovascular implantable electronic device; OR, odds ratio; SBP, systolic blood pressure; SIRS, systemic inflammatory response syndrome.",
"\n\n^a^Medical comorbid conditions included the following: diabetes mellitus, chronic kidney disease, heart failure, liver disease, chronic pulmonary obstructive disease, prosthetic valve replacement, hematologic cancer, and immunosuppression.",
"\n\n^b^SIRS is defined as ≥ of the following: temperature ≥38.3°C or ≤36°C, pulse rate \\>90/min, respirations \\>20/min, and white blood cell count \\>12 000/μL or \\<4000/μL.\n\n^c^Significant at *P* \\< .05.",
"\n\n^d^Abnormal laboratory values included the following: creatinine \\>1.2 mg/dL, erythrocyte sedimentation rate \\>22 mm/h, C-reactive protein \\>3.0 mg/L, blood urea nitrogen \\>20 mg/dL, and aspartate aminotransferase \\>40 U/L.\n\nAntimicrobial Management and Operative Culture Yield {#s9}\n----------------------------------------------------\n\nManagement, operative culture yield and outcomes in all 95 patients with CIED pocket infection are summarized in [Table 4](#T4){ref-type=\"table\"}. ",
"At admission, providers elected to withhold antimicrobials until device removal in slightly more than half of patients in the non-BSI group (57.3%) and in 1 of 25 in the BSI group. ",
"In this particular patient, no SIRS criteria were recorded at admission or laboratory abnormalities at presentation. ",
"Antibiotic therapy was started in this patient once blood culture results returned positive.",
"\n\n###### \n\nManagement, operative culture yield and outcomes of the 95 patients with CIED pocket infection\n\n Patients, No. (%)",
"^a^ \n ----------------------------------------------------------------------- ---------------------- ------------- -----------\n CIED removal 68 (100) 25 (92.5) .02\n Antibiotics withheld until device removal 38 (55.8) 1/25 (4)^b^ \\<.001^c^\n Duration off antibiotic therapy until device removal, median (IQR), d 2 (1--7) 0 (0--2.7) \\<.001^c^\n Clinical deterioration while hospitalized and off antibiotic therapy 0 0 ...\n Complications at time of removal^d^ 1 (1.4)^d^ 1/25 (4)^d^ .56\n CIED infection relapse or recurrence^e^ 2/66 (3.0) 3/25 (12) .10\n Culture yield from CIED devices \n Positive Gram stain 12 (31.5) 7 (23.3) .50\n Significant microbial growth from intraoperative cultures 48 (70.5) 22/25 (80) .08\n Antibiotics withheld before CIED removal (n = 39) 31 (79.4)^f^ ... ...\n Antibiotics received before CIED removal (n = 29) 17 (58.6)^f^ ... ...\n\nAbbreviations: BSI, bloodstream infection; CIED, cardiovascular implantable electronic device; IQR, interquartile range.",
"\n\n^a^Data represent No. (%) ",
"of patients, unless otherwise specified.",
"\n\n^b^Only 25 patients in the BSI group underwent device removal.",
"\n\n^c^Significant at *P* \\< .01.",
"\n\n^d^One patient in the non-BSI had a mechanical complication and the other in the BSI had complications related to infection.",
"\n\n^e^A total of 66 patients in the non-BSI and 25 in the BSI group underwent device reimplantation after treatment of the index episode.",
"\n\n^f^ *P* = .06 for comparison between these 2 non-BSI subgroups.",
"\n\nThe median time off antibiotic therapy until device removal in the non-BSI group was 2 (IQR, 1--7) days. ",
"None of the patients in the non-BSI group showed clinical progression or deterioration while off antibiotic therapy. ",
"The device was removed in all patients in the non-BSI group and 25 of 27 (92.5%) in the BSI group. ",
"Of the 2 patients whose device was retained, 1 died of infection complications before the scheduled procedure and the other declined surgical intervention and was kept on life-long antibiotic suppression. ",
"Both received antimicrobials at the time of initial presentation.",
"\n\nOf the patients undergoing device removal, complications were reported in 1 patient in the non-BSI group, who had superior vena cava rupture, and in 1 in the BSI group, who experienced cardiac arrhythmia and subsequently died. ",
"However, none of the patients in whom antibiotic therapy was initially withheld had infection-related complications. ",
"No infection relapse or recurrence, after reimplantation of a new device, was reported in patients in whom initial antibiotic therapy was withheld until extraction of the infected device.",
"\n\nOverall, in the non-BSI group, positive intraoperative device/tissue cultures were more common in patients in whom antibiotics were initially withheld until device extraction (79.4% vs 58.6% in their antibiotic-treated counterparts), however, this difference was not statistically significant (*P* = .061).",
"\n\nMicrobiology {#s10}\n------------\n\nOrganisms isolated in blood and operative cultures from the 95 cases of CIED pocket and BSI are shown in [Table 5](#T5){ref-type=\"table\"}. ",
"The median time to blood culture positivity was 16 (IQR, 11.5--35.5) hours, and the median duration of BSI was 2.5 (1--4.25) days. ",
"In patients with positive blood and intraoperative cultures, concordance was reported in 80% (20 of 25) of cases. ",
"Of the remaining 5 cases, 2 had discrepant results between blood and intraoperative cultures; and 3 had negative intraoperative culture results. ",
"All 5 of these cases received antibiotics at the time of presentation.",
"\n\n###### \n\nMicrobiology of the 95 Cases of CIED Pocket Infection\n\n Positive Cultures, No. ",
" \n ------------------------- ------------------------ ---- ----\n CoNS 20 7 5\n *Staphylococcus aureus* 12 16 13\n *Corynebacterium* sp. ",
" 1 1 1\n *Enterococcus faecalis* 0 3 0\n *Cutibacterium acnes* 9 0 0\n *Pseudoclavibacter* sp. ",
" 1 0 0\n Gram-negative bacteria 2 0 1\n ≥2 Organisms 3 0 0\n\nAbbreviations: BSI, bloodstream infection; CIED, cardiovascular implantable electronic device; CoNS, coagulase- negative *Staphylococcus* species\n\n^a^Intraoperative cultures obtained in the 25 patients with BSI who underwent device removal.",
"\n\nDISCUSSION {#s11}\n==========\n\nOur study findings suggest that patients with CIED pocket infection who present with SIRS criteria (fever or hypothermia, tachycardia, tachypnea, and leukocytosis or leukopenia) and/or hypotension (systolic blood pressure \\<90 mm Hg or a \\>40--mm Hg drop from baseline) are more likely to have underlying BSI. ",
"Consequently, patients presenting with these features should be started on empiric antimicrobial therapy once blood cultures have been obtained. ",
"On the contrary, the absence of these features suggests that underlying BSI is less likely, and it may be reasonable to withhold starting empiric antimicrobial therapy. ",
"In our cohort, none of the patients who were deemed clinically stable (absence of SIRS criteria or hypotension) and in whom empiric antibiotics were withheld (for a median of 2 days) experienced clinical progression or hemodynamic deterioration while awaiting device extraction. ",
"Furthermore, in the non-BSI group, the percentage of cases with significant microbial growth reported from intraoperative cultures was higher in patients in whom antibiotic therapy was withheld until device removal (79.4% vs 58.6% among those in whom it was not withheld; *P* = .06). ",
"The statistical non-significance of this comparison is likely due to the small sample size.",
"\n\nSimilar observations have been reported for hip and knee prosthetic joint infections, in which discontinuation of antibiotic therapy for ≥14 days before surgery led to higher culture sensitivity in patients with planned resection arthroplasty \\[[@CIT0013]\\]. ",
"Based on these observations, withholding antibiotic therapy for suspected prosthetic joint infections, in patients who do not have BSI, is now a standard clinical practice at our institution and others. ",
"We believe a similar case can be made for patients with CIED pocket infections, without BSI, in whom device extraction is planned in the next 2--3 days.",
"\n\nPatients with a longer duration of symptoms at initial presentation were more likely to have infection limited to the generator pocket on univariate analysis. ",
"However, this association was not significant in multivariable analysis, probably owing to small sample size. ",
"The paucity of systemic symptoms and the absence of BSI in patients presenting with longer duration of symptoms may be, in part, due to infection with more indolent organisms, as demonstrated by higher prevalence of CoNS in this group of patients. ",
"Interestingly, compared with the non-BSI group, patients with BSI were more likely to have pocket infections due to *S. aureus*, suggesting that infection with this particular organism might lead to higher risk of endovascular infection. ",
"Although an association between *S. aureus* BSI from a remote site of infection and hematogenous seeding of CIED leads has been observed earlier \\[[@CIT0014]\\], this particular association between pocket infection with *S. aureus* and higher risk of associated BSI is novel and not reported earlier.",
"\n\nPredictors of BSI in patients presenting with CIED pocket infection have not been specifically evaluated in prior publications. ",
"Le et al \\[[@CIT0015]\\] analyzed risk factors for CIED-related infective endocarditis (CIED-IE) in an earlier investigation and reported that use of immunomodulator therapy and hemodialysis were associated with a higher risk of CIED-IE. ",
"These patients were more likely to present with leukocytosis, fever, or malaise compared to non CIED-IE cases. ",
"Interestingly, patients with CIED-IE were less likely to present with infection at the generator pocket site, suggesting that the predominant mechanism of IE in these cases was hematogenous seeding of device leads or heart valves from distant sources of bacteremia.",
"\n\nDistinguishing patients with CIED pocket infection who may have underlying BSI from those who do not, before availability of blood culture results, is crucial, because patients with BSI have poorer outcomes, and delays in starting empiric antibiotics and device extraction can have detrimental effects \\[[@CIT0016]\\].",
"\n\nIn current practice, empiric antimicrobial therapy before device removal is based on personal discretion of the treating physician without much evidence to guide the decision-making process. ",
"This approach is concerning from a stewardship perspective, as routine administration of empiric antibiotics in stable patients, with no risk factors for underlying BSI, can negatively impact the yield of blood and operative specimen cultures. ",
"Consequently, pathogen-specific therapy is not an option in patients with negative cultures, who are often treated with broad-spectrum antimicrobial therapy \\[[@CIT0009]\\]. ",
"Such approach opposes current efforts to minimize antibiotic overuse in an attempt to reduce antimicrobial resistance.",
"\n\nBased on our study findings and review of the published literature, we recommend obtaining 2 sets of blood cultures in all patients regardless of clinical presentation. ",
"Patients with ≥1 predictor of underlying BSI should be started on empiric antibiotic therapy directed against staphylococci. ",
"Whether or not addition of aerobic gram-negative coverage is necessary should be determined on a case-by-case basis. ",
"For patients without any positive predictors of underlying BSI, antibiotic therapy may be withheld to optimize the intraoperative culture yield. ",
"All patients with confirmed CIED infection, regardless of clinical presentation, should be evaluated as soon as possible for complete device extraction \\[[@CIT0016]\\].",
"\n\nOur study is retrospective, and its design has inherent limitations. ",
"Considering the large tertiary academic nature of our practice, there is also a possibility of referral bias. ",
"It is certainly possible that patients referred to our medical center were sicker and more likely to have received prior courses of antimicrobial therapy and prior management, such as partial removal of cardiac device. ",
"This might skew our data compared with overall cases of CIED infection, which might affect subsequent interpretation of results. ",
"Owing to the small sample size, we were unable to develop a robust scoring system to predict underlying BSI in patients presenting with CIED pocket infection.",
"\n\nIn conclusion, our study findings suggest that patients with CIED pocket infection who are afebrile, hemodynamically stable, and have normal leukocyte counts at admission are unlikely to have underlying BSI. ",
"Therefore, it may be reasonable to withhold antimicrobial therapy in these patients to optimize the intraoperative culture yield, provided that device extraction is planned in the next 2--3 days.",
"\n\n***Financial support.*** ",
"Dr. Sohail reports receiving funds from TYRX Inc. and Medtronic for prior research unrelated to this study administered according to a sponsored research agreement between Mayo Clinic and study sponsor that prospectively defined the scope of the research effort and corresponding budget; and honoraria/consulting fees from Medtronic Inc., Spectranetics, Boston Scientific Corp., and Aziyo Biologics, Inc.\n\n***Potential conflicts of interest.*** ",
"P. A. F. reports honoraria/consultant fees from Medtronic, Guidant, and Astra Zeneca; research grant from Medtronic, Astra Zeneca via Beth Israel, Guidant, St Jude, Bard; and intellectual property rights with Bard EP, Hewlett Packard, and Medical Positioning. ",
"L. M. B. reports royalty payments (authorship) from UpToDate (\\<\\$20 000) and consultant payments from Boston Scientific; \\<20,000. ",
"M. R. S. reports receiving funds from Medtronic for prior research unrelated to this study and honoraria/consulting fees from Medtronic, Spectranetics, Boston Scientific, and Aziyo Biologics (all \\<\\$20 000).",
"\n\nAll other authors report no conflicts.",
"All authors have submitted the ICMJE Form for Disclosure of Potential Conflicts of Interest. ",
"Conflicts that the editors consider relevant to the content of the manuscript have been disclosed.",
"\n"
] | {
"pile_set_name": "PubMed Central"
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"Q:\n\nDecomposition of bilinear function into symmetric and skew-symmetric in $\\mathbb{k}$ with $\\operatorname{char}\\mathbb{k}=2$\n\nI know that if $\\varphi:V\\times V\\to \\mathbb{k}$ with $\\operatorname{char}\\mathbb{k}\\neq 2$ is bilinear function then $\\varphi=a+b$, where $a, b$ are symmetric and skew symmetric bilinear functions. ",
"\nIt can be done in the following way: $$\\varphi(x,y)=\\dfrac{\\varphi(x,y)+\\varphi(y,x)}{2}+\\dfrac{\\varphi(x,y)-\\varphi(y,x)}{2}=a(x,y)+b(x,y).$$\nBut is this claim true if $\\operatorname{char}\\mathbb{k}=2$? ",
"I guess not because in this field I cannot divide by $2$.\nLet's do it rigorously. ",
"Suppose we can write $\\varphi(x,y)=a(x,y)+b(x,y)$ where $a(x,y), b(x,y)$ are symmetric and skew-symmetric functions, respectively. ",
"\nThen one can show that: $\\varphi(x,y)=-\\varphi(y,x)$ and $\\varphi(x,y)=\\varphi(y,x)$ which means that $\\varphi$ is simultaneously symmetric and skew-symmetric. ",
"But where is the contradiction?",
"\nWould be very thankful for any comments!",
"\n\nA:\n\nNote that in such a field, the symmetric forms and the skew forms are the same because $-k = k$. Hence if $b(x,y) = -b(y,x)$ then in fact $b(x,y) = b(y,x)$ and\nso $b$ is symmetric.",
"\nSo to answer the question, any non symmetric bilinear form cannot be represented as the sum of a symmetric form and skew form.",
"\nFor example, $\\phi(x,y) = x_1 y_2$. We see that $\\phi(e_1,e_2) = 1$ but\n$\\phi(e_2,e_1) = 0$.\n\n"
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"A great winter work parka. ",
"I bought the wrong size through work rep and cannot return. ",
"The one I use is great - you can stand around all day and not get cold. ",
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"Chipping Barnet (UK Parliament constituency)\n\nChipping Barnet is a constituency created in 1974 represented in the House of Commons of the UK Parliament since 2005 by Theresa Villiers of the Conservative Party. ",
"It is part of the London Borough of Barnet.",
"\n\nConstituency profile\nThe constituency is slightly more elevated than the surrounding areas. ",
"Barnet was once a narrow projection of Hertfordshire into the county of Middlesex, and consisted of an agricultural market town. ",
" The town became well connected to central London by the London Underground network and is today prime suburbia, with many of its properties semi-detached with substantial gardens as well as having many small parks and nature reserves. ",
" The area is largely bereft of tower blocks of social housing original purpose.",
"\n\nIt has been held by a Conservative since its creation for the February 1974 general election, and withstood the Labour landslide in 1997 by just over 2% (1,035 votes). ",
" The 2015 result gave the seat the 92nd most marginal majority of the Conservative Party's 331 seats by percentage of majority. ",
"In 2017, the Conservatives came closer than ever before to losing Chipping Barnet, with incumbent Theresa Villiers retaining the seat by just 353 votes, with Labour now holding the majority of councillors in the constituency, in Brunswick Park, Coppetts, East Barnet, and Underhill. ",
"However, as of the 2018 Barnet London Borough Council elections, the Conservatives now hold a majority of councillors in Brunswick Park, High Barnet, Oakleigh, and Totteridge.",
"\n\nIn the 2019 general election, the seat was seen as an important potential gain for the Labour Party, due to Villiers' small majority and high profile (as the Environment Secretary) and the seat's vote to remain in the 2016 European Union membership referendum, though it was reported that many voters were reluctant to vote for the Labour candidate over concerns about antisemitism in the party. ",
"Nevertheless, Villiers retained the seat with an increased majority.",
"\n\nBoundaries\n1974–1997: The London Borough of Barnet wards of Arkley, Brunswick Park, East Barnet, Hadley, and Totteridge.",
"\n\n1997–2010: As above plus Friern Barnet.",
"\n\n2010–present: The London Borough of Barnet wards of Brunswick Park, Coppetts, East Barnet, High Barnet, Oakleigh, Totteridge, and Underhill.",
"\n\nFor the 2010 general election, following a review of parliamentary representation and as a consequence of changes to ward boundaries, the Boundary Commission for England recommended that parts of Underhill ward and Coppetts ward be transferred to Chipping Barnet from the constituencies of Hendon and Finchley and Golders Green respectively. ",
"It also recommended that a small part of Mill Hill ward be transferred from Chipping Barnet to Hendon.",
"\n\nMembers of Parliament\n\nElections\n\nElections in the 2010s\n\nElections in the 2000s\n\nElections in the 1990s\n\nElections in the 1980s\n\nElections in the 1970s\n\nSee also \n List of Parliamentary constituencies in Greater London\n\nNotes and references\nNotes \n \nReferences\n\nExternal links \nnomis Constituency Profile for Chipping Barnet — presenting data from the ONS annual population survey and other official statistics.",
"\nPolitics Resources (Election results from 1922 onwards)\nElectoral Calculus (Election results from 1955 onwards)\n\nCategory:Politics of the London Borough of Barnet\nCategory:Parliamentary constituencies in London\nCategory:United Kingdom Parliamentary constituencies established in 1974"
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"Vladimir Turiyansky\n\nVladimir Lvovich Turiyansky is a Russian poet, composer, and bard.",
"\n\nAfter studying for three years in the Moscow State Art and Cultural University, Turiyansky worked as a geologist, going on expeditions to Siberia. ",
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"\n\nIn 1959, Turiyansky began writing songs, primarily around his poems.",
"\n\nTuriyansky now lives in Moscow, Russia.",
"\n\nReferences\n\nCategory:1935 births\nCategory:Living people\nCategory:Russian bards\nCategory:Russian geologists\nCategory:Russian guitarists\nCategory:Russian male composers\nCategory:Russian male musicians\nCategory:Russian male poets\nCategory:Russian male singer-songwriters"
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"Compartment-directed physical examination of the knee can predict articular cartilage abnormalities disclosed by needle arthroscopy.",
"\nTo determine whether physical examination maneuvers that focus on each knee compartment and assess crepitus at several distinct sites can specifically disclose articular cartilage abnormalities in the compartment being assessed. ",
"Twenty patients with knee pain were examined before needle arthroscopy. ",
"Crepitus was sought from the patellofemoral compartment, medial tibiofemoral compartment, and lateral tibiofemoral compartment. ",
"Any crepitus felt in the distal tibia during a tibiofemoral stress maneuver was recorded as transmitted bony crepitus (TBC). ",
"Needle arthroscopy assessed articular cartilage (5 sites) and both menisci in each knee. ",
"Crepitus by conventional assessment revealed patellar cartilage disruption (69% sensitive, 50% specific) and abnormalities of tibiofemoral cartilage (67% sensitive, 40% specific) but could not indicate their location. ",
"Tibiofemoral crepitus found cartilage disruption in the compartment at a sensitivity of 22% and a specificity of 100%, and with added tibiofemoral stress, a sensitivity of 65% and a specificity of 94% (the one \"false positive\" had bare bone in the other compartment). ",
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"VANCOUVER -- The judge in the first-degree murder trial for Rocky Rambo Wei Nam Kam, on trial for the September 2017 murders of a Vancouver couple in their Marpole home, has released video of police interrogating the then 25-year-old two months after the killings.",
"\n\nDuring eight hours of questioning, police asked Kam about evidence linking him to the crime, including DNA and surveillance video. ",
"Kam repeatedly tells interrogators, \"I have nothing to say.\"",
"\n\nNear the end of the video, a police offer said to Kam, \"I can see you’re emotional. ",
"Why aren’t you taking this chance to at least express remorse? ",
"Maybe today you’re not ready to talk about the whole ugly story. ",
"But at least say you’re sorry.”",
"\n\nKam replied, \"For what?\"",
"\n\nThe officer responded, \"For what? ",
"You’re an animal.\" ",
"She then walked out of the interrogation. ",
"With the room empty, Kam is recorded speaking in Chinese, roughly translated to \"This is f**king crazy. ",
"I didn’t kill anyone. ",
"How can I be sorry? ",
"I can only feel sorry if I killed someone.\" ",
"Kam has pleaded not guilty.",
"\n\nFast forward to his first-degree murder trial, and Kam has testified on the witness stand about killing Dianna Mah-Jones and her husband Richard Jones, but claims he thought he was playing a video game. ",
"His lawyers are arguing he did not plan the crime in advance, was in a dissociative state and could not form intent for first-degree murder.",
"\n\nCrown is now cross-examining Kam, zeroing in on the items he purchased in the weeks leading up to the murders, including a hatchet, gloves, a knife and twine that were found at the crime scene. ",
"Kam said he bought the hatchet after seeing an axe throwing scene on the TV show Modern Family, the gloves to hold the hatchet, the twine to make a craft shaped like a horse, and the knife to cut the twine. ",
"He insists he didn’t buy them because he planned to kill someone.",
"\n\nCrown pointed out Kam was very thrifty and rarely spent money, and questioned why he would buy all these items for frivolous reasons like axe throwing and crafts. ",
"Kam said he bought the cheapest versions of the items he wanted.",
"\n\nThe prosecution also asked Kam about web searches he made before the murders for a stun gun, drugs to incapacitate people, and crime scene cleanup. ",
"Kam denied the searches show he was planning a murder.",
"\n\nProsecutors said their cross-examination of Kam will extend into Tuesday, and after his testimony is complete, defence is expected to call a psychologist to the stand."
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"/*\n * PanmirrorTheme.java\n *\n * Copyright (C) 2020 by RStudio, PBC\n *\n * Unless you have received this program directly from RStudio pursuant\n * to the terms of a commercial license agreement with RStudio, then\n * this program is licensed to you under the terms of version 3 of the\n * GNU Affero General Public License. ",
"This program is distributed WITHOUT\n * ANY EXPRESS OR IMPLIED WARRANTY, INCLUDING THOSE OF NON-INFRINGEMENT,\n * MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. ",
"Please refer to the\n * AGPL (http://www.gnu.org/licenses/agpl-3.0.txt) for more details.",
"\n *\n */\n\npackage org.rstudio.studio.client.panmirror.theme;\n\nimport jsinterop.annotations.",
"JsType;\n\n@JsType\npublic class PanmirrorTheme\n{\n public boolean darkMode;\n public boolean solarizedMode;\n public String cursorColor;\n public String selectionColor;\n public String nodeSelectionColor;\n public String backgroundColor;\n public String metadataBackgroundColor;\n public String chunkBackgroundColor;\n public String spanBackgroundColor;\n public String divBackgroundColor;\n public String commentColor;\n public String commentFontStyle;\n public String commentBackgroundColor;\n public String gutterBackgroundColor;\n public String gutterTextColor;\n public String textColor;\n public String surfaceWidgetTextColor;\n public String lightTextColor;\n public String linkTextColor;\n public String placeholderTextColor;\n public String invisibleTextColor;\n public String markupTextColor;\n public String findTextBackgroundColor;\n public String findTextBorderColor;\n public String borderBackgroundColor;\n public String blockBorderColor;\n public String focusOutlineColor;\n public String paneBorderColor;\n public String fixedWidthFont;\n public double fixedWidthFontSizePt;\n public String proportionalFont;\n public double proportionalFontSizePt;\n public PanmirrorThemeCode code;\n}\n"
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"Bonjour, J'ai trouvé un article sur l'Etudiant susceptible de vous intéresser. ",
"Pour le découvrir, cliquez ici : https://www.letudiant.fr/educpros/actualite/le-palmares-2018-des-ecoles-d-ingenieurs.html A bientôt sur letudiant.fr !",
"\n\nLe palmarès 2018 des écoles d'ingénieurs de l'Etudiant se dote cette année d'un nouveau critère : l'ouverture à de nouveaux publics. ",
"De quoi chambouler le classement ? ",
"Revue de détails.",
"\n\nComme lors sa précédente édition, l'Etudiant publie en 2018 un palmarès général, somme des notes obtenues par les écoles dans trois grandes familles d'indicateurs : l'excellence académique, l'ouverture internationale et la proximité avec les entreprises. ",
"Nouveauté cette année, un nouveau critère accordant un à trois points bonus leur a été adjoint : l'ouverture à de nouveaux publics.",
"\n\nConstruit pour valoriser les initiatives en faveur des jeunes rencontrant des difficultés dans le financement de leurs études, ce nouvel indicateur tient compte de la part d'élèves en apprentissage (ce qui leur permet d'être salariés parallèlement à leurs études), des montants des frais de scolarité, du taux d'élèves boursiers, ainsi que de la possibilité de bénéficier d'une rémunération dans certains établissements publics comme Polytechnique, l'École nationale de météorologie, ou encore Arts et Métiers.",
"\n\nDans le palmarès, les écoles sont ordonnées selon leur nombre de points au sein de quatre groupes : A+ (de 62 à 49 points), A (de 48 à 37 points), B (de 36 à 27 points) et C (moins de 27 points), afin de les comparer et de les situer dans le paysage des formations d'ingénieurs en un coup d'œil.",
"\n\nLes formations des groupe A+ et A\n\nÉcoles Total des points en 2018 Excellence académique Ouverture internationale Proximité avec les entreprises Ouverture à de nouveaux publics Groupe A+ École polytechnique, Palaiseau 62 20 20 20 2 CentraleSupélec (diplôme Centrale) 59 18 20 20 1 École des ponts ParisTech, Marne-la-Vallée 57 18 18 20 1 CentraleSupélec (diplôme Supélec) 55 18 16 20 1 Mines ParisTech 54 18 14 20 2 Centrale Lyon 53 18 15 18 2 Centrale Nantes 16 16 19 2 Télécom ParisTech 20 12 20 1 Insa Lyon 50 16 15 17 2 IMT Atlantique, Brest, Nantes 49 15 13 19 2 ISAE-Supaero, Toulouse 19 11 17 2 Groupe A Mines Nancy 48 15 17 14 2 Centrale Lille 46 14 14 16 2 ENSAE ParisTech, Palaiseau 15 11 18 2 UTC, Compiègne 15 11 18 2 ESPCI Paris 45 20 11 13 1 AgroParisTech 44 17 7 18 2 ENSTA ParisTech, Palaiseau 18 11 14 1 Grenoble INP-Ensimag 18 11 13 2 Arts et Métiers (1) 43 12 11 18 2 Mines Saint-Étienne 17 9 15 2 Centrale Marseille 42 15 13 12 2 ENSCP Chimie ParisTech 17 12 11 2 Grenoble INP-Phelma 41 19 12 8 2 INSA Toulouse 16 9 14 2 ENSEEIHT-INP Toulouse 40 16 10 12 2 Grenoble INP-Ense³ 16 10 12 2 ECPM, Strasbourg 38 17 11 8 2 IMT Lille-Douai 37 11 8 16 2 Insa Rouen 12 11 12 2 Télécom SudParis, Évry 13 10 12 2 UTT, Troyes 12 9 14 2\n\nL'IMT Atlantique se rapproche de l'élite\n\nResserré par rapport au palmarès 2017, le groupe A+ consacre les toutes meilleures écoles d'ingénieurs françaises, brillant notamment par la qualité de leurs formations et leurs excellents rapports avec le monde professionnel. ",
"Leurs étudiants, triés sur le volet après des concours très sélectifs, sont fortement recherchés par les grandes entreprises et les meilleurs laboratoires.",
"\n\nSix grandes écoles basées en Île-de-France accaparent la majorité des premières places : Polytechnique (avec 62 points sur 63) devance d'une courte tête CentraleSupélec (et ses deux programmes, encore distincts, amenés à fusionner l'année prochaine), l'École des ponts ParisTech, les Mines ParisTech et Télécom ParisTech. ",
"Elles sont accompagnées par d'excellentes écoles régionales : Centrale Lyon et Nantes, ISAE-Supaero, à Toulouse, Insa Lyon, premier établissement postbac de notre palmarès, dont la réputation et les performances n'ont rien à envier aux grandes post-prépas, ainsi que IMT Atlantique. ",
"Issue de la fusion toute récente des Mines de Nantes et de Télécom Bretagne, la nouvelle école a su tirer parti des forces de ses composantes d'origine pour atteindre une taille critique et se rapprocher de l'élite.",
"\n\nDans les INP, les écoles partagent leurs ressources\n\n\n\nJuste au-dessous, le groupe A rassemble 21 formations publiques, dont une partie de grandes généralistes, qui se retrouvent sur l'ensemble du territoire et sont issues de grands réseaux : Centrale (Lille, Marseille), les Mines et Télécom (Saint-Étienne, Lille-Douai, SudParis), Insa (Toulouse, Rouen) et UT (Compiègne, Troyes). ",
"De grandes écoles spécialisées en font aussi partie, comme l'ESPCI Paris et Chimie ParisTech pour la physique et la chimie, AgroParisTech pour l'agriculture ou l'ENSAE ParisTech pour les statistiques.",
"\n\nEnfin, les meilleures écoles INP (Ensimag, Phelma, ENSEEIHT, Ense...) complètent le groupe A. Spécialisées dans des domaines très différents, elles se sont organisées en réseau, partagent leurs ressources et proposent des parcours croisés, permettant de suivre des cours dans plusieurs écoles du groupe, afin d'obtenir une double compétence. ",
"Cette stratégie leur permet de gagner en visibilité et d'apparaître dans de grands palmarès internationaux, comme les pages thématiques du classement de Shanghai.",
"\n\nUn tiers des diplômés de l'ESTACA commence sa carrière à l'étranger\n\nBeaucoup plus diversifié en termes de taille, de statut (public ou privé), de spécialisation et de recrutement, le groupe B réunit 71 formations de très bon niveau, aux atouts différents, selon qu'elles misent sur l'excellence académique ou les relations avec les entreprises. ",
"C'est dans ce domaine que brillent les meilleures écoles privées du palmarès, telles que l'ESILV, l'EPF, l'ESIEE Paris, ou l'Estaca. ",
"Proposant des cursus orientés vers les transports (aéronautique, automobile...), un secteur affichant de grands besoins en ingénieurs, cette dernière place un tiers de ses étudiants à l'étranger.",
"\n\nLes écoles de taille modeste des grands réseaux déjà évoqués (INP, Insa, Mines, Télécom) se retrouvent, elles aussi, dans ce groupe. ",
"À l'image de leurs grandes sœurs, elles offrent des formations sérieuses et de qualité. ",
"Plus académiques, les écoles universitaires s'appuient sur la qualité de leur corps professoral, majoritairement composé d'enseignants-chercheurs, afin de convaincre les futurs étudiants.",
"\n\nLa moitié des étudiants de l'ICAM sont en alternance\n\n\n\nLes 61 écoles du groupe C, le dernier de notre palmarès, échappent dans leur grande majorité à la logique de classement, du fait de leurs spécificités.",
"\n\nCertaines d'entre elles se révèlent ainsi extrêmement spécialisées et cherchent avant tout à répondre aux besoins spécifiques de leur secteur d'activité, avec succès. ",
"C'est le cas de l'École supérieure du bois ou de l'École nationale de météo. ",
"Elles apparaissent comme des références dans leur domaine, mais s'adressent aux étudiants ayant déjà une idée précise de leur future carrière.",
"\n\nD'autres écoles, elles aussi spécialisées, mais dans des domaines plus vastes : le numérique (ESME Sudria par exemple), l'agriculture (UniLaSalle, ESA), le génie maritime (SeaTech), le BTP (ISA BTP, ESITC Caen), la biologie (EBI), ou encore les sciences cognitives (ENSC)...\n\nEnfin, le groupe C rassemble de grandes écoles généralistes, à l'image du Cesi, du Cnam et l'Icam, émaillées sur tout le territoire. ",
"Ces formations, proches des entreprises locales, soutiennent l'apprentissage ouvrent leur recrutement à différents profils. ",
"Ainsi, la moitié des étudiants de l'Icam sont des alternants.",
"\n\nMéthodologie\n\nLe palmarès concerne 164 écoles délivrant le diplôme d'ingénieur, habilitées par la CTI (Commission des titres d'ingénieurs). ",
"Celles-ci recrutent après le bac pour cinq ans ou après un bac + 2 pour trois ans. ",
"Afin de constituer ce classement, l'Etudiant s'est appuyé sur les données certifiées de la CTI et de la Cdefi (Conférence des directeurs d'écoles françaises d'ingénieurs), ainsi que, pour certaines écoles, sur un questionnaire envoyé par l'Etudiant au printemps 2017. ",
"Les données portent sur l'année scolaire 2016-2017 et les effectifs de la rentrée 2017. ",
"Jusqu'en novembre, nous avons vérifié, actualisé et recoupé les informations communiquées par les établissements.",
"\n\n\n\nCette enquête a permis d'établir une cinquantaine d'indicateurs classants. ",
"Douze d'entre eux nous ont permis de créer trois profils de référence : excellence académique, ouverture internationale et proximité des entreprises. ",
"L'addition de ces trois profils et d'un critère supplémentaire (ouverture à de nouveaux profils) constitue la base de notre palmarès général. ",
"Les écoles y sont ordonnées selon leur nombre de points et sont rattachées à quatre groupes de force décroissante : A+ (de 62 à 49 points), A (de 48 à 37 points), B (de 36 à 27 points) et C (moins de 27 points)."
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[
"---\nauthor:\n- 'Frédéric <span style=\"font-variant:small-caps;\">Jean</span>[^1] [^2]'\ndate: 'Lectures given at the CIMPA School “Géométrie sous-riemannienne”, Beirut, Lebanon, 2012'\ntitle: 'Control of Nonholonomic Systems and Sub-Riemannian Geometry'\n---\n\nNonholonomic systems are control systems which depend linearly on the control. ",
"Their underlying geometry is the sub-Riemannian geometry, which plays for these systems the same role as Euclidean geometry does for linear systems. ",
"In particular the usual notions of approximations at the first order, that are essential for control purposes, have to be defined in terms of this geometry. ",
"The aim of these notes is to present these notions of approximation and their link with the metric tangent structure in sub-Riemannian geometry.",
"\n\nThe notes are organized as follows. ",
"In Section \\[se:geomnhs\\] we introduce the basic definitions on nonholonomic systems and sub-Riemannian geometry. ",
"Section \\[se:contro\\] is devoted to the study of the controllability of nonholonomic systems, and to the topological properties of sub-Riemannian distances. ",
"Section \\[se:1storder\\] provides a detailed exposition of the notions of first-order approximation, including nonholonomic orders, privileged coordinates, nilpotent approximations, and distance estimates such as the Ball-Box Theorem. ",
"We then see in Section \\[se:tgtstruct\\] how these notions allow us to describe the tangent structure to a Carnot-Carathéodory space (the metric space defined by a sub-Riemannian distance). ",
"Finally, we present in the appendix some results on flows in connection with the Hausdorff formula (Section \\[se:flowsvf\\]), and some proofs on privileged coordinates (Section \\[se:proofpriv\\]).",
"\n\nGeometry of nonholonomic systems {#se:geomnhs}\n================================\n\nThroughout these notes we work in a smooth $n$-dimensional manifold $M$. However most of our considerations are local, so $M$ can also be thought of as an open subset of $\\R^n$.\n\nNonholonomic systems {#se:snh}\n--------------------\n\nA *nonholonomic system* on $M$ is a control system which is of the form $$\\tag{$\\Sigma$}\n\\label{eq:snh}\n\\dot{q}=\\sum_{i=1}^{m}u_iX_i(q),\\ q\\in M,\\ \\\nu=(u_1,\\dots,u_m)\\in\\mathbb{R}^m,$$ where $X_1,\\dots,X_m$ are $C^\\infty$ vector fields on $M$. To give a meaning to such a control system, we have to define what are its solutions, that is, its trajectories.",
"\n\nA *trajectory of* is a path $\\gamma: [0,T] \\to M$ for which there exists a function $u(\\cdot) \\in\nL^1([0,T],\\R^m)$ such that $\\gamma$ is a solution of the ordinary differential equation: $$\\dot{q}(t)=\\sum_{i=1}^{m}u_i (t) X_i (q(t)), \\quad \\hbox{for a.e.\\ } t\n\\in [0,T].$$ Such a function $u(\\cdot)$ is called a *control* associated with $\\gamma$.\n\nEquivalently, a trajectory is an absolutely continuous path $\\gamma$ on $M$ such that $\\dot \\gamma (t) \\in \\Delta (\\gamma (t))$ for almost every $t\\in\n[0,T]$, where we have set, for every $q\\in M$, $$\\label{eq:deltap}\n \\Delta (q) = \\mathrm{span}\n\\left\\{X_1(q),\\dots,X_m(q)\\right\\} \\subset T_qM.$$ Note that the rank of the vector spaces $\\Delta(q)$ is a function of $q$, which may be non constant. ",
"If it is constant, $\\Delta$ defines a distribution on $M$, that is, a subbundle of $TM$.\n\n\\[ex:unicycle\\] The most typical example of nonholonomic system is the simplified kinematic model of a unicycle. ",
"In this model, a configuration $q=(x,y,\\theta)$ of the unicycle is described by the planar coordinates $(x,y)$ of the contact point of the wheel with the ground, and by the angle $\\theta$ of orientation of the wheel with respect to the $x$-axis. ",
"The space of configurations is then the manifold $\\R^2\n\\times S^1$.\n\nThe wheel is subject to the constraint of rolling without slipping, which writes as $\\dot x \\sin \\theta - \\dot y \\cos \\theta =0$, or, equivalently as $\\dot q \\in \\ker \\omega(q)$, where $\\omega$ is the one-form $\\sin \\theta dx - \\cos \\theta dy$. Hence the set $\\Delta$ of is $\\ker \\omega$.\n\nChoosing as controls the tangential velocity $u_1$ and the angular velocity $u_2$, we obtain the nonholonomic system $\\dot q = u_1 X_1(q) + u_2 X_2(q)$ on $\\R^2 \\times S^1$, where $X_1=\\cos \\theta \\partial_x + \\sin \\theta \\partial_y$, and $X_2= \\partial_\\theta$.\n\nLet us mention here a few properties of the trajectories of (for more details, see [@rif12]).",
"\n\n- Fix $p \\in M$ and $T>0$. For every control $u(\\cdot) \\in\n L^1([0,T],\\R^m)$, there exists $\\tau \\in (0,T]$ such that the Cauchy problem $$\\label{eq:cauchpb}\n \\left\\{\n \\begin{array}[c]{l}\n \\dot{q}(t)=\\sum_{i=1}^{m}u_i (t) X_i (q(t)) \\quad \\hbox{for a.e.\\ }t \\in\n [0,\\tau], \\\\\n q(0) =p,\n \\end{array}\n \\right.$$ has a unique solution denoted by $\\gamma_u$ or $\\gamma(\\cdot ; p,\n u)$. It is called the *trajectory issued from $p$ associated with $u$*.",
"\n\n- If the rank of $X_1,\\dots,X_m$ is constant and equal to $m$ on $M$, every trajectory is associated with a unique control. ",
"Otherwise different controls can be associated with the same trajectory. ",
"In this case it will sometimes be useful to consider among these controls only the ones which minimize the $L^1$ norm $\\int \\|u(t)\\|dt$. By convexity, this defines a unique control with which the trajectory is associated.",
"\n\n- Any time-reparameterization of a trajectory is still a trajectory: if $\\gamma: [0,T] \\to M$ is a trajectory associated with a control $u$, and $\\alpha:\n [0,S] \\to [0,T]$ is a $C^1$-diffeomorphism, then $\\gamma \\circ\n \\alpha : [0,S] \\to M$ is a trajectory associated with the control $\\alpha'(s) u \\left( \\alpha(s)\\right)$. In particular, one can reverse time along $\\gamma$: the resulting path $\\gamma(T-s)$, $s\\in [0,T]$, is a trajectory associated with the control $-u(T-s)$.\n\nIn this context, the first question is the one of the controllability: can we join any two points by a trajectory? ",
"This suggests to introduce the following definition.",
"\n\n\\[de:Ap\\] The *attainable set from $p \\in M$* is defined to be the set $\\mathcal{A}_p$ of points attained by a trajectory of issued from $p$.\n\nThe question above then becomes: is the attainable set from any point equal to the whole manifold $M$? ",
"We will answer this question in Section \\[se:contro\\].",
"\n\nIn the case where the answer is positive, next issues are notably the motion planning (i.e. find a trajectory joining two given points) and the stabilization (i.e. design the control as a function $u(q)$ of the state in such a way that the resulting differential equation is stable). ",
"The usual way to deal with these problems is to use a first-order approximation of the system. ",
"The underlying idea is the following. ",
"Consider a nonlinear control system in $\\R^n$, $$\\dot x = f(x,u), \\quad x \\in \\R^n, \\ u \\in \\R^m,$$ and a pair $(\\bar x,\\bar u) \\in \\R^{m+n}$ such that $f(\\bar x,\\bar\nu)=0$. The linearized system around this equilibrium pair is defined to be the linear control system: $$\\dot {\\delta x} = \\frac{\\partial f}{\\partial x}(\\bar x,\\bar u) \\delta\nx + \\frac{\\partial f}{\\partial u}(\\bar x,\\bar u) \\delta u, \\quad\n\\delta x \\in \\R^n, \\ \\delta u \\in \\R^m.$$ If this linearized system is controllable, so is the nonlinear one near $\\bar x$. In this case the solutions to the motion planning and stabilization problems for the linearized system may be used to construct solutions of the corresponding problems for the nonlinear system (see for instance [@kha95]). ",
"Thus, locally, the study of the control system amounts to the one of the linearized system.",
"\n\nDoes this strategy apply to nonholonomic systems? ",
"Consider a nonholonomic system defined on an open subset $M$ of $\\R^n$. For every $\\bar q \\in M$, the pair $(\\bar q, 0)$ is an equilibrium pair and the corresponding linearized system is $$\\dot {\\delta x} = \\sum_{i=1}^{m} \\delta u_i X_i(\\bar q), \\quad\n\\delta x \\in \\R^n, \\ \\delta u \\in \\R^m.$$ For this linearized system, the attainable set from a point $\\delta q$ is obviously the affine subset $$\\delta q + \\Delta (\\bar q) = \\delta q + \\mathrm{span}\n\\left\\{X_1(\\bar q),\\dots,X_m(\\bar q)\\right\\}.$$ Thus, except in the very special case where $\\mathrm{rank} \\Delta (\\bar q)\n=n$, the linearized system is not controllable and the strategy above does not apply, whereas nonholonomic systems may be controllable (and generically they are), as we will see Section \\[se:contro\\].",
"\n\nThis may be explained as follows. ",
"The linearization is a first-order approximation with respect to a Euclidean (or a Riemannian) distance. ",
"However for nonholonomic systems the underlying distance is a sub-Riemannian one and it behaves very differently from a Euclidean one. ",
"Thus, the local behaviour should be understood through the study of a first-order approximation with respect to this sub-Riemannian distance, not through the linearized system.",
"\n\nWe will introduce now the sub-Riemannian distances. ",
"In Section \\[se:1storder\\] we will see how to construct first-order approximations with respect to this kind of distances, and how to use them for motion planning for instance.",
"\n\nSub-Riemannian distance {#se:srdistance}\n-----------------------\n\nA nonholonomic system induces a distance on $M$ in the following way. ",
"We first define the [*sub-Riemannian metric*]{} associated with to be the function $g : TM \\to \\overline{\\R}$ given by $$\\label{eq:srmetric}\n g(q,v) = \\inf \\left\\{ u_1^2+ \\cdots + u_m^2 \\ : \\\n \\sum_{i=1}^{m}u_iX_i(q)=v \\right\\},$$ for $q\\in M$ and $v \\in T_qM$, where we adopt the convention that $\\inf \\emptyset = + \\infty$. This function $g$ is smooth and satisfies:\n\n- $g(q,v)= +\\infty$ if $v \\not\\in \\Delta (q)$,\n\n- $g$ restricted to $\\Delta (q)$ is a positive definite quadratic form.",
"\n\nSuch a metric allows to define a distance in the same way as in Riemannian geometry.",
"\n\n\\[de:srdist\\] The *length* of an absolutely continuous path $\\gamma (t)$, $t \\in [0,T]$, is $${\\mathrm{length}} (\\gamma) = \\int_0^T \\sqrt{g \\left(\\gamma (t), \\dot \\gamma\n(t)\\right)} dt,$$ and the [*sub-Riemannian distance*]{} on $M$ associated with the nonholonomic system is defined by $$d(p,q) = \\inf {\\mathrm{length}} (\\gamma),$$ where the infimum is taken over all absolutely continuous paths $\\gamma$ joining $p$ to $q$.\n\nNote that only trajectories of may have a finite length. ",
"In particular, if no trajectory joins $p$ to $q$, then $d(p,q)=+\\infty$. We will see below in Corollary \\[le:d=dist\\] that, under an extra assumption on the nonholonomic system, $d$ is actually a distance function.",
"\n\nWhen $\\gamma$ is a trajectory, its length is also equal to $$\\min \\int_0^T \\|u(t)\\| dt,$$ the minimum being taken over all control $u(\\cdot)$ associated with $\\gamma$. As already noticed, this minimum is attained at a unique control which could be defined as *the* control associated with $\\gamma$.\n\nAn important feature of the length of a path is that it is independent of the parametrization of the path. ",
"As a consequence, the sub-Riemannian distance $d(p,q)$ may also be understood as the minimal time needed for the nonholonomic system to go from $p$ to $q$ with bounded controls, that is, $$\\label{eq:tpsmin}\n d(p,q) = \\inf \\left\\{\nT\\geq 0 \\ : \\\n\\begin{array}[c]{c}\n \\exists \\hbox{ a trajectory } \\gamma_u: [0,T] \\to M \\hbox{ s.t.}\\\\\n\\gamma_u(0) = p, \\ \\gamma_u(T)=q,\\\\\n\\hbox{ and } \\|u(t)\\| \\leq 1 \\hbox{ for a.e.\\ } t \\in [0,T]\n\\end{array}\n\\right\\} .$$ This formulation justifies the assertion made in Section \\[se:snh\\]: for nonholonomic systems, first-order approximations with respect to the time should be understood as first-order approximations with respect to the sub-Riemannian distance.",
"\n\nAnother consequence of is that $d(p,q)$ is the solution of a time-optimal control problem. ",
"It then results from standard existence theorems (see for instance [@Lee1967] or [@rif12]) that, when $p$ and $q$ are sufficiently close and $d(p,q) <\\infty$, there exists a trajectory $\\gamma$ joining $p$ to $q$ such that $${\\mathrm{length}} (\\gamma) = d(p,q) .$$ Such a trajectory is called a *minimizing trajectory*.",
"\n\n\\[re:velocity1\\] Any reparameterization of a minimizing trajectory is also minimizing. ",
"Therefore any pair of close enough points can be joined by a minimizing trajectory *of velocity one*, that is, a trajectory $\\gamma$ such that $g(\\gamma(t),\\dot \\gamma(t)) = 1$ for a.e.",
" $t$. As a consequence, there exists a control $u(\\cdot)$ associated with $\\gamma$ such that $\\|u(t)\\|=1$ a.e. ",
"Every sub-arc of such a trajectory $\\gamma$ is also clearly minimizing, hence the equality $d(p,\\gamma(t)) = t$ holds along $\\gamma$.\n\nSub-Riemannian manifolds\n------------------------\n\nThe distance $d$ defined in Section \\[se:srdistance\\] does not always meet the classical notion of sub-Riemannian distance arising from a sub-Riemannian manifold. ",
"Let us recall the latter definition.",
"\n\nA [*sub-Riemannian manifold*]{} $(M,D,g_R)$ is a smooth manifold $M$ endowed with a [*sub-Riemannian structure*]{} $(D,g_R)$, where:\n\n- $D$ is a distribution on $M$, that is a subbundle of $TM$;\n\n- $g_R$ is a Riemannian metric on $D$, that is a smooth function $ g_R : D \\rightarrow \\R$ whose restrictions to $D(q)$ are positive definite quadratic forms.",
"\n\nThe [*sub-Riemannian metric*]{} associated with $(D,g_R)$ is the function $g_{SR} : TM \\to \\overline{\\R}$ given by $$\\label{eq:srmetric2}\n g_{SR}(q,v) =\n\\left\\{\n \\begin{array}[c]{ll}\n g_R(q,v) & \\hbox{if } v\\in D(q), \\\\\n+ \\infty & \\hbox{otherwise}.",
"\n \\end{array} \\right.$$ The [*sub-Riemannian distance*]{} $d_{SR}$ on $M$ is then defined from the metric $g_{SR}$ as $d$ is defined from the metric $g$ in Section \\[se:srdistance\\].",
"\n\nWhat is the difference between the two constructions, that is, between the definitions and of a sub-Riemannian metric?",
"\n\nConsider a [*sub-Riemannian structure*]{} $(D,g_R)$. Locally, on some open subset $U$, there exist vector fields $X_1, \\dots, X_m$ whose values at each point $q \\in U$ form an orthonormal basis of $D(q)$ for the quadratic form $g_R$; the metric $g_{SR}$ associated with $(D,g_R)$ then coincides with the metric $g$ associated with $X_1, \\dots, X_m$. Thus, locally, there is a one-to-one correspondence between sub-Riemannian structures and nonholonomic systems for which the rank of $\\Delta (q) = \\mathrm{span}\n\\left\\{X_1(q),\\dots,X_m(q)\\right\\}$ is constant.",
"\n\nHowever this correspondence does not hold globally since, for topological reasons, a distribution of rank $m$ may not always be generated by $m$ vector fields on the whole $M$. Conversely, the vector fields $X_1,\\dots,X_m$ of a nonholonomic system do not always generate a linear space $\\Delta(q)$ of constant rank equal to $m$. It may even be impossible, again for topological reasons (for instance, on an even dimensional sphere).",
"\n\nA way to conciliate both notions is to generalize the definition of sub-Riemannian structure.",
"\n\nA [*generalized sub-Riemannian structure*]{} on $M$ is a triple $(E, \\sigma,\ng_R)$ where\n\n- $E$ is a vector bundle over $M$;\n\n- $\\sigma : E \\rightarrow TM$ is a morphism of vector bundles;\n\n- $g_R$ is a Riemannian metric on $E$.\n\nWith a generalized sub-Riemannian structure a metric is associated which is defined by $$g_{SR}(q,v) = \\inf \\{g(q,u) \\ : \\ u \\in E(q), \\ \\sigma(u) = v \\},\n\\quad \\hbox{for } q \\in M, \\ v \\in T_qM.$$ The generalized sub-Riemannian distance $d_{SR}$ on $M$ is then defined from this metric $g_{SR}$ as $d$ is defined from the metric $g$.\n\nThis definition of sub-Riemannian distance actually contains the two notions of distance we have introduced before.",
"\n\n- Take $E = M \\times \\R^m$, $\\sigma : E \\rightarrow TM$, $\\sigma (q,u)= \\sum_{i=1}^m u_i X_i (q)$ and $g_R$ the Euclidean metric on $\\R^m$. The resulting generalized sub-Riemannian distance is the distance associated with the nonholonomic system .",
"\n\n- Take $E=D$, where $D$ is a distribution on $M$, $\\sigma : D\n \\hookrightarrow TM$ the inclusion, and $g_R$ a Riemannian metric on $D$. We recover the distance associated with the sub-Riemannian structure $(D,g_R)$.\n\nLocally, a generalized sub-Riemannian structure can always be defined by a single finite family $X_1, \\dots, X_m$ of vector fields, and so by a nonholonomic system (without rank condition). ",
"It actually appears that this is also true globally (see [@Agrachev2011a], or [@Drager2012] for the fact that a submodule of $TM$ is finitely generated): any generalized sub-Riemannian distance may be associated with a nonholonomic system.",
"\n\nIn these notes, we will always consider a sub-Riemannian distance $d$ associated with a nonholonomic system. ",
"However, as we just noticed, all the results actually hold for a generalized sub-Riemannian distance.",
"\n\nControllability {#se:contro}\n===============\n\nConsider a nonholonomic system $$\\tag{$\\Sigma$}\n\\dot{q}=\\sum_{i=1}^{m}u_iX_i(q),$$ on a smooth $n$-dimensional manifold $M$. This section is concerned with the question of *controllability*: is the attainable set $\\mathcal{A}_p$ from any point $p$ equal to the whole manifold $M$? ",
"We will see next the implications on the sub-Riemannian distance $d$ and on the topology of the metric space $(M,d)$.\n\nThe Chow-Rashevsky Theorem {#se:chow}\n--------------------------\n\nThe controllability of is mainly characterized by the properties of the Lie algebra generated by $X_1, \\dots, X_m$. We first introduce notions and definitions on this subject.",
"\n\nLet $VF(M)$ denote the set of smooth vector fields on $M$. We define $\\Delta^1$ to be the linear subspace of $VF(M)$ generated by $X_1, \\dots, X_m$, $$\\Delta^1 = \\mathrm{span} \\{ X_1, \\dots, X_m\\}.$$ For $s\\geq 1$, define $\\Delta^{s+1}= \\Delta^s + [\\Delta^1,\\Delta^s]$, where we have set $[\\Delta^1,\\Delta^s]=\n\\mathrm{span} \\{ [X,Y] \\ : \\ X \\in \\Delta^1, \\ Y \\in \\Delta^s \\}$. The *Lie algebra generated by* $X_1, \\dots, X_m$ is defined to be $\\mathit{Lie} (X_1, \\dots, X_m)= \\bigcup_{s\\geq 1} \\Delta^s$. Due to the Jacobi identity, $\\mathit{Lie} (X_1, \\dots, X_m)$ is the smallest linear subspace of $VF(M)$ which both contains $X_1,\\dots, X_m$ and is invariant by Lie brackets.",
"\n\nLet us denote by $I=i_1 \\cdots i_k$ a multi-index of $\\{1,\\dots,m\\}$, and by $|I|=k$ the length of $I$. We set $$X_I=[X_{i_1},[ \\dots, [X_{i_{k-1}},X_{i_k}] \\dots].$$ With these notations, $\\Delta^s = \\mathrm{span} \\{ X_I \\ : \\ |I| \\leq\ns \\}$.\n\nFor $q\\in M$, we set $\\mathit{Lie} (X_1, \\dots, X_m)(q) = \\{ X(q)\n: \\, X \\in \\mathit{Lie} (X_1, \\dots, X_m) \\}$, and, for $s\\geq 1$, $\\Delta^s (q) = \\{ X(q) \\ : \\ X \\in \\Delta^s \\}$. By definition these sets are linear subspaces of $T_qM$.\n\nWe say that (or the vector fields $X_1,\\dots,X_m$) satisfies [*Chow’s Condition*]{} if $$\\mathit{Lie} (X_1, \\dots, X_m)(q)=T_qM, \\qquad \\forall q\n\\in M.$$\n\nEquivalently, for any $q \\in M$, there exists an integer $r=r(q)$ such that $\\dim \\Delta^r (q)=n$.\n\nThis property is also known as the Lie algebra rank condition (LARC), and as the Hörmander condition (in the context of PDE).",
"\n\n\\[le:Apopen\\] If satisfies Chow’s Condition, then for every $p\\in M$, the set $\\mathcal{A}_p$ is a neighbourhood of $p$.\n\nWe work in a small neighbourhood $U\n\\subset M$ of $p$ that we identify with a neighbourhood of $0$ in $\\R^n$ .",
"\n\nLet $\\phi_t^i=\\exp(tX_i)$ be the flow of the vector field $X_i$, $i=1,\\dots,m$. Every curve $t \\mapsto \\phi^i_t(q)$ is a trajectory of and we have $$\\phi_t^i = \\mathrm{id} + t X_i + o(t).$$ For every multi-index $I$ of $\\{1,\\dots,m\\}$, we define the local diffeomorphisms $\\phi^I_t$ on $U$ by induction on the length $|I|$ of $I$: if $I=iJ$, then $$\\phi_t^{iJ}=[\\phi_t^i,\\phi_t^J]\n:=\\phi_{-t}^J \\circ \\phi_{-t}^i\n\\circ \\phi_t^J \\circ\\phi_t^i.$$ By construction, $\\phi^I_t (q)$ is the endpoint of a trajectory of issued from $q$. Moreover, on a neighbourhood of $p$ there holds $$\\label{eq:phiI}\n\\phi^I_t =\\mathrm{id} + t^{|I|} X_{I} + o(t^{|I|}).$$ We postpone the proof of this formula to the Appendix (Proposition \\[le:quasich\\]).",
"\n\nTo obtain a diffeomorphism whose derivative with respect to the time is exactly $X_I$, we set $$\\psi^I_t = \\left\\{\n\\begin{array}[c]{ll}\n\\phi^I_{t^{1/|I|}} & \\hbox{if } t \\geq 0, \\\\[2mm]\n\\phi^I_{-|t|^{1/|I|}} & \\hbox{if } t < 0 \\hbox{ and } |I| \\hbox{\nis\n odd}, \\\\[2mm]\n{[} \\phi^J_{|t|^{1/|I|}},\\phi^i_{|t|^{1/|I|}} {]} & \\hbox{if } t\n< 0 \\hbox{\n and } |I| \\hbox{ is even},\n\\end{array}\n\\right.$$ where $I=iJ$. Thus $$\\begin{aligned}\n\\label{eq:psii}\n\\psi^I_t =\\mathrm{id} + t X_{I} + o(t),\\end{aligned}$$ and $\\psi^I_t (q)$ is the endpoint of a trajectory of issued from $q$.\n\nLet us choose now commutators $X_{I_1}, \\dots,X_{I_n}$ whose values at $p$ span $T_pM$. This is possible thanks to Chow’s Condition. ",
"We introduce the map $\\varphi$ defined on a small neighbourhood $\\Omega$ of $0$ in $\\R^n$ by $$\\varphi (t_1, \\dots , t_n)= \\psi^{I_n}_{t_n} \\circ \\cdots \\circ\n\\psi^{I_1}_{t_1} (p) \\in M.$$ We conclude from (\\[eq:psii\\]) that this map is $C^1$ near $0$ and has an invertible derivative at $0$, which implies that it is a local $C^1$-diffeomorphism. ",
"Therefore $\\varphi(\\Omega)$ contains a neighbourhood of $p$.\n\nNow, for every $t \\in \\Omega$, $\\varphi(t)$ is the endpoint of a concatenation of trajectories of , the first one being issued from $p$. It is then the endpoint of a trajectory starting from $p$. Therefore $\\varphi(\\Omega) \\subset \\mathcal{A}_p$, which implies that $\\mathcal{A}_p$ is a neighbourhood of $p$.\n\n\\[thmCR\\] If $M$ is connected and if satisfies Chow’s Condition, then any two points of $M$ can be joined by a trajectory of .",
"\n\nLet $p\\in M$. If $q\\in \\mathcal{A}_p$, then $p \\in \\mathcal{A}_q$. As a consequence, $\\mathcal{A}_p=\\mathcal{A}_q$ for any $q\\in M$ and the lemma above implies that $\\mathcal{A}_p$ is an open set. ",
"Hence the manifold $M$ is covered by the union of the sets $\\mathcal{A}_p$ that are pairwise disjointed. ",
"Since $M$ is connected, there is only one such open set.",
"\n\nThis theorem appears also as a consequence of the Orbit Theorem (Sussmann, Stefan [@ste74; @sus73]): each set $\\mathcal{A}_p$ is a connected immersed submanifold of $M$ and, at each point $q \\in \\mathcal{A}_p$, $\\mathit{Lie}(X_1, \\dots, X_m)(q) \\subset\nT_q\\mathcal{A}_p$. Moreover, when the rank of the Lie algebra is constant on $M$, both spaces are equal, i.e. $\\mathit{Lie}(X_1, \\dots, X_m)(q)= T_q\\mathcal{A}_p$.\n\nThus, when the Lie algebra generated by $X_1, \\dots, X_m$ has constant rank, Chow’s Condition is not restrictive: it is indeed satisfied on each $\\mathcal{A}_p$ by the restriction of the vector fields $X_1,\\dots,X_m$ to the manifold $\\mathcal{A}_p$.\n\nThe converse of Chow’s theorem is false in general. ",
"Consider for instance the nonholonomic system in $\\R^3$ defined by $X_1=\\partial_x$, $X_2=\\partial_y + f(x)\\partial_z$ where $f(x)=e^{-1/x^2}$ for positive $x$ and $f(x)=0$ otherwise. ",
"The associated sub-Riemannian distance is finite whereas $X_1,\\dots,X_m$ do not satisfy Chow’s Condition.",
"\n\nHowever, for an analytic nonholonomic system (i.e. when $M$ and the vector fields $X_1,\\dots,X_m$ are in the analytic category), Chow’s Condition is equivalent to the controllability of (see [@nag66; @sus74]).",
"\n\n\\[re:orbit\\] Our proof of Theorem \\[thmCR\\] also shows that, under the assumptions of the theorem, for every point $p \\in M$ the set $$\\left\\{ \\exp(t_{i_1} {X}_{i_1}) \\circ \\cdots \\circ \\exp(t_{i_k}\n{X}_{i_k})(p) \\ : \\ k\\in \\N, \\ t_{i_j} \\in \\R, \\ i_j \\in \\{ 1,\n\\dots, m\\} \\right\\}$$ is equal to the whole $M$. This set is often called the *orbit* at $p$ of the vector fields $X_1, \\dots, X_m$.\n\nTopological structure of $(M,d)$ {#se:topo}\n--------------------------------\n\nThe proof of Lemma \\[le:Apopen\\] gives a little bit more than the openness of $\\mathcal{A}_p$. For $\\eps$ small enough, any $\\phi_t^i(q)$, $0\\leq t \\leq \\eps$, is a trajectory of length $\\eps$. Thus $\\varphi (t_1, \\dots,t_n)$ is the endpoint of a trajectory of length less than $N\n\\big( |t_1|^{1/|I_1|} + \\cdots + |t_n|^{1/|I_n|} \\big)$, where $N$ counts the maximal number of concatenations involved in the $\\psi^{I_i}_{t}$’s. ",
"This gives an upper bound for the distance, $$\\begin{aligned}\n\\label{eq:ineqd}\nd \\big( p, \\varphi(t) \\big) \\leq N \\big( |t_1|^{1/|I_1|} + \\cdots +\n|t_n|^{1/|I_n|} \\big).\\end{aligned}$$\n\nThis kind of estimates of the distance in terms of local coordinates plays an important role in sub-Riemannian geometry, as we will see in Section \\[se:distance\\]. ",
"However here $(t_1,\\dots,t_n)$ are not smooth local coordinates, as $\\varphi$ is only a $C^1$-diffeomorphism, not a smooth diffeomorphism.",
"\n\nLet us try to replace $(t_1,\\dots,t_n)$ by smooth local coordinates. ",
"Choose local coordinates $(y_1,\\dots,y_n)$ centered at $p$ such that $\\frac{\\partial}{\\partial y_i}|_p=X_{I_i}(p)$. The map $\\varphi^y=y \\circ \\varphi$ is a $C^1$-diffeomorphism between neighbourhoods of $0$ in $\\R^n$, and its differential at 0 is $d\\varphi^y_0= \\mathrm{Id}_{\\R^n}$.\n\nDenoting by $\\| \\cdot \\|_{\\R^n}$ the Euclidean norm on $\\R^n$, we obtain, for $\\|t \\|_{\\R^n}$ small enough, $y_i(t)=t_i + o(\\|t\n\\|_{\\R^n})$. The inequality (\\[eq:ineqd\\]) becomes $$d(p,q^y) \\leq N' \\|y \\|_{\\R^n}^{1/r},$$ where $q^y$ denotes the point of coordinates $y$, and $r=\\max_i\n|I_i|$. This inequality allows to compare $d$ to a Riemannian distance.",
"\n\nLet $g_R$ be a Riemannian metric on $M$, and $d_R$ the associated Riemannian distance. ",
"On a compact neighbourhood of $p$, there exists a constant $c>0$ such that $g(X_i,X_i)(q) \\leq c^{-1}$, which implies $c d_R(p,q) \\leq d(p,q)$. Moreover we have $d_R(p,q^y) \\geq \\mathit{Cst} \\|y \\|_{\\R^n}$. We have then obtained a first estimate to the sub-Riemannian distance.",
"\n\n\\[th:RvsSR\\] Assume satisfies Chow’s Condition. ",
"For any Riemannian metric $g_R$, we have, for $q$ close enough to $p$, $$c d_R(p,q) \\leq d(p,q) \\leq C d_R(p,q)^{1/r},$$ where $c,C$ are positive constants and $r$ is an integer such that $\\Delta_p^r=T_pM$.\n\nIf we choose for $g_R$ a Riemannian metric which is compatible with $g$, that is, which satisfies $g_R|_\\Delta=g$, then by construction $d_R(p,q) \\leq d(p,q)$.\n\n\\[le:d=dist\\] Under the hypotheses of Theorem \\[thmCR\\], $d$ is a distance function on $M$, i.e.,\n\n(i) $d$ is a function from $M \\times M$ to $[0,\\infty)$;\n\n(ii) $d(p,q) = d(q,p)$ (symmetry);\n\n(iii) $d(p,q)=0$ if and only if $p=q$;\n\n(iv) $d(p,q) + d(p,q') \\leq d(p,q')$ (triangle inequality).",
"\n\nBy Chow-Rashevsky’s theorem (Theorem \\[thmCR\\]), the distance between any pair of points is finite, which gives *(i)*. ",
"The symmetry of the distance results from the fact that, if $\\gamma(s)$, $s\\in [0,T]$, is a trajectory joining $p$ to $q$, then $s \\mapsto \\gamma(T-s)$ is a trajectory of same length joining $q$ to $p$. Point *(iii)* follows directly from Theorem \\[th:RvsSR\\]. ",
"Finally, the triangle inequality is a consequence of the following remark. ",
"If $\\gamma(s)$, $s\\in [0,T]$, is a trajectory joining $p$ to $q$ and $\\gamma'(s)$, $s\\in [0,T']$, is a trajectory joining $q$ to $q'$, then the concatenation $\\gamma * \\gamma'$, defined by $$\\gamma * \\gamma' (s) =\n\\left\\{\n \\begin{array}[c]{ll}\n \\gamma(s) & \\hbox{if } s\\in [0,T], \\\\\n\\gamma'(s-T) & \\hbox{if } s\\in [T,T+T'], \\\\\n \\end{array}\n\\right.$$ is a trajectory joining $p$ to $q'$ whose length satisfies $${\\mathrm{length}} (\\gamma * \\gamma') = {\\mathrm{length}} (\\gamma) +\n{\\mathrm{length}} (\\gamma').$$\n\nA second consequence of Theorem \\[th:RvsSR\\] is that the sub-Riemannian distance $d$ is $1/r$-Hölder with respect to any Riemannian distance, and so continuous.",
"\n\n\\[le:toposr\\] If satisfies Chow’s Condition, then the topology of the metric space $(M,d)$ coincides with the topology of $M$ as a smooth manifold.",
"\n\nFirst-order approximations {#se:1storder}\n==========================\n\nConsider a nonholonomic system : $\\dot{q}=\\sum_{i=1}^{m}u_iX_i(q)$ on a manifold $M$ satisfying Chow’s Condition, and denote by $d$ the induced sub-Riemannian distance. ",
"As we have seen in Section \\[se:snh\\], the infinitesimal behaviour of this system should be captured by an approximation to the first-order with respect to $d$. In this section we will then provide notion of first-order approximation and construct the basis of an infinitesimal calculus adapted to nonholonomic systems. ",
"To this aim, a fundamental role will be played by the concept of noholonomic order of a function at a point. ",
"We will then see that approximations to the first-order appear as nilpotent approximations, in the sense that $X_1, \\dots, X_m$ are approximated by vector fields that generate a nilpotent Lie algebra.",
"\n\nThe whole section is concerned with local objects. ",
"Henceforth, throughout the section we fix a point $p\\in M$ and an open neighbourhood $U$ of $p$ that we identify with a neighbourhood of $0$ in $\\R^n$ through some local coordinates.",
"\n\nNonholonomic orders {#se:nhorders}\n-------------------\n\nLet $f: M \\to \\R$ be a continuous function. ",
"The *nonholonomic order of $f$ at $p$*, denoted by $\\mathrm{ord}_p(f)$, is the real number defined by $$\\mathrm{ord}_p(f) = \\sup \\left\\{ s \\in \\R \\ : \\ f(q) = O \\big( d(p,q)^s \\big) \\right\\}.$$\n\nThis order is always nonnegative. ",
"Moreover $\\mathrm{ord}_p(f) = 0$ if $f(p) \\neq 0$, and $\\mathrm{ord}_p(f)\n= +\\infty$ if $f(p) \\equiv 0$.\n\n\\[ex:euclidean\\] When $M =\\R^n$, $m=n$, and $X_i=\\partial_{x_i}$, the sub-Riemannian distance is simply the Euclidean distance on $\\R^n$. In this case, nonholonomic orders coincide with the standard ones. ",
"Namely, $\\mathrm{ord}_0(f)$ is the smallest degree of monomials having nonzero coefficient in the Taylor series $$f(x) \\sim \\sum c_\\alpha x_1^{\\alpha_1} \\dots x_n^{\\alpha_n}$$ of $f$ at $0$. We will see below that there exists in general an analogous characterization of nonholonomic orders.",
"\n\nLet $C^\\infty (p)$ denote the set of germs of smooth functions at $p$. For $f \\in C^\\infty (p)$, we call [*nonholonomic derivatives of order 1 of $f$*]{} the Lie derivatives $X_1f, \\dots , X_mf$. We call further $X_i(X_jf)$, $X_i(X_j(X_kf))$,… the [*nonholonomic derivatives of $f$ of order 2, 3,…*]{} The nonholonomic derivative of order 0 of $f$ at $p$ is $f(p)$.\n\n\\[le:order\\] Let $f \\in C^\\infty (p)$. Then $\\mathrm{ord}_p(f)$ is equal to the biggest integer $k$ such that all nonholonomic derivatives of $f$ of order smaller than $k$ vanish at $p$. Moreover, $$f(q) = O \\big( d(p,q)^{\\mathrm{ord}_p(f)} \\big).$$\n\nThe proposition results from the following two assertions:\n\n1. ",
" if $\\ell$ is an integer such that $\\ell <\\mathrm{ord}_p(f)$, then all nonholonomic derivatives of $f$ of order $\\leq \\ell$ vanish at $p$;\n\n2. ",
" if $\\ell$ is an integer such that all nonholonomic derivatives of $f$ of order $\\leq \\ell$ vanish at $p$, then $f(q) = O \\big( d(p,q)^{\\ell + 1} \\big)$.\n\nLet us first prove point *(i)*. ",
"Let $\\ell$ be an integer such that $\\ell <\\mathrm{ord}_p(f)$. We write a nonholonomic derivative of $f$ of order $k \\leq \\ell$ as $$(X_{i_1} \\dots X_{i_k}f)(p) = \\frac{\\partial^k}{\\partial t_1\n\\cdots\n\\partial t_k} f \\big( \\exp(t_k X_{i_k}) \\circ \\cdots \\circ \\exp(t_1 X_{i_1}) (p)\n\\big)\\Big|_{t=0}.$$ The point $q =\\exp(t_k X_{i_k}) \\circ \\cdots \\circ \\exp(t_1 X_{i_1})\n(p)$ is the endpoint of a trajectory of length $|t_1|+\n\\cdots + |t_n|$. Therefore, $d(p,q) \\leq |t_1|+ \\cdots + |t_n|$.\n\nSince $k \\leq \\ell <\\mathrm{ord}_p(f)$, there exists a real number $s>0$ such that $f(q)= O\\big((|t_1|+ \\cdots +\n|t_n|)^{k+s} \\big)$. This implies that $$(X_{i_1} \\dots X_{i_k}f)(p) = \\frac{\\partial^k}{\\partial t_1\n\\cdots\n\\partial t_k} f (q)\\Big|_{t=0} =0.$$ Thus point *(i)* is proved.",
"\n\nThe proof of point *(ii)* goes by induction on $\\ell$. For $\\ell=0$, assume that all nonholonomic derivatives of $f$ of order $\\leq 0$ vanish at $p$, that is $f(p)=0$. Choose any Riemannian metric on $M$ and denote by $d_R$ the associated Riemannian distance on $M$. Since $f$ is smooth, there holds $f(q) \\leq \\mathit{Cst} \\ d_R(p,q)$ near $p$. By Theorem \\[th:RvsSR\\], this implies $f(q) \\leq \\mathit{Cst} \\ d(p,q)$, and so property *(ii)* for $\\ell = 0$.\n\nAssume that, for a given $\\ell \\geq 0$, *(ii)* holds for any function $f$ (induction hypothesis) and take a function $f$ such that all its nonholonomic derivatives of order $<\\ell+1$ vanish at $p$.\n\nObserve that, for $i=1, \\dots,m$, all the nonholonomic derivatives of $X_if$ of order $<\\ell$ vanish at $p$. Indeed, $X_{i_1} \\dots\nX_{i_k}(X_if) = X_{i_1} \\dots X_{i_k}X_if$. Applying the induction hypothesis to $X_if$ leads to $X_if(q)= O \\big( d(p,q)^\\ell \\big)$. In other words, there exist positive constants $C_1, \\dots, C_m$ such that, for $q$ close enough to $p$, $$X_if (q) \\leq C_i d(p,q)^\\ell.$$\n\nFix now a point $q$ near $p$. By Remark \\[re:velocity1\\], there exists a minimizing curve $\\gamma (\\cdot)$ of velocity one joining $p$ to $q$. Therefore $\\gamma$ satisfies $$\\dot \\gamma (t) = \\sum_{i=1}^m u_i(t) X_i \\big(\\gamma(t)\\big)\n\\quad \\hbox{for a.e.\\ }t \\in [0,T], \\qquad \\gamma(0)=p, \\\n\\gamma(T)=q,$$ with $\\sum_i u_i^2 (t) =1$ a.e.",
" and $d\\big(p,\\gamma (t)\\big) = t$ for any $t \\in [0,T]$. In particular $d(p,q)= T$.\n\nTo estimate $f(q)=f\\big(\\gamma(T)\\big)$, we compute the derivative of $f\\big(\\gamma(t)\\big)$ with respect to $t$, $$\\begin{aligned}\n\\frac{d}{dt}f\\big(\\gamma(t)\\big) & = & \\sum_{i=1}^m u_i(t) X_if\n\\big(\\gamma(t)\\big), \\\\\n \\Rightarrow \\ \\left| \\frac{d}{dt}f\\big(\\gamma(t)\\big) \\right| & \\leq\n & \\sum_{i=1}^m\n |u_i(t)| C_i d\\big(p,\\gamma (t)\\big)^\\ell \\leq Ct^\\ell,\\end{aligned}$$ where $C=C_1 + \\cdots +C_m$. Integrating this inequality between 0 and $t$ gives $$\\big| f\\big(\\gamma(t)\\big) \\big| \\leq | f(p)| + \\frac{C}{\\ell+1}\nt^{\\ell+1}.$$ Note that $f(p)=0$, since the nonholonomic derivative of $f$ of order $0$ at $p$ vanishes. ",
"Finally, at $t=T=d(p,q)$, we obtain $$| f(q)| \\leq \\frac{C}{\\ell+1} T^{\\ell+1},$$ which concludes the proof of *(ii)*.",
"\n\nAs a consequence, the nonholonomic order of a smooth (germ of) function is given by the formula $$\\mathrm{ord}_p(f)= \\min \\big\\{ s \\in \\N \\ : \\ \\exists \\ i_1,\\dots,i_s \\in\n\\{1,\\dots,m\\} \\ \\ \\mathrm{s.t.} \\ ",
" \\ (X_{i_1} \\dots X_{i_s}f)(p) \\neq 0\n\\big\\},$$ where as usual we adopt the convention that $\\min \\emptyset = + \\infty$.\n\nIt is clear now that any function in $C^\\infty (p)$ vanishing at $p$ is of order $\\geq 1$. Moreover, the following basic computation rules are satisfied: for every $f,g$ in $C^\\infty (p)$ and every $\\lambda \\in \\R \\setminus \\{0\\}$, $$\\begin{aligned}\n\\mathrm{ord}_p(fg) & \\geq & \\mathrm{ord}_p (f) + \\mathrm{ord}_p (g) ,\\\\\n\\mathrm{ord}_p(\\lambda f) & = & \\mathrm{ord}_p (f),\\\\\n\\mathrm{ord}_p(f+g) & \\geq & \\min \\big( \\mathrm{ord}_p(f),\n\\mathrm{ord}_p (g) \\big).\\end{aligned}$$ Note that the first inequality is actually an equality. ",
"However the proof of this fact requires an additional result (see Proposition \\[le:algorder\\]).",
"\n\nThe notion of nonholonomic order extends to vector fields. ",
"Let $VF\n(p)$ denote the set of germs of smooth vector fields at $p$.\n\nLet $X \\in VF (p)$. The *nonholonomic order of $X$ at $p$*, denoted by $\\mathrm{ord}_p(X)$, is the real number defined by: $$\\mathrm{ord}_p(X) = \\sup \\left\\{ \\sigma \\in \\R \\ : \\ \\mathrm{ord}_p(Xf) \\geq \\sigma + \\mathrm{ord}_p(f),\n\\quad \\forall f \\in C^\\infty (p) \\right\\}.$$ The order of a differential operator is defined in the same way.",
"\n\nNote that $\\mathrm{ord}_p(X) \\in \\Z$ since the order of a smooth function is an integer. ",
"Moreover the null vector field $X\\equiv0$ has infinite order, $\\mathrm{ord}_p(0)=+\\infty$.\n\nSince the order of a function coincides with its order as a differential operator acting by multiplication, we have the following properties. ",
"For every $X, Y \\in VF (p)$ and every $f \\in C^\\infty(p)$, $$\\label{eq:ordcl}\n\\begin{array}{rcl}\n\\mathrm{ord}_p([X,Y]) & \\geq & \\mathrm{ord}_p (X) + \\mathrm{ord}_p (Y) ,\\\\\n\\mathrm{ord}_p(fX) & \\geq & \\mathrm{ord}_p (f) + \\mathrm{ord}_p (X),\\\\\n\\mathrm{ord}_p(X) & \\leq & \\mathrm{ord}_p (Xf) - \\mathrm{ord}_p (f),\\\\\n\\mathrm{ord}_p(X+Y) & \\geq & \\min \\big( \\mathrm{ord}_p(X), \\mathrm{ord}_p (Y) \\big).",
"\n\\end{array}$$ As already noticed for functions, the second inequality is in fact an equality. ",
"This is not the case for the first inequality (take for instance $X=Y$).",
"\n\nAs a consequence of , $X_1, \\dots ,X_m$ are of order $\\geq -1$, $[X_i,X_j]$ of order $\\geq -2$, and more generally, every $X$ in the set $\\Delta^k$ is of order $\\geq -k$.\n\n\\[ex:euclidean2\\] In the Euclidean case (see example \\[ex:euclidean\\]), the nonholonomic order of a constant differential operator is the negative of its usual order. ",
"For instance $\\partial_{x_i}$ is of nonholonomic order $-1$. Actually, in this case, every vector field that does not vanish at $p$ is of nonholonomic order $-1$.\n\n\\[ex:heis\\] Consider the following vector fields on $\\R^{3}$: $$X_1 = {\\partial_x} - \\frac{y}{2} {\\partial_z} \\quad \\mbox{ and } \\quad\nX_2 = {\\partial_y} + \\frac{x}{2} {\\partial_z}.$$ The coordinate functions $x$ and $y$ have order 1 at $0$, whereas $z$ has order 2 at $0$, since $X_1x(0)=X_2y(0)=1$, $X_1z(0)=X_2z(0)=0$, and $X_1X_2z(0)=1/2$. These relations also imply $\\mathrm{ord}_0(X_1) =\n\\mathrm{ord}_0(X_2) = -1$. Finally, the Lie bracket $[X_1,X_2]\n= \\partial_{z}$ is of order $-2$ at $0$ since $[X_1,X_2]z=1$.\n\nWe are now in a position to give a meaning to first-order approximation.",
"\n\nA family of $m$ vector fields $\\widehat{X}_1, \\dots, \\widehat{X}_m$ defined near $p$ is called a [*first-order approximation of $X_1, \\dots, X_m$ at $p$*]{} if the vector fields $X_i - \\widehat{X}_i$, $i=1, \\dots, m$, are of order $\\geq 0$ at $p$.\n\nA consequence of this definition is that the order at $p$ defined by the vector fields $\\widehat{X}_1, \\dots, \\widehat{X}_m$ coincides with the one defined by $X_1, \\dots, X_m$. Hence for any $f \\in C^\\infty (p)$ of order greater than $s-1$, $$(X_{i_1} \\dots X_{i_s}f)(q) = (\\widehat{X}_{i_1} \\dots\n\\widehat{X}_{i_s}f)(q) + O \\left( d(p,q)^{\\mathrm{ord}_p(f)-s+1} \\right).$$\n\nTo go further in the characterization of orders and approximations, we need suitable systems of coordinates.",
"\n\nPrivileged coordinates {#se:privcoor}\n----------------------\n\nWe have introduced in Section \\[se:chow\\] the sets of vector fields $\\Delta^s$, defined by $\\Delta^s = \\mathrm{span} \\{ X_I \\ : \\ |I|\n\\leq s \\}$. Since $X_1,\\dots,X_m$ satisfy Chow’s Condition, the values of these sets at $p$ form a flag of subspaces of $T_pM$, that is, $$\\begin{aligned}\n\\label{eq:flag}\n\\Delta^1(p) \\subset \\Delta^2 (p) \\subset \\cdots \\subset \\Delta^{r-1}(p) \\varsubsetneq \\Delta^r(p)=T_pM,\\end{aligned}$$ where $r=r(p)$ is called the [*degree of nonholonomy at $p$*]{}.",
"\n\nSet $n_i(p)= \\dim \\Delta^i(p)$. The $r$-tuple of integers $(n_1(p), \\dots,\nn_r(p))$ is called the [*growth vector at $p$*]{}. ",
"The first integer $n_1(p)\\leq m $ is the rank of the family $X_1(p),\\dots,X_m(p)$, and the last one $n_r(p)=n$ is the dimension of the manifold $M$.\n\nLet $s \\geq 1$. By abuse of notations, we continue to write $\\Delta^s$ for the map $q\\mapsto \\Delta^s (q)$. This map $\\Delta^s$ is a distribution if and only if $n_s(q)$ is constant on $M$. We then distinguish two kind of points.",
"\n\nThe point $p$ is a [*regular point*]{} if the growth vector is constant in a neighbourhood of $p$. Otherwise, $p$ is a [*singular point*]{}.",
"\n\nThus, near a regular point, all maps $\\Delta^s$ are locally distributions.",
"\n\nThe structure of flag (\\[eq:flag\\]) may also be described by another sequence of integers. ",
"We define the [*weights at $p$*]{}, $w_i=w_i(p)$, $i=1, \\dots, n$, by setting $w_j=s$ if $n_{s-1}(p) < j \\leq n_s(p)$, where $n_0=0$. In other words, we have $$\\begin{gathered}\nw_1=\\cdots=w_{n_1}=1, \\ w_{n_1+1}= \\cdots=w_{n_2}=2, \\dots ,\\\\\nw_{n_{r-1}+1}=\\cdots=w_{n_r}=r.\\end{gathered}$$ The weights at $p$ form an increasing sequence $w_1(p) \\leq \\cdots\n \\leq w_n(p)$ which is constant near $p$ if and only if $p$ is a regular point.",
"\n\n\\[ex:heis2\\] The Heisenberg case in $\\R^3$ given in example \\[ex:heis\\] has a growth vector which is equal to $(2,3)$ at every point. ",
"Therefore all points of $\\R^3$ are regular. ",
"The weights at any point are $w_1=w_2=1$, $w_3=2$.\n\n\\[ex:martinet\\] Consider the following vector fields on $\\R^{3}$, $$X_1 = {\\partial_x} \\quad \\mbox{ and } \\quad\nX_2 = {\\partial_y} + \\frac{x^2}{2} {\\partial_z}.$$ The only nonzero brackets are $$X_{12}= [X_1,X_2] = x{\\partial_z} \\quad \\mbox{ and } \\quad\nX_{112}= [X_1, [X_1,X_2]] = {\\partial_z}.$$ Thus the growth vector is equal to $(2,2,3)$ on the plane $\\{ x=0\\}$, and to $(2,3)$ elsewhere. ",
"As a consequence, the set of singular points is the plane $\\{ x=0\\}$. The weights are $w_1=w_2=1$, $w_3=2$ at regular points, and $w_1=w_2=1$, $w_3=3$ at singular ones.",
"\n\n\\[ex:dnhinfini\\] Consider the vector fields on $\\R^3$ $$X_1=\\partial_x \\quad \\hbox{and} \\quad X_2=\\partial_y + f(x) \\partial_z,$$ where $f$ is a smooth function on $\\R$ which admits every positive integer $n \\in \\N$ as a zero with multiplicity $n$ (such a function exists and can even be chosen in the analytic class thanks to the Weierstrass factorization theorem [@Rudin1970 Th.",
" 15.9]). ",
"Every point $(n,y,z)$ is singular and the weights at this point are $w_1=w_2=1$, $w_3=n+1$. As a consequence the degree of nonholonomy $w_3$ is unbounded on $\\R^3$.\n\nLet us give some basic properties of the growth vector and of the weights.",
"\n\n- At a regular point, the growth vector is a strictly increasing sequence: $n_1(p) < \\cdots < n_r(p)$. Indeed, if $n_s(q)=n_{s+1}(q)$ in a neighbourhood of $p$, then $\\Delta^s$ is locally an involutive distribution and so $s=r$. As a consequence, at a regular point $p$, the jump between two successive weights is never greater than 1, $w_{i+1} - w_i \\leq 1$, and there holds $r(p) \\leq n-m+1$.\n\n- For every $s$, the map $q \\mapsto n_s(q)$ is a lower semi-continuous function from $M$ to $\\N$. Therefore, the set of regular points is open and dense in $M$.\n\n- For every $i=1, \\dots, n$, the weight $w_i(\\cdot)$ is an upper semi-continuous function. ",
"In particular, this is the case for the degree of nonholonomy $r(\\cdot)=w_n(\\cdot)$, that is, $r(q) \\leq r(p)$ for $q$ near $p$. As a consequence $r(\\cdot)$ is bounded on any compact subset of $M$.\n\n- The degree of nonholonomy may be unbounded on $M$ (see example \\[ex:dnhinfini\\] above). ",
"Thus determining if a nonholonomic system is controllable is a non decidable problem: the computation of an infinite number of brackets may be needed to decide if Chow’s Condition is satisfied.",
"\n\n However, in the case of polynomial vector fields on $\\R^n$ (relevant in practice), it can be shown that the degree of nonholonomy is bounded by a universal function of the degree $k$ of the polynomials (see [@gab95b; @gab98]): $$r(x) \\leq 2^{3n^2} n^{2n} k^{2n}.$$\n\nThe meaning of the sequence of weights is best understood in terms of basis of $T_pM$. Choose first vector fields $Y_1, \\dots,Y_{n_1}$ in $\\Delta^1$ whose values at $p$ form a basis of $\\Delta^1(p)$. Choose then vector fields $Y_{n_1+1}, \\dots,Y_{n_2}$ in $\\Delta^2$ such that the values $Y_1(p),\\dots,Y_{n_2}(p)$ form a basis of $\\Delta^2(p)$. For each $s$, choose $Y_{n_{s-1}+1},\n\\dots,Y_{n_s}$ in $\\Delta^s$ such that $Y_1(p),\\dots,Y_{n_s}(p)$ form a basis of $\\Delta^s(p)$. We obtain in this way a family of vector fields $Y_1, \\dots , Y_n$ such that \\[pa:adaptedframe\\] $$\\begin{aligned}\n\\label{eq:adaptedframe}\n\\left\\{\n\\begin{array}{l}\n Y_1(p), \\dots , Y_{n}(p) \\hbox{ is a basis of } T_pM, \\\\\n Y_i \\in \\Delta^{w_i} , \\ i=1, \\dots, n.\n\\end{array}\n\\right.\\end{aligned}$$ A family of $n$ vector fields satisfying (\\[eq:adaptedframe\\]) is called an [*adapted frame at $p$*]{}. ",
"The word “adapted” means here “adapted to the flag (\\[eq:flag\\])”, since the values at $p$ of an adapted frame contain a basis $Y_1(p), \\dots,\nY_{n_s}(p)$ of each subspace $\\Delta^s(p)$ of the flag. ",
"By continuity, at a point $q$ close enough to $p$, the values of $Y_1, \\dots, Y_n$ still form a basis of $T_qM$. However, if $p$ is singular, this basis may not be adapted to the flag (\\[eq:flag\\]) at $q$.\n\nLet us explain now the relation between weights and orders. ",
"We write first the tangent space as a direct sum, $$T_pM= \\Delta^1(p) \\oplus \\Delta^2(p) / \\Delta^1(p) \\oplus \\cdots \\oplus\n\\Delta^r(p)/ \\Delta^{r-1}(p),$$ where $\\Delta^s(p)/\\Delta^{s-1}(p)$ denotes a supplementary of $\\Delta^{s-1}(p)$ in $\\Delta^s(p)$, and take a local system of coordinates $(y_1,\\dots,y_n)$. The dimension of each space $\\Delta^s(p)/\\Delta^{s-1}(p)$ is equal to $n_s-n_{s-1}$, and we can assume that, up to a reordering, we have $dy_j(\\Delta^s(p)/\\Delta^{s-1}(p)) \\neq 0$ for $n_{s-1}< j \\leq n_s$.\n\nTake an integer $j$ such that $0<j \\leq n_1$. From the assumption above, there holds $dy_j(\\Delta^1(p))\\neq 0$, and consequently there exists $X_i$ such that $dy_j(X_i(p)) \\neq 0$. Since $dy_j(X_i)=X_i y_j$ is a first-order nonholonomic derivative of $y_j$, we have $\\mathrm{ord}_p (y_j) \\leq 1=w_j$.\n\nTake now an integer $j$ such that $n_{s-1}< j \\leq n_s$ for $s >1$, that is, $w_j=s$. Since $dy_j(\\Delta^s(p)/\\Delta^{s-1}(p)) \\neq 0$, there exists a vector field $Y$ in $\\Delta^s$ such that $dy_j(Y(p))=(Yy_j)(p) \\neq 0$. By definition of $\\Delta^s$, the Lie derivative $Yy_j$ is a linear combination of nonholonomic derivatives of $y_j$ of order not greater than $s$. One of them must be nonzero, and so $\\mathrm{ord}_p(y_j) \\leq s=w_j$.\n\nFinally, any system of local coordinates $(y_1,\\dots,y_n)$ satisfies $\\mathrm{ord}_p(y_j) \\leq w_j$ up to a reordering (or $\\sum_{i=1}^n \\mathrm{ord}_p(y_i) \\leq \\sum_{i=1}^n\nw_i$ without reordering). ",
"The coordinates with the maximal possible order will play an important role.",
"\n\nA [*system of privileged coordinates at $p$*]{} is a system of local coordinates $(z_1,\\dots,z_n)$ such that $\\mathrm{ord}_p(z_j)=w_j$ for $j=1,\\dots,n$.\n\nNotice that privileged coordinates $(z_1,\\dots,z_n)$ satisfy $$\\begin{aligned}\n\\label{eq:adapted}\ndz_i (\\Delta^{w_i}(p)) \\neq 0, \\quad dz_i (\\Delta^{w_i-1}(p))=0, \\quad\ni=1,\\dots,n,\\end{aligned}$$ or, equivalently, $\\partial_{z_i}|_p$ belongs to $\\Delta^{w_i}(p)$ but not to $\\Delta^{w_i-1}(p)$. Local coordinates satisfying (\\[eq:adapted\\]) are called [*linearly adapted coordinates*]{} (“adapted\" because the differentials at $p$ of the coordinates form a basis of $T^*_pM$ dual to the values of an adapted frame). ",
"Thus privileged coordinates are always linearly adapted coordinates. ",
"The converse is false, as shown in the example below.",
"\n\nTake $X_1=\\partial_x$, $X_2=\\partial_y + (x^2 +y) \\partial_z$ in $\\R^3$. The weights at 0 are $(1,1,3)$ and $(x,y,z)$ are adapted at 0. ",
"But they are not privileged: indeed, the coordinate $z$ is of order $2$ at $0$ since $(X_2X_2 z)(0)=1$.\n\nAs it is suggested by Kupka [@kup96], one can define *privileged functions at $p$* to be the smooth functions $f$ on $U$ such that $$\\mathrm{ord}_p(f) = \\min \\{ s \\in \\N \\ : \\ df(\\Delta^s(p)) \\neq 0 \\}.$$ It results from the discussion above that some local coordinates $(z_1,\\dots,z_n)$ are privileged at $p$ if and only if each $z_i$ is a privileged function at $p$.\n\nLet us now show how to compute orders using privileged coordinates. ",
"We fix a system of privileged coordinates $(z_1,\\dots,z_n)$ at $p$. Given a sequence of integers $\\alpha=(\\alpha_1, \\dots, \\alpha_n)$, we define the weighted degree of the monomial $z^\\alpha=z_1^{\\alpha_1} \\cdots z_n^{\\alpha_n}$ to be $w(\\alpha)=w_1\n\\alpha_1 + \\cdots + w_n \\alpha_n$ and the weighted degree of the monomial vector field $z^\\alpha \\partial_{z_j}$ as $w(\\alpha)-w_j$. The weighted degrees allow to compute the orders of functions and vector fields in a purely algebraic way.",
"\n\n\\[le:algorder\\] For a smooth function $f$ with a Taylor expansion $$f(z) \\sim \\sum_{\\alpha} c_\\alpha z^\\alpha,$$ the order of $f$ is the least weighted degree of monomials having a nonzero coefficient in the Taylor series.",
"\n\nFor a vector field $X$ with a Taylor expansion $$X(z) \\sim \\sum_{\\alpha, j} a_{\\alpha,j} z^\\alpha \\partial_{z_j},$$ the order of $X$ is the least weighted degree of a monomial vector fields having a nonzero coefficient in the Taylor series.",
"\n\nIn other words, when using privileged coordinates, the notion of nonholonomic order amounts to the usual notion of vanishing order at some point, only assigning weights to the variables.",
"\n\nFor $i=1,\\dots,n$, we have $\\partial_{z_i}|_p \\in \\Delta^{w_i}\n(p)$. Then there exist $n$ vector fields $Y_1, \\dots,Y_n$ which form an adapted frame at $p$ and such that $Y_1(p)=\\partial_{z_1}|_p$, …, $Y_n(p)=\\partial_{z_n}|_p$. For every $i$, the vector field $Y_i$ is of order $\\geq -w_i$ at $p$ since it belongs to $\\Delta^{w_i}$. Moreover we have $(Y_iz_i)(p)=1$ and $\\mathrm{ord}_p(z_i)=w_i$. Thus $\\mathrm{ord}_p(Y_i)=-w_i$.\n\nTake a sequence of integers $\\alpha=(\\alpha_1, \\dots, \\alpha_n)$. The monomial $z^\\alpha$ is of order $\\geq w(\\alpha)$ at $p$ and the differential operator $Y^\\alpha=Y^{\\alpha_1}_1 \\cdots\nY^{\\alpha_n}_n$ is of order $\\geq -w(\\alpha)$. Observing that $(Y_iz_j)(p)=0$ if $j\\neq i$, we easily see that $(Y^\\alpha\nz^\\alpha)(p) =\\frac{1}{\\alpha_1! ",
"\\dots \\alpha_n!}\\neq 0$, whence $\\mathrm{ord}_p(z^\\alpha)= w (\\alpha)$.\n\nIn the same way, we obtain that, if $z^\\alpha$, $z^\\beta$ are two different monomials and $\\lambda$, $\\mu$ two nonzero real numbers, then $\\mathrm{ord}_p( \\lambda z^\\alpha + \\mu z^\\beta) = \\min\n\\big( w(\\alpha), w(\\beta) \\big)$. Thus the order of a series is the least weighted degree of monomials actually appearing in the series itself. ",
"This shows the statement on order of functions.",
"\n\nAs a consequence, for any smooth function $f$, the order at $p$ of $\\partial_{z_i}f$ is $\\geq \\mathrm{ord}_p(f) -w_i$. Since moreover $\\partial_{z_i}z_i=1$, we obtain that $\\mathrm{ord}_p(\\partial_{z_i})$ is equal to $-w_i$. The second part of the statement follows.",
"\n\nA notion of homogeneity is also naturally associated with a system of privileged coordinates $(z_1,\\dots,z_n)$ defined on $U$. We define first the one-parameter family of dilations $$\\delta_t : (z_1,\\dots,z_n) \\mapsto (t^{w_1}z_1,\\dots, t^{w_n}z_n),\n\\qquad t \\geq 0.$$ Each dilation $\\delta_t$ is a map from $\\R^n$ to $\\R^n$. By abuse of notations, for $q\\in U$ and $t$ small enough we write $\\delta_t(q)$ instead of $\\delta_t(z(q))$, where $z(q)$ are the coordinates of $q$. A dilation $\\delta_t$ acts also on functions and vector fields by pull-back: $\\delta_t^*f=f \\circ \\delta_t$ and $\\delta_t^*X$ is the vector field such that $(\\delta_t^*X)(\\delta_t^*f)=\\delta_t^*(Xf)$.\n\nA function $f$ is [*homogeneous of degree*]{} $s$ if $\\delta_t^*f=t^sf$. A vector field $X$ is [*homogeneous of degree*]{} $\\sigma$ if $\\delta_t^*X=t^\\sigma X$.\n\nFor a smooth function (resp. ",
"a smooth vector field), this is the same as being a finite sum of monomials (resp. ",
"monomial vector fields) of weighted degree $s$. As a consequence, if a function $f$ is homogeneous of degree $s$, then it is of order $s$ at $p$.\n\nA typical degree $1$ homogeneous function is the so-called [ *pseudo-norm at $p$*]{}, defined by: $$\\label{eq:pseudonorm}\nz \\mapsto \\|z\\|_p = |z_1|^{1/w_1} + \\cdots\n+|z_n|^{1/w_n}.$$ When composed with the coordinates function, the pseudo-norm at $p$ is a (non smooth) function of order $1$, that is, $$\\|z(q)\\|_p = O \\big( d(p,q) \\big).$$ Actually, it results from Proposition \\[le:algorder\\] that the order of a function $f\\in C^\\infty (p)$ is the least integer $s$ such that $f(q)=O (\\|z(q)\\|_p^s)$.\n\n#### Examples of privileged coordinates.",
"\n\nOf course all the results above on algebraic computation of orders hold only if privileged coordinates do exist. ",
"Two types of privileged coordinates are commonly used in the literature.",
"\n\n##### a. Exponential coordinates.",
"\n\nChoose an adapted frame $Y_1, \\dots, Y_n$ at $p$. The inverse of the local diffeomorphism $$(z_1, \\dots, z_n) \\mapsto \\exp(z_1 Y_1 + \\cdots + z_n Y_n)(p)$$ defines a system of local privileged coordinates at $p$, called [*canonical coordinates of the first kind*]{}. ",
"These coordinates are mainly used in the context of hypoelliptic operator and for nilpotent Lie groups with right (or left) invariant sub-Riemannian structure.",
"\n\nThe inverse of the local diffeomorphism $$(z_1, \\dots, z_n) \\mapsto \\exp(z_n Y_n)\\circ \\cdots \\circ \\exp(z_1\nY_1)(p)$$ also defines privileged coordinates at $p$, called [*canonical coordinates of the second kind*]{}. ",
"They are easier to work with than the one of the first kind. ",
"For instance, in these coordinates, the vector field $Y_n$ read as $\\partial_{z_n}$. One can also exchange the order of the flows in the definition to obtain any of the $Y_i$ as $\\partial_{z_i}$. The fact that canonical coordinates of both first and second kind are privileged is proved in Section \\[se:2ndkind\\].",
"\n\nWe leave it to the reader to verify that the diffeomorphism $$(z_1, \\dots, z_n) \\mapsto \\exp(z_n Y_n+ \\cdots +z_{s+1}Y_{s+1})\\circ\n\\exp(z_s Y_s) \\cdots \\circ \\exp(z_1 Y_1)(p)$$ also induces privileged coordinates. ",
"As a matter of fact, any “mix\" between first and second kind canonical coordinates defines privileged coordinates.",
"\n\n##### b. Algebraic coordinates.",
"\n\nThere exist also effective constructions of privileged coordinates (the construction of exponential coordinates is not effective in general since it requires to integrate flows of vector fields). ",
"We present here Bellaïche’s algorithm, but other constructions exist (see [@ste86; @agr87]). ",
"\\[pa:coordalg\\]\n\n1. ",
" Choose an adapted frame $Y_1, \\dots, Y_n$ at $p$.\n\n2. ",
" Choose coordinates $(y_1,\\dots,y_n)$ centered at $p$ such that $\\partial_{y_i}|_p = Y_i(p)$.\n\n3. ",
" For $j=1, \\dots,n$, set $$z_j = y_j - \\sum_{k=2}^{w_j-1} h_k(y_1, \\dots, y_{j-1}),$$ where, for $k=2, \\dots, w_j-1$, $$h_k(y_1, \\dots, y_{j-1})=\n \\sum_{\\stackrel{\\scriptstyle{|\\alpha|=k}}{w(\\alpha)<w_j}} \\!\\!\\!\\!",
"\n Y_1^{\\alpha_1} \\ldots Y_{j-1}^{\\alpha_{j-1}} \\Big(\n y_j -\\sum_{q=2}^{k-1} h_q(y)\\Big)(p) \\\n \\frac{y_1^{\\alpha_1}}{\\alpha_1!} ",
"\\cdots\n \\frac{y_{j-1}^{\\alpha_{j-1}}}{\\alpha_{j-1}!},$$ with $|\\alpha|=\\alpha_1 + \\cdots + \\alpha_n$.\n\nThe fact that coordinates $(z_1, \\dots, z_n)$ are privileged at $p$ will be proved in Section \\[se:coordalg\\].",
"\n\nCoordinates $(y_1,\\dots,y_n)$ are linearly adapted coordinates. ",
"They can be obtained from any original system of coordinates by an affine change. ",
"The privileged coordinates $(z_1, \\dots, z_n)$ are then obtained from $(y_1,\\dots,y_n)$ by an expression of the form $$\\begin{aligned}\n z_1 & = & y_1, \\\\\n z_2 & = & y_2 + \\mathrm{pol}(y_1), \\\\\n & \\vdots & \\\\\n z_n & = & y_n + \\mathrm{pol}(y_1, \\dots, y_{n-1}),\n \\end{aligned}$$ where each $\\mathrm{pol}$ is a polynomial function without constant nor linear terms. ",
"The inverse change of coordinates takes the same triangular form, which makes the use of these coordinates easy for computations.",
"\n\nNilpotent approximation\n-----------------------\n\nFix a system of privileged coordinates $(z_1, \\dots, z_n)$ at $p$. Every vector field $X_i$ is of order $\\geq -1$, hence it has, in $z$ coordinates, a Taylor expansion $$X_i(z) \\sim \\sum_{\\alpha, j} a_{\\alpha,j} z^\\alpha \\partial_{z_j},$$ where $w(\\alpha) \\geq w_j-1$ if $a_{\\alpha,j} \\neq 0$. Grouping together the monomial vector fields of same weighted degree, we express $X_i$ as a series $$X_i = X_i^{(-1)} + X_i^{(0)} + X_i^{(1)} + \\cdots$$ where $X_i^{(s)}$ is a homogeneous vector field of degree $s$.\n\n\\[le:hmnilp\\] Set $\\widehat{X}_i=X_i^{(-1)}$, $i=1,\\dots,m$. The family of vector fields $\\widehat{X}_1, \\dots, \\widehat{X}_m$ is a first-order approximation of $X_1, \\dots, X_m$ at $p$ and generate a nilpotent Lie algebra of step $r=w_n$.\n\nThe fact that the vector fields $\\widehat{X}_1, \\dots, \\widehat{X}_m$ form a first-order approximation of $X_1, \\dots, X_m$ results from their construction.",
"\n\nNote further that any homogeneous vector field of degree smaller than $-w_n$ is zero, as it is easy to check in privileged coordinates. ",
"Moreover, if $X$ and $Y$ are homogeneous of degree respectively $k$ and $l$, then the bracket $[X,Y]$ is homogeneous of degree $k+l$ because $\\delta_t^* [X,Y]= [\\delta_t^*X, \\delta_t^*Y]=t^{k+l}[X,Y]$.\n\nIt follows that every iterated bracket of the vector fields $\\widehat{X}_1,\n\\dots, \\widehat{X}_m$ of length $k$ (i.e. containing $k$ of these vector field) is homogeneous of degree $-k$ and is zero if $k >\nw_n$.\n\nThe family $(\\widehat{X}_1, \\dots, \\widehat{X}_m)$ is called the [ *(homogeneous) nilpotent approximation*]{} of $(X_1, \\dots, X_m)$ at $p$ associated with the coordinates $z$.\n\n\\[ex:unicycle2\\] Consider the vector fields on $\\R^2 \\times S^1$ defining the kinematic model of a unicycle (see example \\[ex:unicycle\\]), that is, $X_1=\\cos \\theta \\partial_x + \\sin \\theta \\partial_y$, $X_2= \\partial_\\theta$. We have $[X_1,X_2]=\\sin \\theta \\partial_x - \\cos \\theta \\partial_y$, so the weights are $(1,1,2)$ at every point. ",
"At $p=0$, the coordinates $(x,\\theta)$ have order $1$ and $y$ has order 2, consequently $(x,\\theta,y)$ is a system of privileged coordinates at $0$. Taking the Taylor expansion of $X_1$ and $X_2$ in the latter coordinates, we obtain the homogeneous components: $$X_1^{(-1)}= \\partial_x + \\theta \\partial_y, \\quad X_1^{(0)}= 0, \\quad\nX_1^{(1)}= - \\frac{\\theta^2}{2} \\partial_x -\n\\frac{\\theta^3}{3!} ",
"\\partial_y, \\quad \\dots$$ and $X_2^{(-1)}=X_2=\\partial_\\theta$. Therefore the homogeneous nilpotent approximation of $(X_1, X_2)$ at $0$ in coordinates $(x,\\theta,y)$ is $$\\widehat{X}_1= \\partial_x + \\theta \\partial_y, \\quad\n\\widehat{X}_2=\\partial_\\theta.$$ We easily check that the Lie brackets of length $3$ of these vectors are zero, that is, $[\\widehat{X}_1,[\\widehat{X}_1,\\widehat{X}_2]] =\n[\\widehat{X}_2,[\\widehat{X}_1,\\widehat{X}_2]]=0$, and so the Lie algebra $\\mathit{Lie} (\\widehat{X}_1,\\widehat{X}_2 )$ is nilpotent of step $2$.\n\nThe homogeneous nilpotent approximation is not intrinsic to the frame $(X_1,\\dots,X_m)$, since it depends on the chosen system of privileged coordinates. ",
"However, if $\\widehat{X}_1, \\dots, \\widehat{X}_m$ and $\\widehat{X}'_1, \\dots,\n\\widehat{X}'_m$ are the nilpotent approximations associated with two different systems of coordinates, then their Lie algebras $\\mathit{Lie} (\\widehat{X}_1, \\dots,\n\\widehat{X}_m )$ and $\\mathit{Lie} (\\widehat{X}'_1, \\dots,\n\\widehat{X}'_m )$ are isomorphic. ",
"If moreover $p$ is a regular point, then $\\mathit{Lie} (\\widehat{X}_1, \\dots, \\widehat{X}_m )$ is isomorphic to the graded nilpotent Lie algebra $$\\mathrm{Gr}(\\Delta)_p = \\Delta (p) \\oplus (\\Delta^2 / \\Delta^1)(p)\n\\oplus \\cdots \\oplus\n(\\Delta^{r-1} / \\Delta^r)(p).$$\n\nThe nilpotent approximation denotes in fact two different objects. ",
"Each $\\widehat{X}_i$ can be seen as a vector field on $\\R^n$ or as the representation in $z$ coordinates of the vector field $z^*\\widehat{X}_i$ defined on a neighbourhood of $p$ in $M$. This will cause no confusion since the nilpotent approximation is associated with a given system of privileged coordinates.",
"\n\nIt is worth to notice the particular form of the nilpotent approximation in privileged coordinates. ",
"Write $\\widehat{X}_i=\\sum_{j=1}^n f_{ij} (z) \\partial_{z_j}$. Since $\\widehat{X}_i$ is homogeneous of degree $-1$ and $\\partial_{z_j}$ of degree $-w_j$, the function $f_{ij}$ is a homogeneous polynomial of weighted degree $w_j-1$. In particular it can not involve variables of weight greater than $w_j-1$, that is, $$\\widehat{X}_i (z)=\\sum_{j=1}^n f_{ij} (z_1, \\dots, z_{n_{w_j}-1})\n\\partial_{z_j}.$$ The nonholonomic control system $\\dot z = \\sum_{i=1}^m u_i \\widehat{X}_i(z)$ associated with the nilpotent approximation is then polynomial and in a triangular form, $$\\dot z_j = \\sum_{i=1}^m u_i f_{ij} (z_1, \\dots, z_{n_{w_j}-1}).$$ Computing the trajectories of a system in such a form is rather easy: given the input function $(u_1(t), \\dots,u_m(t))$, it is possible to compute the coordinates $z_j$ one after the other, only by integration.",
"\n\nAs vector fields on $\\R^n$, $\\widehat{X}_1, \\dots, \\widehat{X}_m$ generate a sub-Riemannian distance on $\\R^n$ which is homogeneous with respect to the dilation $\\delta_t$.\n\n\\[le:nahomo\\]\n\n- The family $( \\widehat{X}_1, \\dots, \\widehat{X}_m )$ satisfies Chow’s Condition on $\\R^n$.\n\n- The growth vector at $0$ of $(\\widehat{X}_1, \\dots, \\widehat{X}_m\n )$ is equal to the one at $p$ of $(X_1, \\dots, X_m)$.\n\nLet $\\widehat{d}$ be the sub-Riemannian distance on $\\R^n$ associated with $( \\widehat{X}_1,\n\\dots, \\widehat{X}_m )$.",
"\n\n- The distance $\\widehat{d}$ is homogeneous of degree 1, $$\\widehat{d} (\\delta_tx, \\delta_ty) = t \\widehat{d} (x,y).$$\n\n- There exists a constant $C >0$ such that, for all $z\n \\in \\R^n$, $$\\frac{1}{C} \\|z\\|_p \\leq \\widehat{d}(0,z) \\leq C \\| z \\|_p,$$\n\nwhere $\\| \\cdot \\|_p$ denotes the pseudo-norm at $p$ (see ).",
"\n\nThrough the coordinates $z$ we identify the neighbourhood $U$ of $p$ in $M$ with a neighbourhood of $0$ in $\\R^n$.\n\nFor every iterated bracket $X_I=[X_{i_k},\\dots,[X_{i_2},X_{i_1}]]$ of the vector fields $X_1,\\dots, X_m$, we set$\\widehat{X}_I=[\\widehat{X}_{i_k},\\dots,[\\widehat{X}_{i_2},\\widehat{X}_{i_1}]]$, and for $k\\geq 1$ we set $\\widehat{\\Delta}^k = \\mathrm{span} \\{ \\widehat{X}_I \\ : \\ |I| \\leq\n k \\}$. As noticed in the proof of Proposition \\[le:hmnilp\\], a bracket $\\widehat{X}_I$ of length $|I|=k$ is homogeneous of weighted degree $-k$, and by construction of the nilpotent approximation, there holds $X_I=\\widehat{X}_I +$ terms of order $>-k$. Therefore, $$\\widehat{X}_I(0) = X_I(p) \\bmod \\mathrm{span} \\{ \\partial_{z_j}\\big|_p \\: : \\: w_j\n< k \\} = X_I(p) \\bmod \\Delta^{k-1}(p).$$ As a consequence, for any integer $k\\geq 1$, we have $$\\label{eq:grvec}\n \\dim \\widehat{\\Delta}^k(0) = \\dim \\Delta^k(p),$$ and property $(ii)$ follows. ",
"Moreover, if $X_{I_1},\\dots,X_{I_n}$ form an adapted frame at $p$, then the family $(\\widehat{X}_{I_1}(0), \\dots, \\widehat{X}_{I_n}(0))$ is of rank $n$, which implies that its determinant is nonzero. ",
"Since the determinant of $X_{I_1},\\dots,X_{I_n}$ is an homogeneous polynomial of weighted degree $0$, it is nonzero everywhere, which implies $(i)$. As for the property $(iii)$, consider the nonholonomic system defined by the nilpotent approximation, that is, $\\dot z = \\sum_{i=1}^m u_i \\widehat{X}_i (z)$. Observe that, if $\\widehat{\\gamma}$ is a trajectory of this system, that is, if $$\\dot{\\widehat{\\gamma}} (t)= \\sum_{i=1}^m u_i \\widehat{X}_i\n\\big(\\widehat{\\gamma}(t)\\big), \\quad t\n\\in [0,T],$$ then the dilated curve $\\delta_{\\lambda} \\widehat{\\gamma}$ satisfies $$\\frac{d}{dt}\\delta_{\\lambda} \\widehat{\\gamma} (t)= \\sum_{i=1}^m \\lambda u_i\n\\widehat{X}_i\n\\big(\\delta_{\\lambda} \\widehat{\\gamma}(t)\\big), \\quad t\n\\in [0,T].$$ Thus $\\delta_{\\lambda} \\widehat{\\gamma}$ is a trajectory of the same system, with extremities $(\\delta_{\\lambda} \\widehat{\\gamma}) (0) = \\delta_{\\lambda}\n(\\widehat{\\gamma} (0))$ and $(\\delta_{\\lambda} \\widehat{\\gamma}) (T) = \\delta_{\\lambda}\n(\\widehat{\\gamma} (T))$, and its length equals $\\lambda {\\mathrm{length}}\n(\\widehat{\\gamma})$. This proves the homogeneity of $\\widehat{d}$. Finally, since $( \\widehat{X}_1, \\dots, \\widehat{X}_m )$ satisfies Chow’s Condition, the distance $\\widehat{d}(0,\\cdot)$ is continuous on $\\R^n$ (see Corollary \\[le:toposr\\]). ",
"We can then choose a real number $C >0$ such that, on the compact set $\\{ \\|z \\| =\n 1 \\}$, we have $1/C \\leq \\widehat{d} (0,z) \\leq C$. ",
"Both functions $\\widehat{d}(0,z)$ and $\\|\n z \\|$ being homogeneous of degree 1, the inequality of Property $(iv)$ follows.",
"\n\nDistance estimates {#se:distance}\n------------------\n\nAs it is the case for Riemannian distances, in general it is impossible to compute analytically a sub-Riemannian distance (it would require to determine all minimizing curves). ",
"This is very important to obtain estimates of the distance, at least locally. ",
"In a Riemannian manifold $(M,g)$, the situation is rather simple: in local coordinates $x$ centered at a point $p$, the Riemannian distance $d_R$ satisfies: $$d_R(q,q')= \\| x(q) - x(q') \\|_{g_p} + o(\\| x(q) \\|_{g_p}+ \\| x(q') \\|_{g_p}),$$ where $\\| \\cdot \\|_{g_p}$ is the Euclidean norm induced by the value $g_p$ of the metric $g$ at $p$. This formula has two consequences: first, it shows that the Riemannian distance behaves at the first-order as the Euclidean distance associated with $\\| \\cdot\n\\|_{g_p}$; secondly, the norm $\\| \\cdot \\|_{g_p}$ gives explicit estimates of $d_R$ near $p$, such as $$\\frac{1}{C} \\| x(q) \\|_{g_p} \\leq d_R(p,q) \\leq C \\| x(q) \\|_{g_p}.$$\n\nIn sub-Riemannian geometry, the two properties above hold, but do not depend on the same function: the first-order behaviour near $p$ is characterized by the distance $\\widehat{d}_p$ defined by a nilpotent approximation at $p$, whereas explicit local estimates of $d(p,\\cdot)$ are given by the pseudo-norm at $p$ $\\| \\cdot \\|_p$ defined in . ",
"We first present the latter estimates, often referred to as the “Ball-Box Theorem”, and then the first-order expansion of $d$ in Theorem \\[le:dvshatd\\].",
"\n\n\\[le:bbox\\] The following statement holds if and only if $z_1, \\dots, z_n$ are privileged coordinates at $p$: there exist constants $C_p$ and $\\eps_p >0$ such that, if $d(p,q^z) < \\eps_p$, then $$\\begin{aligned}\n\\label{eq:distestim}\n\\frac{1}{C_p} \\|z\\|_p \\leq d(p,q^z) \\leq C_p \\| z \\|_p\\end{aligned}$$ (as previously, $q^z$ denotes the point near $p$ with coordinates $z$ and $\\| \\cdot \\|_p$ the pseudo-norm at $p$).",
"\n\nExpressed in a given system of privileged coordinates, the sub-Riemannian balls $B(p,\\eps)$ satisfy, for $\\eps < \\eps_p$, $$\\mathrm{Box} \\big(\\frac{1}{C_p} \\eps \\big) \\subset B(p,\\eps) \\subset\n\\mathrm{Box} \\big(C_p \\eps \\big) ,$$ where $\\mathrm{Box} ( \\eps) = [-\\eps^{w_1},\\eps^{w_1}] \\times \\cdots\n\\times [-\\eps^{w_n},\\eps^{w_n}]$.\n\n\\[pa:bbox\\_nonuniform\\] The constants $C_p$ and $\\eps_p$ depend on the base point $p$. Around a regular point $p_0$, it is possible to construct systems of privileged coordinates depending continuously on the base point $p$. In this case, the corresponding constants $C_p$ and $\\eps_p$ depend continuously on $p$. This is no longer true at a singular point. ",
"In particular, if $p_0$ is singular, the estimate (\\[eq:distestim\\]) does not hold uniformly near $p_0$: we can not choose the constants $C_p$ and $\\eps_p$ independently on $p$ near $p_0$. We will see in section \\[se:desing\\] uniform versions of the Ball-Box Theorem.",
"\n\nThe Ball-Box Theorem is stated in different papers, often under the hypothesis that the point $p$ is regular. ",
"To our knowledge, two valid proofs exist, the ones in [@nag85] and in [@bel96]. ",
"The result also appears without proof in [@gro96] and in [@ger84], and with erroneous proofs in [@mit85] and in [@mon02].",
"\n\nWe present here a proof adapted from the one of Bellaïche (our is much simpler because Bellaïche actually proves a more general result, namely (\\[eq:complete\\_bel\\])). ",
"Basically, the idea is to compare the distances $d$ and $\\widehat{d}$. The main step is Lemma \\[le:x-xhat\\] below, which is essential in other respects to explain the role of nilpotent approximations in control theory.",
"\n\nFix a point $p\\in M$, and a system of privileged coordinates at $p$. Through these coordinates we identify a neighbourhood of $p$ in $M$ with a neighbourhood of 0 in $\\R^n$. As in the preceding subsection, we denote by $\\widehat{X}_1, \\dots, \\widehat{X}_m$ the homogeneous nilpotent approximation of $X_1, \\dots, X_m$ at $p$ (associated with the given system of privileged coordinates) and by $\\widehat{d}$ the induced sub-Riemannian distance on $\\R^n$. Recall also that $r=w_n$ denotes the degree of nonholonomy at $p$.\n\n\\[le:x-xhat\\] There exist constants $C$ and $\\eps >0$ such that, for any $x_0 \\in\n\\R^n$ and any $t \\in \\R^+$ with $\\tau = \\max (\\|x_0\\|_p,t) < \\eps$, we have $$\\| x(t) - \\widehat{x}(t) \\|_p \\leq C \\tau t^{1/r},$$ where $x(\\cdot)$ and $\\widehat{x}(\\cdot)$ are trajectories of the nonholonomic systems defined respectively by $X_1, \\dots, X_m$ and $\\widehat{X}_1, \\dots, \\widehat{X}_m$, starting at the same point $x_0$, associated with the same control function $u(\\cdot)$, and satisfying $\\|u(t)\\|=\n1$ a.e.",
"\n\nThe first step is to prove that $\\| x (t) \\|_p$ and $\\| \\widehat{x} (t) \\|_p \\leq \\mathit{Cst}\\, \\tau$ for small enough $\\tau$, where $\\mathit{Cst}$ is a constant. ",
"Let us do it for $x(t)$, the proof being exactly the same for $\\widehat{x}(t)$.\n\nThe equation of the control system associated with $X_1, \\dots, X_m$ is $$\\dot x_j = \\sum_{i=1}^m u_i \\big( f_{ij}(x) + r_{ij}(x) \\big),\n\\quad j=1, \\dots,n,$$ where $f_{ij}(x) + r_{ij}(x)$ is of order $\\leq w_j -1$ at 0. ",
"Thus, for $j=1, \\dots,n$ and $i=1, \\dots,m$, $|f_{ij}(x) + r_{ij}(x)| \\leq \\mathit{Cst}\\, \\| x \\|_p^{w_j-1}$ when $\\| x \\|_p$ is small enough. ",
"Note that, along the trajectory $x(t)$, $\\|x\\|_p$ is small when $\\tau$ is. ",
"Since $\\|u(t)\\|=\n1$ a.e., ",
"we get: $$\\begin{aligned}\n\\label{eq:dotxj}\n| \\dot x_j | \\leq \\mathit{Cst}\\, \\| x \\|_p^{w_j-1}.\\end{aligned}$$\n\nTo integrate this inequality, choose an integer $N$ such that all $N/w_j$ are even integers and set $\\| x \\|_N= \\big( \\sum_{i=1}^n\n|x_i|^{N/w_i} \\big)^{1/N}$. The function $\\| x \\|_N$ is equivalent to $\\|x\\|_p$ in the norm sense, and it is differentiable except at the origin. ",
"Inequality (\\[eq:dotxj\\]) implies $\\frac{d}{dt}\\| x \\|_N \\leq\n\\mathit{Cst}\\, $, and then, by integration, $$\\| x(t) \\|_N \\leq \\mathit{Cst}\\, t + \\| x (0)\\|_N \\leq \\mathit{Cst}\\,\n\\tau.$$ The functions $\\| x \\|_N$ and $\\|x\\|_p$ being equivalent, we obtain, for a trajectory starting at $x_0$, $\\| x (t)\\|_p \\leq \\mathit{Cst}\\,\n \\tau$ when $\\tau$ is small enough.",
"\n\nThe second step is to prove $|x_j(t) - \\widehat{x}_j(t)| \\leq \\mathit{Cst}\\,\n\\tau^{w_j}t$. The function $x_j - \\widehat{x}_j$ satisfies the differential equation $$\\begin{aligned}\n\\dot x_j - \\dot{\\widehat{x}_j} & = & \\sum_{i=1}^m u_i \\big( f_{ij}(x) -\nf_{ij}(\\widehat{x}) + r_{ij}(x)\n\\big), \\\\\n& = & \\sum_{i=1}^m u_i \\big( \\sum_{\\{ k \\, : \\, w_k <w_j\\}} (x_k\n- \\widehat{x}_k) Q_{ijk}(x, \\widehat{x}) + r_{ij}(x)\n\\big),\\end{aligned}$$ where $Q_{ijk}(x, \\widehat{x})$ is a homogeneous polynomial of weighted degree $w_j-w_k-1$. For $\\|x\\|_p$ and $\\| \\widehat{x} \\|_p$ small enough, we have $$|r_{ij}(x)| \\leq \\mathit{Cst}\\, \\|x\\|_p^{w_j} \\quad \\hbox{and} \\quad\n|Q_{ijk}(x, \\widehat{x})| \\leq \\mathit{Cst}\\, (\\|x\\|_p + \\|\\widehat{x}\\|_p)^{w_j-w_k-1}.$$ Using the inequalities of the first step, we obtain finally, for $\\tau$ small enough, $$\\begin{aligned}\n\\label{eq:xj-hatxj}\n|\\dot x_j (t) - \\dot{\\widehat{x}}_j(t)| \\leq \\mathit{Cst}\\, \\sum_{\\{ k \\, : \\, w_k\n<w_j\\}} |x_k (t)- \\widehat{x}_k (t)| \\tau^{w_j-w_k-1} + \\mathit{Cst}\\, \\tau^{w_j}.\\end{aligned}$$\n\nThis system of inequalities has a triangular form, hence it can be integrated iteratively. ",
"For $w_j=1$, the inequality is $|\\dot\nx_j (t) - \\dot{\\widehat{x}}_j(t)| \\leq \\mathit{Cst}\\, \\tau$, and so $|x_j (t) -\n\\widehat{x}_j(t)| \\leq \\mathit{Cst}\\, \\tau t$. By induction, let $j > n_1$ and assume $|x_k (t) - \\widehat{x}_k(t)| \\leq \\mathit{Cst}\\, \\tau^{w_k} t$ for $k<j$. Inequality (\\[eq:xj-hatxj\\]) implies $$|\\dot x_j (t) - \\dot{\\widehat{x}}_j(t)| \\leq \\mathit{Cst}\\, \\tau^{w_j-1} t + \\mathit{Cst}\\,\n\\tau^{w_j} \\leq \\mathit{Cst}\\, \\tau^{w_j},$$ and so $|x_j (t) - \\widehat{x}_j(t)| \\leq \\mathit{Cst}\\, \\tau^{w_j} t$.\n\nFinally, $$\\| x(t) - \\widehat{x} (t) \\|_p \\leq \\mathit{Cst}\\, \\tau (t^{1/w_1} + \\cdots +\nt^{1/w_n}) \\leq \\mathit{Cst}\\, \\tau t^{1/r},$$ which completes the proof of the lemma.",
"\n\nObserve first that, by definition of order, a system of coordinates $z$ is privileged if and only if $d(p,q^z) \\geq \\mathit{Cst}\\,\n\\| z\\|_p$. What remains to prove is that, if $z$ are privileged coordinates, then $d(p,q^z) \\leq \\mathit{Cst}\\, \\| z\\|_p$.\n\nWe will show that, for $\\|x^0\\|_p$ small enough, $$d(0,x^0) \\leq 2 \\widehat{d} (0,x^0),$$ and so $d(0,x^0) \\leq \\mathit{Cst}\\, \\|x^0\\|_p$ by Lemma \\[le:nahomo\\]. ",
"This will prove Theorem \\[le:bbox\\].",
"\n\nFix $x^0 \\in \\R^n$, $\\|x^0\\|_p < \\eps$. Let $\\widehat{x}_0 (t)$, $t \\in\n[0,T_0]$, be a minimizing curve for $\\widehat{d}$, having velocity one, and joining $x^0$ to 0. ",
"According to Remark \\[re:velocity1\\], such a curve exists, and there exists a control $u_0(\\cdot)$ associated with $\\widehat{x}_0$ such that $\\|u_0(t)\\|=1$ a.e. ",
"Moreover, $T_0=\\widehat{d} (0,x^0)$.\n\nLet $x_0(t)$, $t \\in [0,T_0]$, be the trajectory of the control system associated with $X_1, \\dots, X_m$ starting at $x^0$ and defined by $u_0(\\cdot)$. We have ${\\mathrm{length}} \\big(x_0\n(\\cdot)\\big) \\leq T_0$. Set $x^1 = x_0(T_0)$. By Lemma \\[le:x-xhat\\], $$\\|x^1\\|_p = \\| x_0(T_0) - \\widehat{x}_0 (T_0) \\|_p \\leq C \\tau T_0^{1/r},$$ where $\\tau = \\max (\\|x^0\\|_p, T_0)$. By Lemma \\[le:nahomo\\], $T_0=\\widehat{d} (0,x^0)$ satisfies $T_0 \\geq \\|x^0\\|_p / C'$, so $\\tau \\leq\nC' T_0$, and $$\\widehat{d} (0,x^1) \\leq C'\\|x^1\\|_p \\leq C'' \\widehat{d} (0,x^0)^{1+1/r},$$ with $C''=C'^2 C$.",
"\n\nChoose now $\\widehat{x}_1 (t)$, $t \\in [0,T_1]$, a minimizing curve for $\\widehat{d}$ of velocity one joining $x^1$ to 0. ",
"There exists a control $u_1(\\cdot)$ associated with $\\widehat{x}_1$ such that $\\|u_1(t)\\|=1$ a.e. ",
"Let $x_1(t)$, $t \\in [0,T_1]$, be the trajectory of the control system associated with $X_1, \\dots, X_m$ starting at $x^1$ and defined by $u_1(\\cdot)$. Set $x^2 = x_1(T_1)$. As previously, we have ${\\mathrm{length}} \\big(x_1(\\cdot)\\big)= \\widehat{d} (0,x^1)$ and $\\widehat{d} (0,x^2) \\leq C'' \\widehat{d} (0,x^1)^{1+1/r}$.\n\nRepeating this construction, we obtain a sequence of points $x^0, x^1, x^2,\n\\dots$ such that $\\widehat{d} (0,x^{k+1}) \\leq C'' \\widehat{d}\n(0,x^k)^{1+1/r}$, and a sequence of trajectories $x_k(\\cdot)$ joining $x^k$ to $x^{k+1}$ of length equal to $\\widehat{d} (0,x^k)$.\n\nTaking $\\|x^0\\|_p$ small enough, we can assume $C''\n\\widehat{d}(0,x^0)^{1/r} \\leq 1/2$. We have then $\\widehat{d} (0,x^1) \\leq\n\\widehat{d}(0,x^0)/2$, …, $\\widehat{d} (0,x^{k}) \\leq \\widehat{d}(0,x^0)/2^k$,…Consequently, $x^k$ tends to 0 as $k\\rightarrow +\\infty$. Putting end to end the curves $x_k(\\cdot)$ gives a trajectory joining $x^0$ to 0 of length $\\widehat{d}(0,x^0) + \\widehat{d}(0,x^1) +\n\\cdots \\leq 2 \\widehat{d}(0,x^0)$. This implies $d(0,x^0) \\leq 2\n\\widehat{d}(0,x^0)$, and the proof is complete.",
"\n\nNow, the distance $\\widehat{d}$ on $\\R^n$ induces a distance $\\widehat{d}_p$ on a neighbourhood of $p$ in $M$ by setting $\\widehat{d}_p(q,q') = \\widehat{d}(z(q),z(q'))$. This distance gives the first-order term in the expansion of $d(p,\\cdot)$.\n\n\\[le:dvshatd\\] On a neighbourhood of $p$ in $M$ there holds $$d(p,q) = \\widehat{d}_p(p,q) \\left(1+\n O\\left(\\widehat{d}_p(p,q)\\right)\\right) .$$\n\n\\[re:bboxhatd\\] By Theorem \\[le:bbox\\] and Lemma \\[le:nahomo\\], when $d(p,q)$ is small enough we get the estimate $$\\label{eq:bboxhatd}\n \\frac{1}{C} \\widehat{d}_p(p,q) \\leq d(p,q) \\leq C\n \\widehat{d}_p(p,q),$$ where $C$ is some positive constant. ",
"Theorem \\[le:dvshatd\\] essentially states that this constant can be chosen arbitrarily close to $1$.\n\nFix $\\delta >0$. We have to prove that there exists $\\eps >0$ such that, if $d(p,q) <\n\\eps$, then $$\\label{eq:dvsdhat}\n (1-\\delta) \\widehat{d}_p(p,q) \\leq d(p,q) \\leq (1+\\delta)\n \\widehat{d}_p(p,q).$$\n\nLet $q$ be a point in $M$. Setting $x^0 =q$ in the proof of Theorem \\[le:bbox\\] furnishes a trajectory joining $q$ to 0 which length is equal to $\\sum_{k=0}^\\infty \\widehat{d}(0,x^k)$, the points $x^k$ being such that $\\widehat{d} (0,x^{k+1}) \\leq C'' \\widehat{d}\n(0,x^k)^{1+1/r}$.\n\nFrom , there exists $\\eps >0$ such that $d(p,q) <\n\\eps$ implies $C'' \\widehat{d}(0,x^0)^{1/r} \\leq \\delta / (1 +\n\\delta)$. In this case the trajectory from $q$ to $0$ is of length not greater than $(1 + \\delta) \\widehat{d}(0,x^0)$ and we have $$d(p,q) \\leq (1+\\delta)\n \\widehat{d}_p(p,q).$$ To prove the other inequality in , we use the same argument but reverse the role of $d$ and $\\widehat{d}$. Let $x_0(t)$, $t \\in\n[0,T_0]$, be a minimizing curve for $d$ of velocity one joining $x^0$ to 0, and let $u_0(\\cdot)$ be a control associated with $x_0(\\cdot)$ such that $\\|u_0(t)\\|=1$ a.e. ",
"We have $T_0={d} (0,x^0)$. Let $\\widehat{x}_0(t)$, $t \\in [0,T_0]$, be the trajectory of the control system associated with $\\widehat{X}_1, \\dots,\n\\widehat{X}_m$ starting at $x^0$ and defined by the control $u_0(\\cdot)$. In particular, ${\\mathrm{length}} \\big(x_0\n(\\cdot)\\big) \\leq T_0$.\n\nSet $x^1 = x_0(T_0)$. By Lemma \\[le:x-xhat\\], $$\\|x^1\\|_p = \\| x_0(T_0) - \\widehat{x}_0 (T_0) \\|_p \\leq C \\tau T_0^{1/r},$$ where $\\tau = \\max (\\|x^0\\|_p, T_0)$. Theorem \\[le:bbox\\] implies $\\tau \\leq\nC_p T_0$, and $${d} (0,x^1) \\leq C_p \\|x^1\\|_p \\leq C'' {d} (0,x^0)^{1+1/r},$$ with $C''=C_p^2 C$.",
"\n\nRepeating this construction gives a trajectory of $\\widehat{X}_1, \\dots, \\widehat{X}_m$ joining $q$ to $p$ whose length is equal to $\\sum_{k=0}^\\infty {d}(0,x^k)$, where ${d} (0,x^{k+1}) \\leq C'' {d}\n(0,x^k)^{1+1/r}$.\n\nFor $d(p,q)$ small enough, we have $C'' {d}(0,x^0)^{1/r} \\leq \\delta / (1 -\n\\delta)$ and the trajectory from $q$ to $0$ is of length $\\leq 1/(1 - \\delta) {d}(0,x^0)$, which leads to $$\\widehat{d}_p(p,q) \\leq \\frac{1}{(1 - \\delta)}\n d(p,q).$$ This completes the proof.",
"\n\nApproximate motion planning\n---------------------------\n\nGiven a control system $(\\Sigma)$, the [*motion planning problem*]{} is to steer $(\\Sigma)$ from an initial point to a final point. ",
"For nonholonomic systems, the exact problem is in general unsolvable. ",
"However methods exist for a particular class of systems, namely for nilpotent (or nilpotentizable) systems. ",
"It is then of interest to devise approximate motion planning techniques based on nilpotent approximations. ",
"These techniques are Newton type methods, the nilpotent approximation playing the role of the usual linearization.",
"\n\nPrecisely, consider a nonholonomic control system $$(\\Sigma) \\ : \\qquad \\dot x = \\sum_{i=1}^m u_i X_i(x), \\quad x \\in\n\\R^n,$$ and initial and final points $a$ and $b$ in $\\R^n$. Denote by $\\widehat{X}_1, \\dots, \\widehat{X}_m$ a nilpotent approximation of $X_1, \\dots, X_m$ at $b$. The $k$-step of an approximate motion planning algorithm take the following form ($x^k$ denotes the state of the system, $x^0$ being the initial point $a$):\n\n1. ",
" compute a control $u(t)$, $t \\in [0,T]$, steering the control system associated with $\\widehat{X}_1, \\dots, \\widehat{X}_m$ from $x^k$ to $b$;\n\n2. ",
" compute the trajectory $x(\\cdot)$ of $(\\Sigma)$ with control $u(\\cdot)$ starting from $x^k$;\n\n3. ",
" set $x^{k+1}=x(T)$.\n\nIs this algorithm convergent or, at least, locally convergent? ",
"The answer to the latter question is positive under an extra hypothesis on the control given in point 2 of the algorithm, namely,\n\n- there exists a constant $K$ such that, if $x^k$ and $b$ are close enough, then $$\\int_0^T \\|u(t)\\| dt \\leq K \\widehat{d} (b, x_k).$$\n\nNote that a control corresponding to a minimizing curve for $\\widehat{d}$ satisfies this condition. ",
"Other standards methods using Lie groups (like the one in [@laf91]) or based on the triangular form of the homogeneous nilpotent approximation also satisfy this hypothesis.",
"\n\nThe local convergence is then proved exactly in the same way than Theorem \\[le:bbox\\]. ",
"We normalize first the control, so that $\\|u(t)\\| = 1$ a.e. ",
"Then from Lemmas \\[le:x-xhat\\] and \\[le:nahomo\\], we have $ \\widehat{d}(b,x^{k+1}) \\leq C'' T^{1+1/r},\n$ and using hypothesis (H), we obtain $$\\widehat{d}(b,x^{k+1}) \\leq\nC''K^{1+1/r} \\widehat{d} (b, x_k)^{1+1/r}.$$ If $a$ is close enough to $b$, we have, at each step of the algorithm, $\\widehat{d}(b,x^{k+1}) \\leq \\widehat{d} (b, x_k) /2$, which proves the local convergence of the algorithm. ",
"In other words, [*for each point $b \\in M$, there exists a constant $\\eps_b>0$ such that, if $d(a,b) < \\eps_b$, then the approximate motion planning algorithm steering the system from $a$ to $b$ converges.*]{}",
"\n\nTo obtain a globally convergent algorithm, a natural idea is to iterate the locally convergent one. ",
"This requires the construction of a finite sequence of intermediate goals $b_0=a, b_1, \\dots,\nb_N=b$ such that $d(b_{i-1},b_i) < \\eps_{b_i}$. However the constant $\\eps_b$ depends on $b$ and, as already noticed for Theorem \\[le:bbox\\], it is not possible to have a uniform nonzero constant near singular points. ",
"Thus this method may provide a globally convergent algorithm only when every point is regular.",
"\n\nTangent structure to Carnot-Carathéodory spaces {#se:tgtstruct}\n===============================================\n\nConsider a manifold $M$ endowed with a sub-Riemannian distance $d$ on $M$. The so-defined metric space $(M,d)$ is called a *Carnot-Carathéodory space*. ",
"The notion of first-order approximation introduced in the previous section has a metric interpretation and will allow us to describe the local structure of a Carnot-Carathéodory space.",
"\n\nMetric tangent space {#se:tangentcone}\n--------------------\n\nIn describing the tangent space to a manifold, we essentially look at smaller and smaller neighbourhoods of a given point, the manifold being fixed. ",
"Equivalently, we can look at a fixed neighbourhood, but expanding the manifold. ",
"As noticed by Gromov, this idea can be used to define a notion of tangent space for a general metric space.",
"\n\nIf $\\textsf{X}$ is a metric space with distance $d$, we define $\\lambda\n\\textsf{X}$, for $\\lambda\n>0$, to be the metric space with same underlying set as $\\textsf{X}$ and distance $\\lambda d$. A [*pointed metric space*]{} $(\\textsf{X},x)$ is a metric space with a distinguished point $x$.\n\nLoosely speaking, a metric tangent space to the metric space $\\textsf{X}$ at $x$ is a pointed metric space $(C_x\\textsf{X},y)$ such that $$(C_x\\textsf{X},y) = \\lim_{\\lambda \\rightarrow + \\infty} (\\lambda \\textsf{X},x).$$ Of course, for this definition to make sense, we have to define the limit of pointed metric spaces.",
"\n\nLet us first define the Gromov-Hausdorff distance between metric spaces. ",
"Recall that, in a metric space $\\textsf{X}$, the Hausdorff distance $\\hbox{H-dist} (A,B)$ between two subsets $A$ and $B$ of $\\textsf{X}$ is the infimum of $\\rho>0$ such that any point of $A$ is within a distance $\\rho$ of $B$ and any point of $B$ is within a distance $\\rho$ of $A$. ",
"The Gromov-Hausdorff distance $\\hbox{GH-dist} (X,Y)$ between two metric spaces $\\textsf{X}$ and $\\textsf{Y}$ is the infimum of Hausdorff distances $\\hbox{H-dist}\n(i(\\textsf{X}),j(\\textsf{Y}))$ over all metric spaces $\\textsf{Z}$ and all isometric embeddings $i : \\textsf{X} \\rightarrow \\textsf{Z}$, $j: \\textsf{Y} \\rightarrow \\textsf{Z}$.\n\nThanks to Gromov-Hausdorff distance, one can define the notion of limit of a sequence of pointed metric spaces: $(\\textsf{X}_n,x_n)$ converge to $(\\textsf{X},x)$ if, for any positive $r$, $$\\hbox{GH-dist} \\big( B^{\\textsf{X}_n}(x_n,r), B^\\textsf{X}(x,r) \\big)\n\\rightarrow 0 \\quad \\hbox{as} \\ n \\rightarrow + \\infty$$ where $B^\\textsf{Y}(y,r)$ is considered as a metric space, endowed with the distance of $Y$. Note that all pointed metric spaces isometric to $(\\textsf{X},x)$ are also limit of $(\\textsf{X}_n,x_n)$. However the limit is unique up to an isometry provided the closed balls around the distinguished point are compact [@Burago2001 Sect.",
" 7.4].",
"\n\nFinally, one says that $(\\textsf{X}_\\lambda,x_\\lambda)$ converge to $(\\textsf{X},x)$ when $\\lambda \\to \\infty$ if, for every sequence $\\lambda_n$, $(\\textsf{X}_{\\lambda_n},x_{\\lambda_n})$ converge to $(\\textsf{X},x)$.\n\nA pointed metric space $(C_x\\textsf{X},y)$ is a [*metric tangent space*]{} to the metric space $\\textsf{X}$ at $x$ if $(\\lambda \\textsf{X},x)$ converge to $(C_x\\textsf{X},y)$ as $\\lambda \\rightarrow + \\infty$. If it exists, it is unique up to an isometry provided the closed balls around $x$ in $(\\lambda \\textsf{X},x)$ are compact.",
"\n\nFor a Riemannian metric space $(M,d_R)$ induced by a Riemannian metric $g$ on a manifold $M$, metric tangent spaces at a point $p$ exist and are isometric to the Euclidean space $(T_pM, g_p)$, that is, the standard tangent space endowed with the scalar product defined by the quadratic form $g_p$.\n\nFor a Carnot-Carathéodory space $(M,d)$, the metric tangent space is given by the nilpotent approximation.",
"\n\n\\[th:tangentcone\\] A Carnot-Carathéodory space $(M,d)$ admits metric tangent spaces $(C_pM,y)$ at every point $p\\in M$. The space $C_pM$ is itself a Carnot-Carathéodory space isometric to $(\\R^n,\\widehat{d})$, where $\\widehat{d}$ is the sub-Riemannian distance associated with a homogeneous nilpotent approximation at $p$.\n\nThis theorem, due to Bellaïche, is a consequence of a strong version of Theorem \\[le:bbox\\]: for $q$ and $q'$ in a neighbourhood of $p$, $$\\begin{aligned}\n\\label{eq:complete_bel}\n|d(q,q')- \\widehat{d} (q,q')| \\leq \\mathit{Cst} \\, \\widehat{d}(p,q) d(q,q')^{1/r}.\\end{aligned}$$ In these notes, we present neither the proof of this result, nor the one of Theorem \\[th:tangentcone\\], and we refer the reader to [@bel96].",
"\n\nRecall that $\\widehat{d}$ is not intrinsic to the frame $(X_1,\\dots,X_m)$. Thus Theorem \\[th:tangentcone\\] does not provide an intrinsic characterization of the metric tangent space. ",
"Such characterizations exist for sub-Riemannian manifolds $(M,D,g_R)$ in [@mar00] and [@fal03], and the latter could easily be adapted to the case of a sub-Riemannian geometry associated with a nonholonomic system. ",
"However these characterizations are intrinsic to the differentiable manifold $M$ equipped with the sub-Riemannian structure $(D,g_R)$, or to $M$ equipped with the frame $(X_1,\\dots,X_m)$, not to the metric space $(M,d)$. To our knowledge, the problem of finding a definition of the metric tangent space $C_pM$ depending only on the Carnot-Carathéodory space $(M,d)$ is still open.",
"\n\nThe question we want to address now is: what is the algebraic structure of $C_pM$? ",
"Of course $C_pM$ is not a linear space in general: for instance, $\\widehat{d}$ is homogeneous of degree 1 but with respect to dilations $\\delta_t$ but not with respect to the usual Euclidean dilations. ",
"We will see that $C_pM$ has a natural structure of group, or at least of quotient of groups.",
"\n\nDenote by $G_p$ the group generated by the diffeomorphisms $\\exp(\nt \\widehat{X}_i)$ acting on the left on $\\R^n$. Since $\\mathfrak{g}_p= \\mathit{Lie} (\\widehat{X}_1, \\dots, \\widehat{X}_m)$ is a nilpotent Lie algebra, $G_p = \\exp (\\mathfrak{g}_p)$ is a simply connected group, having $\\mathfrak{g}_p$ as its Lie algebra.",
"\n\nThis Lie algebra $\\mathfrak{g}_p$ splits into homogeneous components $$\\mathfrak{g}_p = \\mathfrak{g}^{-1} \\oplus \\cdots \\oplus \\mathfrak{g}^{-r},$$ where $\\mathfrak{g}^{-s}$ is the set of homogeneous vector fields of degree $-s$, and so $\\mathfrak{g}_p$ is a graded Lie algebra. ",
"The first component $\\mathfrak{g}^{-1}=\n\\mathrm{span} \\langle \\widehat{X}_1, \\dots, \\widehat{X}_m \\rangle$ generates $\\mathfrak{g}_p$ as a Lie algebra. ",
"All these properties imply that $G_p$ is what we call a Carnot group.",
"\n\nA [*Carnot group*]{} is a simply connected Lie group, such that the associated Lie algebra is graded, nilpotent, and generated by its first component.",
"\n\nNote that the dilations $\\delta_t$ act on $\\mathfrak{g}_p$ as a multiplication by $t^{-s}$ on $\\mathfrak{g}^{-s}$. This action extends to $G_p$ by the exponential mapping.",
"\n\n\\[ex:heisgroup\\] The simplest non Abelian Carnot group is the *Heisenberg group* $\\mathbb{H}^3$ which is the connected and simply connected Lie group whose Lie algebra satisfies $$\\mathfrak{g} = \\mathfrak{g}^{-1} \\oplus \\mathfrak{g}^{-2}, \\qquad \\hbox{with } \\dim \\mathfrak{g}^{-1} = 2.$$ As a consequence, $\\dim \\mathbb{H}^3 = 3$. Choosing a basis $X$, $Y$, $Z = [X,Y]$ of $\\mathfrak{g}$, we define coordinates on $\\mathbb{H}^3$ by the exponential mapping $$(x,y,z) \\mapsto \\exp (x X + y Y + z Z).$$ By the Campbell-Hausdorff formula (see Section \\[se:chforflows\\]), the law group on $\\mathbb{H}^3$ in these coordinates is $$(x,y,z) \\cdot (x',y',z') = (x+x', y+y', z+z'+\\frac{1}{2} (xy'-x'y)),$$ which is homogeneous with respect to the dilation $\\delta_t (x,y,z) = (tx,ty,t^2z)$.\n\nFinally, denote by $X_1, X_2$ the left-invariant vector fields on $\\mathbb{H}^3$ whose values at the identity are respectively $X$ and $Y$. In coordinates $(x,y,z)$, these vector fields write as $$X_1 = {\\partial_x} - \\frac{y}{2} {\\partial_z} \\quad \\mbox{ and } \\quad\nX_2 = {\\partial_y} + \\frac{x}{2} {\\partial_z},$$ which are the vector fields of what we have called the Heisenberg case in examples \\[ex:heis\\] and \\[ex:heis2\\].",
"\n\nLet $\\widehat{\\xi}_1, \\dots, \\widehat{\\xi}_m$ be the right-invariant vector fields on $G_p$ such that $\\widehat{\\xi}_i ( \\mathrm{id}) = \\widehat{X}_i$, where $\\mathrm{id}$ is the identity of $G_p$. Equivalently, $$\\widehat{\\xi}_i (g) = \\frac{d}{dt} \\big[ \\exp (t \\widehat{X}_i) g \\big] \\big|_{t=0}.$$ With $(\\widehat{\\xi}_1, \\dots, \\widehat{\\xi}_m)$ is associated a right-invariant sub-Riemannian metric and a sub-Riemannian distance $d_{G_p}$ on $G_p$.\n\nThe action of $G_p$ on $\\R^n$ is smooth and transitive. ",
"Indeed, for every $x \\in\n\\R^n$, the orbit of $x$ under the action of $G_p$ is the set $$\\left\\{ \\exp(t_{i_1} \\widehat{X}_{i_1}) \\circ \\cdots \\circ \\exp(t_{i_k}\n\\widehat{X}_{i_k})(x) \\ : \\ k\\in \\N, \\ t_{i_j} \\in \\R, \\ i_j \\in \\{ 1,\n\\dots, m\\} \\right\\}.$$ By Chow-Rashevsky’s theorem (or more precisely from Remark \\[re:orbit\\]), this set is the whole $\\R^n$ since $( \\widehat{X}_1, \\dots, \\widehat{X}_m )$ satisfies Chow’s Condition on $\\R^n$ (Lemma \\[le:nahomo\\]).",
"\n\nTo understand the algebraic structure of $C_pM$ we will use the following standard result on transitive action of Lie groups (see for instance [@lee03 Th.",
" 9.24]).",
"\n\n\\[th:lee\\] Let $G$ be a Lie group acting on the left smoothly and transitively on a manifold $M$. Let $q \\in M$ and $H$ be the isotropy subgroup of $q$ which is defined by $H=\\{ g \\in G \\ : \\ g \\cdot q = q \\}$. Then $H$ is a closed subgroup of $G$, the left coset space $G/H$ is a manifold of dimension $\\dim G - \\dim\nH$, and the map $F : G/H \\to M$ defined by $F(gH) = g \\cdot q$ is an equivariant diffeomorphism.",
"\n\nLet $H_p$ be the isotropy subgroup of $0 \\in \\R^n$ under the action of $G_p$. According to Theorem \\[th:lee\\], the map $\\phi_p : G_p \\rightarrow \\R^n$, $\\phi_p(g)=g(0)$, induces a diffeomorphism $$\\psi_p : G_p / H_p \\rightarrow \\R^n, \\qquad \\psi_p(gH_p) = g(0).$$\n\nObserve that $H_p$ is invariant under dilations, since $\\delta_t g\n(\\delta_t x) = \\delta_t (g(x))$. Hence $H_p$ is connected and simply connected, and so $H_p = \\exp (\\mathfrak{h}_p)$, where $\\mathfrak{h}_p$ is the Lie sub-algebra of $\\mathfrak{g}_p$ containing the vector fields vanishing at $0$, $$\\mathfrak{h}_p = \\{ Z \\in \\mathfrak{g}_p \\ : \\ Z(0)=0 \\}.$$ As $\\mathfrak{g}_p$, $\\mathfrak{h}_p$ is invariant under dilations and splits into homogeneous components.",
"\n\nNow, the elements $\\widehat{X}_1, \\dots, \\widehat{X}_m$ of $\\mathfrak{g}_p$ act on the left on $G_p / H_p$ with the notation $\\overline{\\xi}_1,\n\\dots, \\overline{\\xi}_m$, $$\\overline{\\xi}_i (gH_p) = \\frac{d}{dt} \\big[ \\exp (t \\widehat{X}_i) g H_p\\big] \\big|_{t=0}.$$ These vector fields define a sub-Riemannian metric and a sub-Riemannian distance $\\overline{d}$ on $G_p / H_p$. We clearly have ${\\psi_p}_*\\overline{\\xi}_i = \\widehat{X}_i$, so $\\psi_p$ maps the sub-Riemannian metric on $G_p / H_p$ associated with $(\\overline{\\xi}_1,\n\\dots, \\overline{\\xi}_m)$ to the one on $\\R^n$ associated with $( \\widehat{X}_1,\n\\dots, \\widehat{X}_m )$. ",
"We summarize this construction by the following result.",
"\n\n\\[th:algstruc\\] The metric tangent space $C_pM$ and $(\\R^n,\\widehat{d})$ are isometric to the coset space $G_p / H_p$ endowed with the sub-Riemannian distance $\\overline{d}$.\n\n\\[ex:grusin\\] Consider the vector fields $X_{1}=\\partial_x$ and $X_2 = x \\partial_y$ on $\\R^2$. The Carnot-Carathéodory space defined by these vector fields is called the *Grušin plane*.",
"\n\nThe only nonzero bracket is $X_{12} = \\partial_y$. Thus, at $p=0$, the weights are $(1,2)$, and $(x,y)$ are privileged coordinates. ",
"Since $X_1$ and $X_2$ are homogeneous with respect to this system of coordinates, we have $\\widehat{X}_1 = X_1$ and $\\widehat{X}_2=\nX_2$. The Lie algebra they generate is $$\\mathfrak{g}_0 = \\mathrm{span} (X_1, X_2, X_{12}) $$ which is of dimension 3, and the group $\\exp (\\mathfrak{g}_0)$ is actually the *Heisenberg group* $\\mathbb{H}^3$ (see example \\[ex:heisgroup\\]). ",
"The Lie sub-algebra $\\mathfrak{h}_0$ of $\\mathfrak{g}_0$ containing the vector fields vanishing at $0$ is $$\\mathfrak{h}_0 = \\mathrm{span} (X_2), $$ which is one-dimensional. ",
"Thus the Grušin plane is isometric to $\\mathbb{H}^3 / \\exp(\\mathfrak{h}_0)$ endowed with the distance $\\overline{d}$.\n\n\\[ex:martinet2\\] Consider the Martinet case, defined on $\\R^{3}$ by $$X_1 = {\\partial_x} \\quad \\mbox{ and } \\quad\nX_2 = {\\partial_y} + \\frac{x^2}{2} {\\partial_z}.$$ As noticed in example \\[ex:martinet\\], at $p=0$, the coordinates $(x,y,z)$ are privileged and, by homogeneity, $\\widehat{X}_1 = X_1$ and $\\widehat{X}_2= X_2$. Moreover the only nonzero bracket are $X_{12} = x\\partial_z$ and $X_{112} = \\partial_z$. Thus, $$\\mathfrak{g}_0 = \\mathrm{span} (X_1, X_2, X_{12},X_{112} ) $$ which is of dimension 4. ",
"The group $\\exp (\\mathfrak{g}_0)$ is called the *Engel group*, and is denoted by $\\mathbb{E}^4$. The Lie sub-algebra $\\mathfrak{h}_0$ of $\\mathfrak{g}_0$ containing the vector fields vanishing at $0$ is $$\\mathfrak{h}_0 = \\mathrm{span} (X_{12}). ",
"$$\n\nWhen the point $p$ is regular, Theorem \\[th:algstruc\\] can be refined thanks to the following result.",
"\n\nIf $p$ is a regular point, then $\\dim G_p=n$.\n\nLet $X_{I_1}, \\dots, X_{I_n}$ be an adapted frame at $p$. Due to the regularity of $p$, $X_{I_1}, \\dots, X_{I_n}$ is also an adapted frame near $p$, so any bracket $X_J$ can be written as $$X_J (z) = \\sum_{\\{i \\, : \\, |I_i| \\leq |J| \\}} a_i(z) X_{I_i} (z),$$ where each $a_i$ is a function of order $\\geq |I_i| - |J|$. Taking the homogeneous terms of degree $- |J|$ in this expression, we obtain $$\\widehat{X}_J (z) = \\sum_{ \\{i \\, : \\, |I_i| = |J|\\}} a_i(0) \\widehat{X}_{I_i}\n(z),$$ and so $\\widehat{X}_J \\in \\mathrm{span} \\langle \\widehat{X}_{I_1},\n\\dots, \\widehat{X}_{I_n}\n\\rangle$. Thus $\\widehat{X}_{I_1}, \\dots, \\widehat{X}_{I_n}$ is a basis of $\\mathfrak{g}_p$, and so $\\dim G_p=n$.\n\nAs a consequence $H_p$ is of dimension zero. ",
"Since $H_p$ is invariant under dilations, $H_p=\\{ \\mathrm{id} \\}$, and hence the mapping $\\phi_p : G_p \\rightarrow \\R^n$, $\\phi_p(g)=g(0)$, is a diffeomorphism. ",
"Moreover ${\\phi_p}_*\\widehat{\\xi}_i=\\widehat{X}_i$, which implies that $\\phi_p$ maps the sub-Riemannian metric on $G_p$ associated with $(\\widehat{\\xi}_1, \\dots, \\widehat{\\xi}_m)$ to the one on $\\R^n$ associated with $( \\widehat{X}_1, \\dots, \\widehat{X}_m )$. ",
"This gives the following result[^3].",
"\n\nWhen $p$ is a regular point, the metric tangent space $C_pM$ and the Carnot-Carathéodory space $(\\R^n,\\widehat{d})$ are isometric to the Carnot group $G_p$ endowed with the right-invariant sub-Riemannian distance $d_{G_p}$.\n\nThus Carnot groups have the same role in sub-Riemannian geometry as Euclidean spaces have in Riemannian geometry. ",
"For this reason they are sometimes referred to as “non Abelian linear spaces”: the internal operation – addition – is replaced by the law group and the external operation – multiplication by a real number – by the dilations. ",
"Note that, when $G_p$ is Abelian (i.e. commutative) then $G_p$ has a linear structure and the sub-Riemannian metric on $G_p$ is a Euclidean metric.",
"\n\n\\[ex:unicycle3\\] In the case of the distance $d$ associated with the unicycle (examples \\[ex:unicycle\\] and \\[ex:unicycle2\\]), the growth vector is $(2,3)$ at every point. ",
"Hence every point $p \\in \\R^2 \\times S^1$ is regular, and the Lie algebra generated by the nilpotent approximation satisfies $$\\mathfrak{g}_p = \\mathfrak{g}^{-1} \\oplus \\mathfrak{g}^{-2}, \\qquad\n\\hbox{with } \\dim \\mathfrak{g}^{-1} = 2.$$ As a consequence, $G_p = \\mathbb{H}^3$ (see example \\[ex:heisgroup\\]), and so the metric tangent space to $(\\R^2 \\times S^1,d)$ at every point $p$ has the structure of the Heisenberg group.",
"\n\nDesingularization and uniform distance estimate {#se:desing}\n-----------------------------------------------\n\nWe have already highlighted in Remark \\[pa:bbox\\_nonuniform\\] that singular points may cause difficulties, in particular because of the loss of uniformity of distance estimates. ",
"Therefore it is necessary to study carefully the behaviour of the distance at such points. ",
"We proceed as it is usual for singularities, that is, we consider a singularity as the projection of a regular object. ",
"To this aim we exploit the algebraic structure of the metric tangent space, which provides a good way of lifting and projecting Carnot-Carathéodory spaces.",
"\n\nLet us begin with nilpotent approximations. ",
"We keep the notations and definitions of the preceding subsection. ",
"At a singular point $p$, we have the following diagram, $$\\begin{array}{ccc}\n (G_p, d_{G_p}) & & \\\\[2mm]\n \\hbox{\\scriptsize $\\pi$ } \\hbox{\\Large $\\downarrow$} &\n \\hbox{\\scriptsize $\\phi_p$} \\hbox{\\Large $\\searrow$} & \\\\[1mm]\n (G_p/H_p, \\overline{d}) & \\stackrel{\\psi_p}{\\widetilde{\\longrightarrow}} &\n (\\R^n,\\widehat{d})\n\\end{array}$$ Since the sub-Riemannian metric on $G_p$ is a right-invariant, every point in the space $(G_p, d_{G_p})$ is regular.",
"\n\nA Carnot-Carathéodory space $(M,d)$ is said to be *equiregular* if every point in $M$ is regular.",
"\n\nThus $(\\R^n,\\widehat{d})$ is the projection of the equiregular space $(G_p, d_{G_p})$. Recall now that $\\widehat{\\xi}_1, \\dots, \\widehat{\\xi}_m$ (resp. ",
"$\\overline{\\xi}_1, \\dots,\n\\overline{\\xi}_m$) are mapped to $\\widehat{X}_1, \\dots, \\widehat{X}_m$ by $\\phi_p$ (resp. ",
"$\\psi_p$). ",
"Working in a system of coordinates, we identify $G_p/H_p$ with $\\R^n$ and $\\overline{\\xi}_i$ with $\\widehat{X}_i$. These coordinates on $\\R^n \\simeq G_p/H_p$, denoted by $x$, induce coordinates $(x,z)\n\\in \\R^N$ on $G_p$ for which we have $$\\begin{aligned}\n\\label{eq:bij}\n\\widehat{\\xi}_i(x,z) = \\widehat{X}_i (x) + \\sum_{j=n+1}^N b_{ij} (x,z)\n\\partial_{z_j}.\\end{aligned}$$\n\nLet $(x(\\cdot),z(\\cdot))$ be a trajectory of the nonholonomic system in $G_p$ defined by $\\widehat{\\xi}_1, \\dots, \\widehat{\\xi}_m$. Then, for every control $u(\\cdot)$ associated with the trajectory, $$(\\dot x (t), \\dot z (t)) = \\sum_{i=1}^m\n u_i(t) \\widehat{\\xi}_i (x,z),$$ It follows from that $x(\\cdot)$ is a trajectory in $\\R^n$ of the system defined by $\\widehat{\\xi}_1, \\dots, \\widehat{\\xi}_m$, which is associated with the same controls $u(\\cdot)$, so that $${\\mathrm{length}} \\big(x(\\cdot)\\big) = {\\mathrm{length}} \\big((x,z)(\\cdot) \\big).$$ Thus $\\widehat{d}$ can be obtained from the sub-Riemannian distance $d_{G_p}$ in $G_p$ by $$\\widehat{d} (q_1, q_2) = \\inf_{\\widetilde{q}_2 \\, \\in \\, q_2H_p} d_{G_p}(\\widetilde{q}_1,\n\\widetilde{q}_2), \\qquad \\hbox{for any } \\widetilde{q}_1 \\in q_1H_p,$$ or, equivalently, $B^{\\widehat{d}} (q_1,\\eps) = \\phi_p \\big(\nB^{d_{G_p}}(\\widetilde{q}_1, \\eps) \\big)$.\n\nWe will use this idea to desingularize the original space $(M,d)$. Choose for $x$ privileged coordinates at $p$, so that $$X_i(x) = \\widehat{X}_i (x) + R_i(x) \\qquad \\hbox{with } \\mathrm{ord}_pR_i \\geq 0.$$ Set $\\widetilde{M}= M \\times \\R^{N-n}$, and in local coordinates $(x,z)$ on $\\widetilde{M}$, define vector fields on a neighbourhood of $(p,0)$ by $$\\xi_i (x,z) = X_i (x) + \\sum_{j=n+1}^N b_{ij} (x,z)\n\\partial_{z_j},$$ with the same functions $b_{ij}$ as in (\\[eq:bij\\]). ",
"Such vector fields are called a *lifting* of the vector fields $X_1,\\dots,X_m$: denoting by $\\pi: \\widetilde{M} \\to M$ the canonical projection, we have $X_i = \\pi_*\\xi_i$ for $i=1, \\dots,m$, that is, $X_i$ is the projection of $\\xi_i$.\n\nWe define in this way a nonholonomic system on an open set $\\widetilde{U} \\subset \\widetilde{M}$ whose nilpotent approximation at $(p,0)$ is $(\\widehat{\\xi}_1,\n\\dots, \\widehat{\\xi}_m)$, by construction. ",
"Unfortunately, $(p,0)$ can be itself a singular point. ",
"Indeed, a point can be singular for a system and regular for the nilpotent approximation taken at this point.",
"\n\nTake the vector fields $X_1=\\partial_{x_1}$, $X_2 =\n\\partial_{x_2} + x_1 \\partial_{x_3} + x_1^2 \\partial_{x_4}$ and $X_3 = \\partial_{x_5} + x_1^{100} \\partial_{x_4}$ on $\\R^5$. The origin 0 is a singular point. ",
"However the nilpotent approximation at 0 is $\\widehat{X}_1=X_1$, $\\widehat{X}_2=X_2$, $\\widehat{X}_3=\\partial_{x_5}$, for which $0$ is not singular.",
"\n\nTo avoid this difficulty, we take a group bigger than $G_p$, namely the free nilpotent group $N_r$ of step $r$ with $m$ generators. ",
"$N_r$ is a Carnot group and its Lie algebra $\\mathfrak{n}_r$ is the free nilpotent Lie algebra of step $r$ with $m$ generators. ",
"The given of $m$ generators $\\alpha_1, \\dots,\n \\alpha_m$ of $\\mathfrak{n}_r$ define on $N_r$ a right-invariant sub-Riemannian distance $d_N$.\n\nThe group $N_r$ can be thought as a group of diffeomorphisms, and so it defines a left action on $\\R^n$. Denoting by $J$ the isotropy subgroup of 0 for this action, we obtain that $(\\R^n, \\widehat{d})$ is isometric to $N_r/J$ endowed with the restriction of the distance $d_N$.\n\nReasoning as above, we are able to lift locally the vector fields $X_1, \\dots, X_m$ on $M$ to vector fields on $M\n \\times \\R^{\\widetilde{n}-n}$, $\\widetilde{n}=\\dim N_r$, having $\\alpha_1, \\dots, \\alpha_m$ for nilpotent approximation at $(p,0)$. Moreover $(p,0)$ is a regular point for the associated nonholonomic system in $M \\times \\R^{\\widetilde{n}-n}$ since $N_r$ is free up to step $r$. We obtain in this way a result of desingularization.",
"\n\nLet $p$ be a point in $M$, $r$ the degree of nonholonomy at $p$, $\\widetilde{n}=\\widetilde{n} (m,r)$ the dimension of the free Lie algebra of step $r$ with $m$ generators, and $\\widetilde{M}= M \\times \\R^{\\widetilde{n}-n}$. Then there exist a neighbourhood $\\widetilde{U} \\subset\n\\widetilde{M}$ of $(p,0)$, a neighbourhood $U \\subset M$ of $p$ with $U \\times \\{0 \\} \\subset \\widetilde{U}$, local coordinates $(x,z)$ on $\\widetilde{U}$, and smooth vector fields on $\\widetilde{U}$, $$\\label{eq:projxi}\n\\xi (x,z) = X_i (x) + \\sum_{j=n+1}^N b_{ij} (x,z)\n\\partial_{z_j} ,$$ such that:\n\n- $\\xi_1, \\dots, \\xi_m$ satisfy Chow’s Condition and have $r$ for degree of nonholonomy everywhere (so the Lie algebra they generate is free up to step $r$);\n\n- every $\\widetilde{q}$ in $\\widetilde{U}$ is regular;\n\n- denoting by $\\pi : \\widetilde{M} \\rightarrow M$ the canonical projection, and by $\\widetilde{d}$ the sub-Riemannian distance defined by $\\xi_1, \\dots, \\xi_m$ on $\\widetilde{U}$, we have $\\pi_*\\xi_i = X_i$, and for $q \\in U$ and $\\eps>0$ small enough, $$B(q,\\eps) = \\pi \\Big( B^{\\widetilde{d}} \\big( (q,0), \\eps \\big)\n \\Big),$$ or, equivalently, $$d(q_1,q_2) = \\inf_{\\widetilde{q}_2 \\, \\in \\, \\pi^{-1} (q_2)} \\widetilde{d}\n \\big( (q_1,0), \\widetilde{q}_2\n \\big).$$\n\nThe lemma still holds if we replace $r$ by any integer greater than the degree of nonholonomy at $p$.\n\nThus any Carnot-Carathéodory space $(M,d)$ is locally the projection of an equiregular Carnot-Carathéodory space $(\\widetilde{M},\\widetilde{d})$. This projection preserves the trajectories, the minimizers, and the distance.",
"\n\n\\[ex:martinet3\\] Consider the vector fields of the Martinet case (see example \\[ex:martinet\\]), defined on $\\R^{3}$ by: $$X_1 = {\\partial_x} \\quad \\mbox{ and } \\quad\nX_2 = {\\partial_y} + \\frac{x^2}{2} {\\partial_z}.$$ Let $\\pi: \\R^4 \\to \\R^3$ be the projection with respect to the last coordinates, $\\pi(x,y,z,w) = (x,y,z)$. Then $X_1$ and $X_2$ are the projections of the vector fields defining the Engel group $\\mathbb{E}^4$ (see example \\[ex:martinet2\\]), $$\\xi_1 = {\\partial_x} \\quad \\mbox{ and } \\quad\n\\xi_2 = {\\partial_y} + \\frac{x^2}{2} {\\partial_z} + x {\\partial_w},$$ that is $\\pi_*\\xi_i = X_i$. Thus, for every pair of points $q_1,q_2 \\in\n\\R^3$, $$d_{\\mathrm{Mart}}(q_1,q_2) = \\inf_{w \\in \\R} d_{\\mathbb{E}^4}\n\\big( (q_1,0), (q_2,w)\n\\big),$$ where $d_{\\mathrm{Mart}}$ and $d_{\\mathbb{E}^4}$ are the sub-Riemannian distance in respectively the Martinet space and the Engel group.",
"\n\n\\[ex:grusin2\\] Consider the vector fields $$X_{1}=\\partial_x, \\qquad X_2 = x \\partial_y,$$ on $\\R^2$, which define the Grušin plane (see example \\[ex:grusin\\]). ",
"Let $\\pi: \\R^3 \\to \\R^2$ be the projection with respect to the last coordinates, $\\pi(x,y,z) = (x,y)$. Then $X_1=\\pi_*\\xi_1$ and $X_2=\\pi_*\\xi_2$, where $$\\xi_1 = {\\partial_x} \\quad \\mbox{ and } \\quad\n\\xi_2 = {\\partial_z} + x {\\partial_z},$$ are, up to a change of coordinates, the vector fields defining the Heisenberg case (see example \\[ex:heis\\]).",
"\n\n#### Application: uniform Ball-Box theorem\n\nThe key feature of regular points is uniformity:\n\n- uniformity of the flag (\\[eq:flag\\]);\n\n- uniformity w.r.t.",
" $p$ of the convergence $(\\lambda (M,d), p)\n \\rightarrow C_pM$ (as explained by Bellaïche [@bel96 Sect.",
" 8], this uniformity is responsible for the group structure of the metric tangent space);\n\n- uniformity of distance estimates (see Remark \\[pa:bbox\\_nonuniform\\]).",
"\n\nIn particular the last property is essential to compute Hausdorff dimensions (see Section \\[se:hausdim\\]) or to prove the global convergence of approximate motion planning algorithms. ",
"Recall what we mean by uniformity in this context: in a neighbourhood of a regular point $p_0$, we can construct privileged coordinates depending continuously on the base point $p$ and such that the distance estimate (\\[eq:distestim\\]) holds with $C_p$ and $\\eps_p$ independent of $p$.\n\nAs already noticed, all these uniformity properties are lost at singular points. ",
"However, using the desingularization of a sub-Riemannian manifold, we are able to give a uniform version of distance estimates.",
"\n\nLet $\\Omega \\subset M$ be a compact set. ",
"We denote by $r_{\\max}$ the maximum of degrees of nonholonomy at points in $\\Omega$. As noticed in Section \\[se:privcoor\\], $r_{\\max}$ is finite. ",
"We assume that $M$ is an oriented manifold, and we choose a volume form $\\omega$ on $M$.\n\nLet $\\mathfrak{X}$ be the set of $n$-tuples $\\mathbf{X}=(X_{I_1}, \\dots, X_{I_n})$ of brackets of length $|I_i| \\leq r_{\\max}$. Since $r_{\\max}$ is finite, $\\mathfrak{X}$ is a finite subset of $\\mathit{Lie}(X_1,\\dots,X_m)^n$. Given $q \\in \\Omega$ and $\\eps\n>0$, we define a function $f_{q,\\eps}: \\mathfrak{X} \\to \\R$ by $$f_{q,\\eps}( \\mathbf{X}) = \\left| \\omega_q \\big( X_{I_1}(q) \\eps^{|I_1|},\n\\dots, X_{I_n}(q) \\eps^{|I_n|} \\big) \\right|.$$ We say that $\\mathbf{X} \\in \\mathfrak{X}$ is an [*adapted frame at $(q,\\eps)$*]{} if it achieves the maximum of $f_{q,\\eps}$ on $\\mathfrak{X}$.\n\nThe values at $q$ of an adapted frame at $(q,\\eps)$ clearly form a basis of $T_qM$. Moreover, $q$ being fixed, the adapted frames at $(q,\\eps)$ are adapted frames at $q$ for $\\eps$ small enough.",
"\n\n\\[th:unifbbox\\] There exist positive constants $K$ and $\\eps_0$ such that, for $q \\in \\Omega$, $\\eps < \\eps_0$, and any adapted frame $\\mathbf{X}$ at $(q,\\eps)$, there holds $$\\mathrm{Box}_\\mathbf{X} (q, \\frac{1}{K}\\eps) \\subset B(q,\\eps) \\subset\n\\mathrm{Box}_\\mathbf{X} (q, K \\eps),$$ where $\\mathrm{Box}_\\mathbf{X} (q, \\eps) = \\{ \\exp(x_1 X_{I_1}) \\circ\n\\cdots \\circ \\exp(x_n X_{I_n})(q) : \\, |x_i| \\leq \\eps^{|I_i|}, \\,\n1 \\leq i \\leq n \\}$.\n\nWe do not give the proof of this theorem in these notes, we refer the reader to [@jea01b].",
"\n\nOf course, when the point $q$ is fixed, the estimate above is equivalent to the one of the Ball-Box theorem for $\\eps$ smaller than some $\\eps_1(q)>0$. However, the main difference is that here $\\eps_0$ does not depend on $q$, whereas in the Ball-Box theorem $\\eps_1=\\eps_1(q)$ can be infinitely close to $0$ as $q$ varies.",
"\n\nAs a direct consequence of the Uniform Ball-Box theorem, we have an estimate of the volume of a small sub-Riemannian ball. ",
"Let $\\mathrm{vol}_\\omega$ be the measure on $M$ associated with $\\omega$, *i.e.* for any measurable subset $A \\subset M$ we set $\\mathrm{vol}_\\omega(A) = \\int_A \\omega$.\n\n\\[th:volballs\\] There exist positive constants $K$ and $\\eps_0$ such that, for all $q \\in \\Omega$ and $\\eps < \\eps_0$, $$\\frac{1}{K} \\max_{\\mathbf{X}} f_{q,\\eps}( \\mathbf{X}) \\leq \\mathrm{vol}_\\omega (B(q,\\eps))\n\\leq K \\max_{\\mathbf{X}} f_{q,\\eps}( \\mathbf{X}),$$ the maximum of $f_{q,\\eps}( \\mathbf{X}) = \\left| \\omega_q \\big( X_{I_1}(q) \\eps^{|I_1|},\n\\dots, X_{I_n}(q) \\eps^{|I_n|} \\big) \\right|$ being taken over all families $\\mathbf{X}=(X_{I_1}, \\dots, X_{I_n})$ of brackets of length $|I_i| \\leq r_{\\max}$.\n\nIf moreover all points in $\\Omega$ are regular, then for all $q \\in \\Omega$ and $\\eps < \\eps_0$, $$\\label{eq:volreg}\n \\frac{1}{K} \\eps^Q \\leq \\mathrm{vol}_\\omega (B(q,\\eps))\n\\leq K \\eps^Q,$$ where $Q = \\sum_{i=1}^n w_i(q)$ does not depend on $q$.\n\nHausdorff dimension {#se:hausdim}\n-------------------\n\nConsider a metric space $(M,d)$ and denote by $\\diam S$ the diameter of a set $S \\subset M$. Let $k \\geq 0$ be a real number. ",
"For every subset $A \\subset M$, we define the *$k$-dimensional Hausdorff measure* ${{\\mathcal H}}^k$ of $A$ as ${{\\mathcal H}}^k(A)\n= \\lim_{\\eps \\to 0^+} {{\\mathcal H}}^k_\\eps(A)$, where $${{\\mathcal H}}^k_\\eps(A) = \\inf \\left\\{ \\sum_{i=1}^\\infty \\left(\\diam S_i\\right)^k\n\\, : \\, A \\subset \\bigcup_{i=1}^\\infty S_i, \\ S_i \\hbox{ closed set}, \\ \\diam S_i \\leq \\eps \\right\\},$$ and the *$k$-dimensional spherical Hausdorff measure* $\\ss^k$ of $A$ as $\\ss^k(A)\n= \\lim_{\\eps \\to 0^+} \\ss^k_\\eps(A)$, where $$\\ss^k_\\eps(A) = \\inf \\left\\{ \\sum_{i=1}^\\infty \\left(\\diam\nS_i\\right)^k \\, : \\, A \\subset \\bigcup_{i=1}^\\infty S_i, \\ S_i \\hbox{ is\n a ball}, \\ \\diam\nS_i \\leq \\eps \\right\\}.$$\n\nIn the Euclidean space $\\R^n$, $k$-dimensional Hausdorff measures are often defined as $2^{-k}\\alpha(k){{\\mathcal H}}^k$ and $2^{-k}\\alpha(k)\\ss^k$, where $\\alpha(k)$ is defined from the usual gamma function as $\\alpha(k) = \\Gamma (\\frac{1}{2})^k / \\Gamma\n(\\frac{k}{2}+1)$. This normalization factor is necessary for the $n$-dimensional Hausdorff measure and the Lebesgue measure to coincide on $\\R^n$.\n\nFor a given set $A \\subset M$, ${{\\mathcal H}}^k(A)$ is a decreasing function of $k$, infinite when $k$ is smaller than a certain value, and zero when $k$ is greater than this value. ",
"We call *Hausdorff dimension* of $A$ the real number $$\\dim_{{\\mathcal H}}A = \\sup \\{ k \\, : \\, {{\\mathcal H}}^k (A) = \\infty \\} = \\inf \\{ k \\,\n: \\, {{\\mathcal H}}^k (A) = 0 \\}.$$ Note that ${{\\mathcal H}}^k \\leq \\ss^k \\leq 2^k {{\\mathcal H}}^k$, so the Hausdorff dimension can be defined equivalently from Hausdorff or spherical Hausdorff measures.",
"\n\nThere exist only few results on Hausdorff measures in sub-Riemannian geometry, except for specific cases [@Agrachev2010; @Ghezzi2012]. ",
"The most general result is the following one.",
"\n\n\\[th:dimhausreg\\] Let $(M,d)$ be an equiregular Carnot-Carathéodory space and $p$ a point in $M$. Then the Hausdorff dimension of a small enough ball $B(p,r)$ is $\\dim_{{\\mathcal H}}B(p,r) = Q$, where $$Q= \\sum_{i=1}^n w_i(p) = \\sum_{i\\geq 1} i \\left( \\dim \\Delta^i(p) - \\dim \\Delta^{i-1}(p) \\right)$$ does not depend on $p$. Moreover ${{\\mathcal H}}^{Q} (B(p,r))$ is finite.",
"\n\nFix a volume form $\\omega$ on $B(p,r)$ (it is possible for a small enough $r$), and denote by $\\mathrm{vol}_\\omega$ the associated measure. ",
"It results from Corollary \\[th:volballs\\] that, for $q \\in B(p,r)$ and $\\eps$ small enough, $$\\label{eq:volequireg}\n\\frac{1}{K} \\eps^Q \\leq \\mathrm{vol}_\\omega (B(q,\\eps))\n\\leq K \\eps^Q.$$\n\nDefine $N_\\eps$ to be the maximal number of disjoints balls of radius $\\eps$ included in $B(p,r)$, and consider such a family $B(q_i,\\eps)$, $i=,\\dots,N_\\eps$, of disjoints balls. ",
"By , $$\\frac{1}{K} \\eps^Q N_\\eps \\leq \\mathrm{vol}_\\omega (B(p,r)) \\quad \\Rightarrow \\quad N_\\eps \\leq K \\eps^{-Q} \\mathrm{vol}_\\omega (B(p,r)).$$ On the other hand the union $\\bigcup_i B(q_i,2\\eps)$ covers $B(p,r)$, and by Theorem \\[th:unifbbox\\] every ball $B(q_i,2\\eps)$ is of diameter $\\geq \\frac{4}{K} \\eps$ if $\\eps$ is small enough. ",
"This implies $$\\ss^Q (B(p,r)) \\leq \\liminf_{\\eps \\to 0} N_\\eps \\left(\\frac{4 \\eps}{K}\\right)^Q < \\infty.$$ Therefore $\\dim_{{\\mathcal H}}B(p,r) \\leq Q$.\n\nConversely, let $\\bigcup_i B(q_i,r_i)$ be a covering of $B(p,r)$ with balls of diameter not greater than $\\eps$. If $\\eps$ is small enough, every $r_i$ is smaller than $\\eps_0$ and there holds $$\\mathrm{vol}_\\omega (B(p,r)) \\leq \\sum_i \\mathrm{vol}_\\omega (B(q_i,r_i)) \\leq K\n\\sum_i r_i^Q.$$ As a consequence, we have $\\ss^Q (B(p,r)) \\geq \\mathrm{vol}_\\omega (B(p,r))\n/K$, which in turn implies $\\dim_{{\\mathcal H}}B(p,r) \\geq Q$. This ends the proof.",
"\n\nWhen $(M,d)$ is not equiregular, the Hausdorff dimension of balls centered at singular points behaves in a different way. ",
"Let us show it on an example.",
"\n\nConsider the Martinet space (see Example \\[ex:martinet\\]), that is, $\\R^3$ endowed with the sub-Riemannian distance associated with the vector fields $$X_1 = {\\partial_x} \\quad \\mbox{ and } \\quad\nX_2 = {\\partial_y} + \\frac{x^2}{2} {\\partial_z}.$$ A point $q=(x,y,z)$ is regular if $x\\neq 0$ and in this case $\\sum_i\nw_i(q)=4$, otherwise it is singular and $\\sum_i w_i(q)=5$.\n\n\\[le:dimhausmartinet\\] Let $p$ a point in the Martinet space.",
"\n\n- If $p$ is regular, then $\\dim_{{\\mathcal H}}B(p,r) = 4$, and ${{\\mathcal H}}^4\n (B(p,r))$ is finite.",
"\n\n- If $p$ is singular, then $\\dim_{{\\mathcal H}}B(p,r) = 4$, but ${{\\mathcal H}}^4\n (B(p,r))$ is not finite.",
"\n\nWhen $p$ is regular, the result is a direct consequence of Theorem \\[th:dimhausreg\\]. ",
"Let us consider a singular point $p$ and a radius $r>0$. Since regular points form an open set, $B(p,r)$ contains small balls centered at regular points, and thus $\\dim_{{\\mathcal H}}B(p,r) \\geq 4$. Moreover, it results from Corollary \\[th:volballs\\] that, for $q=(x,y,z)$ close enough from $p$ and for $\\eps >0$ small enough,\n\n$$\\label{eq:volmart}\n\\frac{1}{K} \\eps^4 \\max (|x|, \\eps) \\leq \\mathrm{vol}_\\omega (B(q,\\eps))\n\\leq K \\eps^4 \\max (|x|, \\eps).$$\n\nWe proceed as in the proof of Theorem \\[th:dimhausreg\\]. ",
"Define $N_\\eps$ to be the maximal number of disjoints balls of radius $\\eps$ included in $B(p,r)$, and consider such a family $B(q_i,\\eps)$, $i=,\\dots,N_\\eps$, of disjoints balls, with $q_i=(x_i,y_i,z_i)$. Notice that the first coordinate $x$ is of nonholonomic order $\\leq 1$ at any point; this implies that there exists a constant $K'>0$ such that $$B(q_i,\\eps) \\subset B(p,r) \\cap \\{ q=(x,y,z) \\ : \\ |x-x_i| \\leq K' \\eps \\}.$$ As a consequence, for an integer $k$, every ball $B(q_i,\\eps)$ such that $(k-1) \\eps \\leq |x_i| < k \\eps$ is included in the set $B(p,r)\n\\cap \\{ q=(x,y,z) \\ : \\ |x| \\in ((k-1-K') \\eps, (k+K')\\eps] \\}$. The volume of the latter set is smaller than $K'' \\eps$, where $K''$ is a constant (depending neither on $k$ nor $\\eps$). ",
"Then it results from that $$K N_\\eps(k) k \\eps^5 \\leq K'' \\eps,$$ where $N_\\eps(k)$ is the number of points $q_i$ such that $(k-1) \\eps\n\\leq |x_i| < k \\eps$. The Ball-Box Theorem implies that $N_\\eps(k)=0$ when $k>K'r/\\eps$, and hence $$N_\\eps = \\sum_{k=1}^{\\lceil K'r/\\eps\\rceil} N_\\eps(k) \\leq\n\\frac{\\mathrm{const}}{\\eps^4} \\sum_{k=1}^{\\lceil K'r/\\eps\\rceil}\n\\frac{1}{k} \\leq \\frac{\\mathrm{const}}{\\eps^4} \\log \\left(\\frac{1}{\\eps}\\right),$$ where $\\lceil t \\rceil$ denotes the integer part of a number $t$. Now the union $\\bigcup_i B(q_i,2\\eps)$ covers $B(p,r)$ and every ball $B(q_i,2\\eps)$ is of diameter $\\geq \\frac{4}{K} \\eps$ if $\\eps$ is small enough. ",
"This implies that, for any real number $s > 4$, $$\\ss^s (B(p,r)) \\leq \\lim_{\\eps \\to 0}\\left(\\frac{4 \\eps}{K}\\right)^s\nN_\\eps \\leq \\lim_{\\eps \\to 0} \\mathrm{const}\\, \\eps^{s-4} \\log\n\\left(\\frac{1}{\\eps}\\right) =0.$$ Consequently $\\dim_{{\\mathcal H}}B(p,r) \\leq 4$, and hence $\\dim_{{\\mathcal H}}B(p,r) =4$, since the converse inequality holds.",
"\n\nWe are left to show that ${{\\mathcal H}}^4 (B(p,r))$, or equivalently $\\ss^4\n(B(p,r))$, is not finite. ",
"Let $\\bigcup_i B(q_i,r_i)$ be a covering of $B(p,r)$ with balls of diameter not greater than $\\eps$. For an integer $k\\geq 1$, denote by $I_k$ the set of indices such that $\\bigcup_{i\\in I_k} B(q_i,r_i)$ is a covering of the set $B(p,r) \\cap\n\\{ q=(x,y,z) \\ : \\ |x| \\in ((k-1) \\eps, k\\eps] \\}$. Thus $$\\sum_{i\\in I_k} \\mathrm{vol}_\\omega (B(q_i,r_i)) \\geq \\mathrm{const} \\, \\eps.$$ On the other hand $i\\in I_k$ implies $\\mathrm{vol}_\\omega (B(q_i,r_i)) \\leq\n\\mathrm{const} \\ r_i^4 k \\eps$, and so $$\\sum_{i\\in I_k} r_i^4 \\geq \\frac{\\mathrm{const}}{k}.$$ Summing up over $k$, we obtain, for a small enough $\\eps$, $$\\sum_{i} r_i^4 \\geq \\mathrm{const}\\, \\log \\left(\\frac{1}{\\eps}\\right).$$ As a consequence, $\\ss^4_\\eps (B(p,r)) \\geq \\mathrm{const}\\, \\log\n(\\frac{1}{\\eps})$, and so $\\ss^4_\\eps (B(p,r))=\\infty$. This ends the proof.",
"\n\n[**Appendix**]{}\n\nFlows of vector fields {#se:flowsvf}\n======================\n\nThis section is dedicated to the proof of Campbell-Hausdorff type formulas for flows of vector fields. ",
"The result in Section \\[se:chforflows\\] has been used in Section \\[se:chow\\], the one in Section \\[se:pushforward\\] will be necessary for the next section.",
"\n\nLet $U$ be an open subset of $\\R^n$ and $VF(U)$ the set of smooth vector fields on $U$. Given a vector field $X \\in VF(U)$, we denote its flow by $\\exp(tX)$.\n\nCampbell-Hausdorff formula for flows {#se:chforflows}\n------------------------------------\n\nWe will need in this section the Campbell-Hausdorff formula which we recall briefly here (for a more detailed presentation see for instance [@bou72 Ch.",
" II]). ",
"Let $x$ and $y$ be two non commutative indeterminates, and $[x,y]=xy-yx$ their commutator, also denoted by $[x,y]=(\\ad x)y$. The length of an iterated commutator $(\\ad x_1) \\cdots (\\ad x_{k-1}) x_k$, where each $x_1,\\dots, x_k$ equals $x$ or $y$, is defined to be the number of occurrences $k$ of $x$ and $y$. Define also $e^x$ and $e^y$ to be the series $\\sum_{k \\geq 0}\n\\frac{x^{k}}{k!}$ and $\\sum_{k \\geq 0} \\frac{y^{k}}{k!}$. Then we have $e^x e^y = e^{H(x,y)}$ in the sense of formal power series, where $$\\begin{aligned}\n\\label{eq:chseries}\nH(x,y) = x+y+ \\frac{1}{2} [x,y] + R(x,y), $$ and $R(x,y)$ is a series whose terms are linear combination of iterated commutators of $x$ and $y$ of length greater than 2. ",
"For an integer $N$ we denote by $H_N(x,y)$ the partial sum of $H(x,y)$ containing only iterated commutators of length not greater than $N$. In particular, $H_1=x+y$ and $H_2= x+y+ \\frac{1}{2} [x,y]$.\n\nConsider now two vector fields $X,Y \\in VF(U)$. Given $t \\in \\R$ and an integer $N$, $H_N(tY,tX)$ is a smooth vector field on $U$ which writes as $\\sum_{i=1}^N t^i Y_i$, where the vector fields $Y_1,\\dots, Y_N$ belong to the Lie algebra generated by $X$ and $Y$.\n\n\\[le:taylor\\_CH\\] Let $p\\in M$. There exist positive constants $\\delta$ and $C$ such that $|t| < \\delta$ implies $$\\left\\| \\exp(tX) \\circ \\exp (tY) (p) - \\exp(H_N(tY,tX))(p)\\right\\| \\leq C |t|^{N+1}.$$\n\nSet $\\psi(t)=\\exp(tX) \\circ \\exp (tY) (p)$, which is a function defined and $C^\\infty$ in a neighbourhood of $0 \\in \\R$, and let $(x_1,\\dots,x_n)$ be a system of local coordinates on a neighbourhood of $p$ in $U$. We will compute the Taylor expansion of every component $x_i(\\psi(t))$, for $i=1, \\dots,n$. To do this, we introduce the function $\\phi(t,s)= \\exp(tX) \\circ \\exp (sY) (p)$, so that $\\psi(t) = \\phi(t,t)$, and we compute the partial derivatives of $x_i \\circ \\phi$ at $0 \\in \\R^2$. We have: $$\\frac{\\partial x_i \\circ \\phi}{\\partial t} (t,s) = \\frac{d}{dt}\\left[ x_i \\circ \\exp(tX) \\right] \\left(\\exp(sY)(p)\\right) = Xx_i(\\phi(t,s)),$$ where $Xx_i$ denotes the Lie derivative of $x_i$ along $X$. Repeating this computation, we obtain for any integer $k$, $$\\frac{\\partial^k x_i \\circ \\phi}{\\partial t^k} (t,s) = X^kx_i(\\phi(t,s)).$$ In the same way, we have: $$\\begin{gathered}\n\\frac{\\partial^{k+l} x_i \\circ \\phi}{\\partial s^l \\partial t^k} (0,0) = \\frac{\\partial^{l}}{\\partial s^l} \\frac{\\partial^{k} x_i \\circ \\phi}{ \\partial t^k} (0,s)\\big|_{s=0} \\\\\n= \\frac{\\partial^{l}}{\\partial s^l} \\left[ X^kx_i(\\exp (sY) (p))\\right]\\big|_{s=0} = Y^l X^kx_i(p).\\end{gathered}$$ We then deduce that the formal Taylor series of $x_i(\\psi(t)) = x_i(\\phi(t,t))$ at $0$ is $$\\sum_{k,l\\geq 0} \\frac{t^{k+l}}{k! ",
"l!} ",
"Y^l X^kx_i(p) = \\left[ \\sum_{l\\geq 0} \\frac{t^l}{l!} ",
"Y^l \\right] \\left[ \\sum_{k\\geq 0} \\frac{t^k}{k!} ",
"X^k \\right] x_i(p),$$ where $X$ and $Y$ are considered as derivation operators. ",
"From the Campbell-Hausdorff formula, the product of the formal series $e^{tY} = \\sum_{l\\geq 0} \\frac{t^l}{l!} ",
"Y^l$ with $e^{tX} = \\sum_{k\\geq 0} \\frac{t^k}{k!} ",
"X^k $ is equal to the series $e^{H(tY,tX)}$. As a consequence, the Taylor expansion of $x_i(\\psi(t))$ up to degree $N$ is given by the terms of degree $\\leq N$ in the series $e^{H(tY,tX)}x_i (p)$, which coincide with the terms of degree $\\leq N$ in the series $e^{H_N(tY,tX)}x_i (p)$.\n\nOn the other hand, it results from Lemma \\[le:dlexpY(t)\\] below that $e^{H_N(tY,tX)}x_i (p)$ is the Taylor series at $0$ of the function $t \\mapsto x_i \\circ \\exp(H_N(tY,tX))(p)$. Thus $$x_i \\circ \\psi(t) - x_i \\circ \\exp(H_N(tY,tX))(p)= O(|t|^{N+1})$$ for every coordinate $x_i$, and the lemma follows.",
"\n\n\\[le:dlexpY(t)\\] Let $Y_1,\\dots, Y_{\\ell}$ be vector fields on $U$, $f:U \\to \\R$ a smooth function, and $p \\in U$. The formal Taylor series at $0 \\in \\R^\\ell$ of the function $(z_1,\\dots,z_\\ell) \\mapsto f(\\exp(\\sum_i z_i Y_i)(p))$ is given by $$\\sum_{k\\geq 0} \\frac{1}{k!} (",
"\\sum_i z_i Y_i)^k f (p) = e^{\\sum_i z_i Y_i}f(p).$$ The formal Taylor series at $0 \\in \\R$ of the function $t \\mapsto f(\\exp(Y(t))(p))$, where $Y(t)=\\sum_{i=1}^\\ell t^i Y_i$, is given by $$\\label{eq:dlexpY(t)}\n\\sum_{k\\geq 0} \\frac{1}{k!} ",
"Y(t)^k f (p)= e^{Y(t)}f(p).$$\n\nThe second statement is obviously a consequence of the first one, so it is sufficient to prove the latter. ",
"We introduce the functions $g(z)=\nf(\\exp(\\sum_i z_i Y_i)(p))$ and $G(t,z)=g(tz)$ which are well-defined and smooth on a neighbourhood of $0$ in $\\R^\\ell$, respectively $\\R \\times\n\\R^\\ell$. We are looking for the Taylor series of $g$ at $0$.\n\nSince $G(t,z)= f(\\exp(t \\sum_i z_i Y_i)(p))$, we have, for any integer $k \\geq 0$, $$\\frac{\\partial^k G}{\\partial t^k}(0,z) = (\\sum_i z_i Y_i)^k f (p).$$ On the other hand $G(t,z)=g(tz)$, and hence the previous derivative can also be computed as $$\\frac{\\partial^k G}{\\partial t^k}(0,z) = \\sum_{\\alpha_1 + \\cdots + \\alpha_\\ell=k} \\frac{k!}{\\alpha_1! ",
"\\cdots \\alpha_\\ell !} ",
"z_1^{\\alpha_1} \\cdots z_\\ell^{\\alpha_\\ell} \\frac{\\partial^k g}{\\partial z_1^{\\alpha_1} \\cdots \\partial z_\\ell^{\\alpha_\\ell}}(0) .$$ Combining both expressions, we obtain $$\\sum_{\\alpha_1 + \\cdots + \\alpha_\\ell=k} \\frac{z_1^{\\alpha_1} \\cdots z_\\ell^{\\alpha_\\ell}}{\\alpha_1! ",
"\\cdots \\alpha_\\ell !} ",
" \\frac{\\partial^k g}{\\partial z_1^{\\alpha_1} \\cdots \\partial z_\\ell^{\\alpha_\\ell}}(0) =\\frac{1}{k!} ",
"\\left(\\sum_i z_i Y_i\\right)^k f (p),$$ and the lemma follows.",
"\n\nLemma \\[le:taylor\\_CH\\] can be extended in two ways. ",
"First, since the vector fields and their flows are smooth on $U$, the estimate holds uniformly with respect to $p$. Second, by Lemma \\[le:dlexpY(t)\\], the vector fields $tX$ and $tY$ may be replaced by the one-parameter families of vector fields $X(t)= tX_1 + \\cdots + t^{k} X_k$ and $Y(t)=t Y_1 + \\cdots + t^\\ell Y_\\ell$, where $X_1,\\dots,X_{k}$ and $Y_1,\\dots, Y_{\\ell}$ are vector fields on $U$. As an example, $H_N(tY,tX)$ is of this form. ",
"To summarize, a slight change in the proof of Lemma \\[le:taylor\\_CH\\] actually shows the following result.",
"\n\n\\[le:corochform\\] Let $K \\subset U$ be a compact. ",
"There exist two positive constants $\\delta,C$ such that, if $p \\in K$ and $|t| < \\delta$, then: $$\\left\\| \\exp(X(t))\\circ \\exp(Y(t))(p) - \\exp(H_N(Y(t),Y(t)))(p)\\right\\| \\leq C |t|^{N+1}.$$\n\nWe are now in a position to prove formula , that we used in the proof of Lemma \\[le:Apopen\\]. ",
"Let $X_1,\\dots,X_m$ be $m$ elements of $VF(U)$. For every multi-index $I \\in \\{1,\\dots,m\\}^k$, $k\\in \\N$, we define the local diffeomorphisms $\\phi^I_t$ on $U$ by induction on the length $|I|$ of $I$. Let $\\phi^i_t = \\exp(tX_i)$ and set, if $I=iJ$, $$\\phi_t^{iJ}=\\phi_{-t}^J \\circ \\phi_{-t}^i\n\\circ \\phi_t^J \\circ\\phi_t^i.$$\n\n\\[le:quasich\\] Let $K \\subset U$ be a compact and $I$ a multi-index. ",
"There exist two positive constants $\\delta,C$ such that, if $p \\in K$ and $|t| < \\delta$, then $$\\begin{aligned}\n\\label{eq:phIt}\n \\left\\| \\phi^I_t (p) - p - t^{|I|} X_{I} (p) \\right\\| & \\leq & C |t|^{|I|+1}.\\end{aligned}$$\n\nFor $\\delta >0$ small enough, the mapping $(p,t) \\mapsto \\phi^I_t(p)$ is defined and $C^\\infty$ on $K \\times (-\\delta,\\delta)$. As a consequence, we are reduced to prove for a fixed $p\\in K$. When $|I|=1$, $\\phi^I_t (p)=\\exp(tX_i)(p)$ for some $i\\in \\{1,\\dots,m\\}$, which is equal to $p + t X_i(p) +O(|t|^2)$.\n\nNow, let $I$ be a multi-index and $N > |I|$ an integer. ",
"Corollary \\[le:corochform\\] implies that $\\phi_t^{I}(p)= \\exp(H_N^{I}(t))(p) + O(t^{N+1})$ where the series $H_N^I(t)$ is defined by induction: if $I=iJ$, then $$H_N^{iJ}(t)=H_N( H_N(tX_i,H_N^J(t)),H_N(-tX_i,-H_N^J(t))).$$ Applying iteratively, we can write $H_N^I(t) = t^{|I|} X_I + t^{|I|+1} R_I(t)$, the latter term being a one-parameter vector field. ",
"As a consequence, $$\\phi_t^{I}(p)= p + t^{|I|} X_I (p)+ \\hbox{terms of degree greater than } {|I|+1},$$ which completes the proof.",
"\n\nPush-forward formula {#se:pushforward}\n--------------------\n\nGiven two vector fields $X, Y \\in VF(U)$, we write $(\\ad X)Y$ for $[X,Y]$, $(\\ad X)^2Y$ for $(\\ad X)((\\ad X)Y)$, etc.",
"\n\n\\[th:quasich\\] Let $K \\subset U$ be a compact, $N$ a positive integer, and $X$, $Y$, $Y_1,\\dots,Y_\\ell$ vector fields in $VF(U)$. There exist two positive constants $\\delta,C$ such that, if $p \\in K$, $t \\in \\R$ and $z\\in \\R^\\ell$ satisfy $|t| < \\delta$ and $\\|z\\| < \\delta$, then $$\\begin{aligned}\n & \\left\\| \\exp(tY)_*X (p) - \\sum_{k = 0}^N {\\genfrac{}{}{}{0}{t^{k}}{k!}} (",
"\\ad Y)^{k}X (p) \\right\\| \\leq C |t|^{N+1}, & \\nonumber \\\\[2mm]\n & \\left\\| \\exp(\\sum_{i=1}^\\ell z_i Y_i)_*X (p) - \\sum_{k = 0}^N {\\genfrac{}{}{}{0}{1}{k!}} ",
"\\left(\\ad \\sum_{i=1}^\\ell z_i Y_i\\right)^{k}X (p) \\right\\| \\leq C \\|z\\|^{N+1}, &\\nonumber\\end{aligned}$$ where $\\exp(tY)_*X= d (\\exp(tY)) \\circ X \\circ \\exp(-tY)$ denotes the push-forward of the vector field $X$ by the diffeomorphism $\\exp(tY)$.\n\nLet us begin with the first inequality. ",
"Set $\\phi_p(t) = \\exp(tY)_*X (p)$. For $\\delta >0$ small enough, the mapping $(p,t) \\mapsto \\phi_p(t)$ is defined and $C^\\infty$ on $K \\times (-\\delta,\\delta)$. As a consequence, there exists a constant $C>0$ such that, for every $p\\in K$ and $|t| < \\delta$, we have $$\\begin{aligned}\n \\left\\| \\phi_p(t) - \\sum_{k = 0}^N {\\genfrac{}{}{}{0}{t^{k}}{k!}} ",
"\\frac{d^k \\phi_p}{d t^k}(0) \\right\\| & \\leq & C |t|^{N+1}. ",
"\\nonumber\\end{aligned}$$ It remains to prove that $\\frac{d^k \\phi_p}{d t^k}(0) = (\\ad Y)^{k}X (p)$ for any integer $k$. Note first that $\\phi_p(0)=X(p)$, and that $$\\frac{d \\phi_p}{d t}(0) = \\frac{d }{d t} \\left[ \\exp(tY)_*X \\right]\\big|_{t=0}(p)$$ is by definition equal to $-L_YX(p)$, where $L_YX$ is the Lie derivative of $X$ along $Y$ (see for instance [@boo86]). ",
"Since $L_YX(p)=(\\ad Y) X(p)$, the cases $k=0$ and $k=1$ are done.",
"\n\nWe need now to compute $\\frac{d \\phi_p}{d t}(t)$ at $t\\neq 0$. Let us write $\\phi_p(t+s)$ as $\\exp(tY)_*\\exp(sY)_*X (p)$. We have $$\\begin{gathered}\n\\frac{d \\phi_p}{d t}(t) = \\frac{d \\phi_p(t+s)}{d s}\\mid_{s=0}= \\exp(tY)_*\\frac{d }{d s}\\left[ \\exp(sY)_*X \\right]\\big|_{s=0}(p) \\\\\n=\\exp(tY)_* ((\\ad Y) X)(p).\\end{gathered}$$ This derivative has the same form as $\\phi_p(t)$, $X$ being replaced by $(\\ad Y)X$. Iterating the argument above, we obtain by induction $\\frac{d^k \\phi_p}{d t^k}(0) = (\\ad Y)^{k}X (p)$, and the first inequality of the proposition is proved.",
"\n\nAs for the second inequality, the same reasoning applies and we only need to compute the partial derivatives at $0 \\in \\R^\\ell$ of the function $\\widetilde{\\phi}(z)=\\exp(\\sum_{i} z_i Y_i)_*X (p)$. This can be done as in the proof of Lemma \\[le:dlexpY(t)\\]. ",
"The proposition follows.",
"\n\nDifferent systems of privileged coordinates {#se:proofpriv}\n===========================================\n\nThis appendix is devoted to the proof that the examples of coordinates introduced in Section \\[se:privcoor\\] are actually privileged coordinates.",
"\n\nCanonical coordinates of the second kind {#se:2ndkind}\n----------------------------------------\n\nLet $p\\in M$ and $Y_1,\\dots,Y_n$ an adapted frame at $p$ (see , page ). ",
"The map $$\\phi : (z_1,\\dots,z_n) \\mapsto \\exp(z_n Y_n)\\circ \\cdots \\circ \\exp(z_1\nY_1)(p)$$ is a local diffeomorphism near $0 \\in \\R^n$ and its inverse defines some coordinates called *canonical coordinates of the second kind near $p$*.",
"\n\nThe following result is due to Hermes [@her91].",
"\n\n\\[ccan\\] Canonical coordinates of the second kind are privileged at $p$.\n\nFor sake of simplicity, we will write the compositions of maps as products; for instance, we write $$\\phi(z)= \\exp(z_1 Y_1)\n \\cdots \\exp(z_n Y_n)(p).$$\n\nFirst, let us recall that $\\phi$ is a local diffeomorphism at $z=0$ because its differential at $0$ is an isomorphism. ",
"This results from $$\\frac{\\partial \\phi}{\\partial\nz_i}(0)=\n\\frac{d}{dt}\\left(\\phi(0,\n\\dots,t,\\dots,0)\\right)\\big|_{t=0}=\\frac{d}{dt}\\left(\n\\exp(tY_i)(p)\\right)\\big|_{t=0}=Y_i(p),$$ for $i=1,\\dots,n$. This computation also reads as $\\phi_*\n\\frac{\\partial}{\\partial z_i} (p) = Y_i(p)$, which implies $Y_i\nz_i(p)=1$ (as in Section \\[se:nhorders\\], $Y_iz_i$ denotes the Lie derivative of the function $z_i$ along the vector field $Y_i$). ",
"Hence the order of $z_i$ at $p$ is not greater than $w_i$.\n\nIt remains to show that the order of $z_i$ at $p$ is at least $w_i$ for each $i=1,\\dots, n$. This is a direct consequence of the following assertion.",
"\n\n**Claim.** *",
"Let $X$ be one of the vector fields $X_1, \\dots,\n X_m$. Then, for $i=1,\\dots, n$, the Taylor expansion at $z=0$ of the function $a_i(z) = Xz_i \\left(\\phi(z)\\right)$ is a sum of homogeneous polynomials in the coordinates $z$ of weighted degree $\\geq w_i-1$.*\n\nFrom the very definition of $a_i(z)$, we have $$\\begin{aligned}\n\\label{eq:ai(z)}\n X(\\phi(z)) & = & \\sum_{i=1}^n a_i(z) \\frac{\\partial \\phi}{\\partial\n z_i} (z).\\end{aligned}$$ Given $z$, let $\\varphi$ be the diffeomorphism defined on a neighbourhood of $p$ by $\\varphi(q) = \\exp(z_1 Y_1)\n \\cdots \\exp(z_n Y_n)(q)$. In particular, $\\varphi(p)=\\phi(z)$. In order to obtain an equality in $T_pM$, we apply the isomorphism $(d \\varphi_p)^{-1}$ to both sides of , and we get $$\\begin{aligned}\n(\\varphi^{-1})_* X (p) & = & \\sum_{i=1}^n a_i(z) \\, (\\varphi^{-1})_*\\frac{\\partial }{\\partial\n z_i} (p).\\end{aligned}$$ This equality is of the form $W= \\sum_{i=1}^n a_i V_i$, where the vectors $W=W(z)$ and $V_i=V_i(z)$, $i=1, \\dots,n$, belong to $T_pM$. If we denote by $b=b(z)\\in\\R^n$ the coordinates of $W$ in the basis $(Y_1(p), \\dots, Y_n(p))$ of $T_pM$, and by $P=P(z)$ the $(n\\times n)$-matrix of the coordinates of $V_1,\\dots,V_n$ in the same basis, then the vector $a(z)=(a_1(z), \\dots,a_n(z))$ appears as the solution of $Pa=b$.\n\nNote first that $P(0)$ equals the identity matrix $\\mathbb{I}$. Therefore both matrices $P(z)$ and $P(z)^{-1}$ are equal to $\\mathbb{I} + $homogeneous terms of positive degree. ",
"Hence the Taylor expansion of $a_i(z)$ and the one of $b_i(z)$ have the same homogeneous terms of lower degree.",
"\n\nOn the other hand, since $\\varphi^{-1} = \\exp(-z_n Y_n) \\cdots \\exp(-z_1 Y_1)$, we have $$\\begin{aligned}\nW(z) & = & \\exp(-z_n\nY_n)_* \\cdots \\exp(- z_1 Y_1)_* X (p).\\end{aligned}$$ Let us choose an integer $N$ bigger than all the weights $w_i$, and apply Proposition \\[th:quasich\\] to $\\exp(- z_1 Y_1)_* X$, then to $\\exp(- z_2 Y_2)_* (\\ad Y_1)^{l_1}X$, and so on, $$\\begin{aligned}\nW(z) & = & \\exp(-z_n\nY_n)_* \\cdots \\exp(- z_2 Y_2)_*\\sum_{l_1= 0}^N \\frac{(-z_1)^{l_1}}{l_1!} (",
"\\ad Y_1)^{l_1}X (p) \\\\\n& & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + O(|z_1|^{N+1})\\\\\n& \\vdots & \\\\\n& = & \\sum_{l_1,\\dots,l_{n} = 0}^N \\frac{(-z_1)^{l_1}}{l_1!} ",
"\\cdots\n\\frac{(-z_n)^{l_n}}{l_n!} \\:",
"\n(\\ad Y_n)^{l_n}\\cdots (\\ad Y_1)^{l_1}X (p) + O(|z|^{N+1}).\\end{aligned}$$ Hence every coordinate $b_i(z)$ of $W(z)$ satisfies $$\\begin{aligned}\n\\label{eq:bi}\nb_i(z) & = & \\sum_{l_1,\\dots,l_{n} = 0}^N\n \\frac{(-z_1)^{l_1}}{l_1!} ",
"\\cdots\n\\frac{(-z_n)^{l_n}}{l_n!} ",
"\\beta_i^l + O(|z|^{N+1}),\\end{aligned}$$ $\\beta_i^l$ being the $i$th coordinate in the basis $(Y_1(p), \\dots, Y_n(p))$ of the vector $(\\ad Y_n)^{l_n}\\cdots (\\ad Y_1)^{l_1}X (p)$. The latter vector belongs to $\\Delta^w(p)$, where $w=1+l_1w_1 +\\cdots+l_nw_n$ (recall that $X \\in \\Delta^1$ and $Y_i \\in \\Delta^{w_i}$). ",
"Since $(Y_1,\n\\dots, Y_n)$ is an adapted frame at $p$, $\\beta_i^l$ is zero when $1+l_1w_1 +\\cdots+l_nw_n < w_i$. It follows that $b_i(z)$ – and then $a_i(z)$ – contains only homogeneous terms of weighted degree greater than or equal to $w_i -1$. This ends the proofs of both the claim and the lemma.",
"\n\nCanonical coordinates of the first kind {#se:1stkind}\n---------------------------------------\n\nLet $p\\in M$ and $Y_1,\\dots,Y_n$ an adapted frame at $p$. The map $$\\widetilde{\\phi} : (z_1,\\dots,z_n) \\mapsto \\exp(z_1 Y_1 + \\cdots + z_n\nY_n)(p)$$ is a local diffeomorphism near $0 \\in \\R^n$ since its differential at $0$ is an isomorphism. ",
"This results from $$\\frac{\\partial \\widetilde{\\phi}}{\\partial\nz_i}(0)=\n\\frac{d}{dt}\\left(\\widetilde{\\phi}(0,\n\\dots,t,\\dots,0)\\right)\\big|_{t=0}=\\frac{d}{dt}\\left(\n\\exp(tY_i)(p)\\right)\\big|_{t=0}=Y_i(p),$$ for $i=1,\\dots,n$. The inverse of $\\widetilde{\\phi}$ defines some local coordinates near $p$ called *canonical coordinates of the first kind*.",
"\n\n\\[le:ccan1\\] Canonical coordinates of the first kind are privileged at $p$.\n\nThe first proof of this lemma appeared in [@rot76], with a different formulation. ",
"The proof we present here is rather different.",
"\n\nThe proof follows exactly the same lines as the one of Lemma \\[ccan\\], replacing $\\phi$ by $\\widetilde{\\phi}$, and $\\varphi$ by $\\widetilde{\\varphi} = \\exp(\\sum_j z_j Y_j)$. We are left to compute the coordinates $\\widetilde{b_i}(z)$, $i=1, \\dots,n$, of the vector $\\widetilde{W}(z)=(\\widetilde{\\varphi}^{-1})_* X (p)$ in the basis $(Y_1(p), \\dots, Y_n(p))$ of $T_pM$. It results directly from Proposition \\[th:quasich\\] that $$\\begin{aligned}\n\\widetilde{W}(z) & = & \\sum_{k = 0}^N {\\genfrac{}{}{}{0}{1}{k!}} ",
"\\left(\\ad \\sum_{i=1}^\\ell z_j Y_j \\right)^{k}X (p) + O(|z|^{N+1}),\\\\\n& = & \\sum_{k=0}^N \\sum_{l_1+\\dots +l_{n} = k} a_l z_1^{l_1} \\cdots\nz_n^{l_n} Z_l(p) + O(|z|^{N+1}),\\end{aligned}$$ where $Z_l$ belongs to $\\Delta^w(p)$, with $w=1+l_1w_1 +\\cdots+l_nw_n$. Thus every coordinate $\\widetilde{b_i}(z)$ of $\\widetilde{W}(z)$ satisfies $$\\begin{aligned}\n\\widetilde{b_i}(z) & = & \\sum_{k=0}^N \\sum_{l_1+\\dots +l_{n} = k} a_l z_1^{l_1} \\cdots\nz_n^{l_n} \\widetilde{\\beta}_i^l + O(|z|^{N+1}),\\end{aligned}$$ $\\widetilde{\\beta}_i^l$ being the $i$th coordinate of $Z_l$ in the basis $(Y_1(p), \\dots, Y_n(p))$. This expression is similar to , and the same conclusion follows.",
"\n\nAlgebraic coordinates {#se:coordalg}\n---------------------\n\nLet us recall the construction of the algebraic coordinates $(z_1, \\dots, z_n)$ given in page . ",
"Let $Y_1, \\dots, Y_n$ be an adapted frame at $p$, and $(y_1,\\dots,y_n)$ be local coordinates centered at $p$ such that $\\partial_{y_i}|_p = Y_i(p)$. For $j=1, \\dots,n$, we set $$\\label{eq:coordalg}\nz_j = y_j - \\sum_{k=2}^{w_j-1} h_k(y_1, \\dots, y_{j-1}),$$ where, for $k=2, \\dots, w_j-1$, $$h_k(y_1, \\dots, y_{j-1})=\n\\sum_{\\stackrel{\\scriptstyle{|\\alpha|=k}}{w(\\alpha)<w_j}} \\!\\!\\!\\!",
"\nY_1^{\\alpha_1} \\ldots Y_{j-1}^{\\alpha_{j-1}} \\Big(\ny_j -\\sum_{q=2}^{k-1} h_q(y)\\Big)(p) \\\n\\frac{y_1^{\\alpha_1}}{\\alpha_1!} ",
"\\cdots\n\\frac{y_{j-1}^{\\alpha_{j-1}}}{\\alpha_{j-1}!},$$ with $|\\alpha|=\\alpha_1 + \\cdots + \\alpha_n$.\n\n\\[le:coordalg\\] The algebraic coordinates $(z_1, \\dots, z_n)$ are privileged at $p$.\n\nThe proof of the lemma is based on the following result.",
"\n\n\\[le:cnsorder\\] A function $f$ is of order $\\geq s$ at $p$ if and only if $$(Y_1^{\\alpha_1} \\cdots Y_n^{\\alpha_n}f)(p)=0$$ for all $\\alpha$ such that $w(\\alpha) <s$.\n\nLet $f$ be a function of order $\\geq s$ at $p$. Using the rules , we have $\\mathrm{ord}_p(Y_i)\\geq -w_i$ for $i=1,\\dots,n$, and hence $\\mathrm{ord}_p(Y_1^{\\alpha_1} \\cdots Y_n^{\\alpha_n}) > -s$ for every $\\alpha=(\\alpha_1,\\dots,\\alpha_n)$ such that $w(\\alpha) <s$. Consequently, for such an $\\alpha$ the function $Y_1^{\\alpha_1} \\cdots Y_n^{\\alpha_n}f$ is of positive order, and so vanishes at $p$.\n\nConversely, let $f$ be a function of order $< s$ at $p$. We introduce the canonical coordinates of the second kind $(x_1,\\dots,x_n)$ defined by means of the adapted basis $Y_1,\\dots,Y_n$. Proposition \\[le:algorder\\] implies that there exists $\\alpha$ such that $w(\\alpha)=\\mathrm{ord}_p (f) <s$ and $(\\partial_{x_1}^{\\alpha_1} \\cdots \\partial_{x_n}^{\\alpha_n}f)(p)\\neq 0$. Moreover, every vector field $Y_i$, $i=1,\\dots,n$, writes in coordinates $x$ as $$\\sum_{j=1}^n Y_i^j (x) \\partial_{x_j}, \\quad \\hbox{where } \\mathrm{ord}_p (Y_i^j) \\geq w_j-w_i.$$ There also holds $Y_i^j (0) = \\delta_{ij}$ since $Y_i(p) = \\partial_{x_i}$. As a consequence, $$Y_1^{\\alpha_1} \\cdots Y_n^{\\alpha_n} (p) = \\partial_{x_1}^{\\alpha_1} \\cdots \\partial_{x_n}^{\\alpha_n}(p) + \\sum_{w(\\beta)<w(\\alpha)} a_\\beta \\partial_{x_1}^{\\beta_1} \\cdots \\partial_{x_n}^{\\beta_n}(p),$$ and thus $(Y_1^{\\alpha_1} \\cdots Y_n^{\\alpha_n}f)(p) = (\\partial_{x_1}^{\\alpha_1} \\cdots \\partial_{x_n}^{\\alpha_n}f)(p)\\neq 0$ since $w(\\alpha)=\\mathrm{ord}_p (f)$. This ends the proof.",
"\n\nLet $i \\in \\{1,\\dots,n\\}$. Note first that $Y_iz_i(p)=Y_iy_i(p)=1$, which implies $\\mathrm{ord}_p (z_i) \\leq w_i$. It remains to show that $\\mathrm{ord}_p (z_i) \\geq w_i$. For this we will use the criterion of Lemma \\[le:cnsorder\\].",
"\n\nLet $\\alpha$ such that $w(\\alpha) <w_i$ (and so $|\\alpha|<w_i$). ",
"Using the expression of $z_i$, we obtain $$\\begin{gathered}\n\\label{eq:yalpha}\n Y_1^{\\alpha_1} \\cdots Y_n^{\\alpha_n} z_i = Y_1^{\\alpha_1} \\cdots\n Y_n^{\\alpha_n} \\left(y_i - \\sum_{k=2}^{w_i-1} h_k(y)\\right) \\\\\n = Y_1^{\\alpha_1} \\cdots Y_n^{\\alpha_n} \\left(y_i -\n \\sum_{k=2}^{|\\alpha|-1} h_k(y)\\right) - Y_1^{\\alpha_1} \\cdots\n Y_n^{\\alpha_n} \\left(\\sum_{k=|\\alpha|}^{w_i-1} h_k(y)\\right).\\end{gathered}$$ The functions $h_k$ are given by $$h_k(y)=\n\\sum_{\\stackrel{\\scriptstyle{|\\beta|=k}}{w(\\beta)<w_i}} \\!\\!\\!\\!",
"\nY_1^{\\beta_1} \\ldots Y_{i-1}^{\\beta_{i-1}} \\Big(\ny_i- \\sum_{q=2}^{k-1} h_q(y)\\Big)(p) \\\n\\frac{y_1^{\\beta_1}}{\\beta_1!} ",
"\\cdots\n\\frac{y_{i-1}^{\\beta_{i-1}}}{\\beta_{i-1}!}.$$ Therefore, we clearly have $\\left(Y_1^{\\alpha_1} \\cdots Y_n^{\\alpha_n}\n h_k\\right)(p)=0$ if $k>|\\alpha|$, and $$\\left(Y_1^{\\alpha_1} \\cdots Y_n^{\\alpha_n} h_{|\\alpha|} \\right)(p)=\nY_1^{\\alpha_1} \\cdots Y_n^{\\alpha_n} \\left(y_i -\n \\sum_{k=2}^{|\\alpha|-1} h_k(y)\\right) (p).$$ Plugging this expression into , we obtain $(Y_1^{\\alpha_1} \\cdots Y_n^{\\alpha_n} z_i)(p)=0$, which ends the proof.",
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"\n\n[^1]: ENSTA ParisTech, UMA, 828 Boulevard des Maréchaux 91762 Palaiseau, France and Team GECO, INRIA Saclay – Île-de-France, [frederic.jean@ensta-paristech.fr]{}\n\n[^2]: This work was supported by the ANR project [*GCM*]{}, program “Blanche”, project number NT09\\_504490, and by the Commission of the European Communities under the 7th Framework Programme Marie Curie Initial Training Network (FP7-PEOPLE-2010-ITN), project SADCO, contract number 264735.",
"\n\n[^3]: This result appeared first in [@mit85], but with an erroneous proof. ",
"The presentation given here is inspired from the one of [@bel96].",
"\n"
] | {
"pile_set_name": "ArXiv"
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"Alternate life history phases of a common seaweed have distinct microbial surface communities.",
"\nMacroalgal life histories are complex, often involving the alternation of distinct free-living life history phases that differ in morphology, longevity and ploidy. ",
"The surfaces of marine macroalgae support diverse microbial biofilms, yet the degree of microbial variation between alternate phases is unknown. ",
"We quantified bacterial (16S rRNA gene) and microeukaryote (18S rRNA gene) communities on the surface of the common intertidal seaweed, Mastocarpus spp., ",
"which alternates between gametophyte (foliose, haploid) and sporophyte (encrusting, diploid) life history phases. ",
"A large portion (97%) of bacterial taxa on the surface Mastocarpus was also present in samples from the environment, indicating that macroalgal surface communities are largely assembled from the surrounding seawater. ",
"Still, changes in the relative abundance of bacterial taxa result in significantly different communities on alternate Mastocarpus life history phases, rocky substrate and seawater at all intertidal elevations. ",
"For microeukaryote assemblages, only high intertidal samples had significant differences between life history phases although sporophytes were not different from the rocky substrate at this elevation; gametophytes and sporophytes did not differ in microeukaryote communities in the mid and low zones. ",
"By sequencing three host genes, we identified three cryptic species of Mastocarpus in our data set, which co-occur in the mid-to-low intertidal zone. ",
"In these samples, M. alaskensis sporophytes harboured distinct bacterial communities compared to M. agardhii and M. intermedius sporophytes, which were not distinguishable. ",
"Conversely, microeukaryote communities did not differ among species."
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"Pike Township, Stark County, Ohio\n\nPike Township is one of the seventeen townships of Stark County, Ohio, United States. ",
" The 2000 census found 4,088 people in the township, 3,282 of whom lived in the unincorporated portions of the township.",
"\n\nGeography\nLocated in the southern part of the county, it borders the following townships:\nCanton Township - north\nOsnaburg Township - northeast corner\nSandy Township - east\nRose Township, Carroll County - southeast corner\nSandy Township, Tuscarawas County - south\nLawrence Township, Tuscarawas County - southwest\nBethlehem Township - west\nPerry Township - northwest corner\n\nThe village of East Sparta is located in southeastern Pike Township.",
"\n\nName and history\nIn 1806, George Young and his wife Catherine moved to what is now known as East Sparta, becoming the first permanent settlers of Pike Township. ",
"However, there are inconstancies in government and family records that instead put him in Pike in 1800, which would make him the first settler of Stark County.",
"\n\nIn 1815, the township was named after Zebulon Pike. ",
" It is one of eight Pike Townships statewide.",
"\n\nGovernment\n\nThe township is governed by a three-member board of trustees, who are elected in November of odd-numbered years to a four-year term beginning on the following January 1. ",
"Two are elected in the year after the presidential election and one is elected in the year before it. ",
"There is also an elected township fiscal officer, who serves a four-year term beginning on April 1 of the year after the election, which is held in November of the year before the presidential election. ",
"Vacancies in the fiscal officership or on the board of trustees are filled by the remaining trustees.",
"\n\nReferences\n\nExternal links\nTownship website\nCounty website\n\nCategory:Townships in Stark County, Ohio\nCategory:Townships in Ohio\nCategory:1815 establishments in Ohio"
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"Every Type of New Orleans Neighbor You Will Have\n\nWe’ve all heard stories from our friends and coworkers of the cray cray neighbors they have, and we have all had to deal with our fair share of them living in New Orleans at that. ",
"It’s one of those things that make this city so charming, right?",
"\n\nThe Neighborhood Watch\n\nAlway suspicious, always watching. ",
"They keep tabs on your hood and are bound to alert you if anything fishy, and even sometimes not so fishy, is going on. ",
"You can count on Ms. Yvonne to be on her porch takin’ names. ",
"This is a good neighbor to have, but you will want to stay on her good side.",
"\n\nThe Cat Lady\n\nYou literally counted up to 20 cats on their porch one day when you were out on a run, and you just had to know. ",
"Now, that you think about it, you actually aren’t sure if you have seen humans inhabiting this house.",
"\n\nThe Airbnb\n\nWhichever side of the argument you are on, it’s all too obvious when you live next to a permanent Airbnb. ",
"Bachelorette parties and people wearing Mardi Gras beads when it’s not Mardi Gras are in and out of this place. ",
"Empty lime green hand grenade cups from last night decorate the porch. ",
"If it isn’t already obvious enough, there’s the lock box hanging on the front gate.",
"\n\nThe Drug Dealer\n\nWe don’t want to be the ones to have to say it, but we have all been there. ",
"There’s people rolling up at all hours of the day and night for a very quick visit. ",
"The neighbor doesn’t appear to leave the house often, and on the couple of occasions you have spoken, they try to discreetly find out if you yourself may be in the market for some illegal goods. ",
"They might as well have a sign in front of their house. ",
"Guess the NOPD has better things to do.",
"\n\nThe Artist\n\nMaybe they are a painter, a musician or someone who types poetry on a corner in the French Quarter as their day job. ",
"Whatever it may be, you can’t really figure out why this person seems to not have a care in the world. ",
"Deep down, you envy them and secretly wish you could have that little, yet be so happy.",
"\n\nThe Frat House\n\nIt seems like there are at least 10 bros clad in their worst New Orleans fashion in proximity to the house at all times. ",
"There’s never a time when you’ve passed by that you don’t smell marijuana seeping out of the house and see empty bottles of cheap liquor strewn across the porch. ",
"College-looking girls arrive at all hours of the night in packs, and we really don’t know why because frankly this place looks like it could really use a deep cleaning.",
"\n\nThe Collector\n\nBasically a nice word for hoarder, you aren’t really sure who actually lives here, but their screen porch is about to bust at the seams with all of the old furniture, lamps and books stacked up from the floor to the ceiling. ",
"One person’s trash is another’s treasure, #AmIRite?",
"\n\nThe One Who is A Little Too Nice\n\nWe’ve all had one. ",
"They yell “hey baby,” and come in hot in for a hug as you head out the door with headphones in for a run. ",
"They have even gone as far as to kiss your cheek or come onto your porch uninvited. ",
"We don’t know you like that Mr. Maurice. ",
"We want to be nice to you, but we can’t because you are just too friendly and we don’t know what it all means.",
"\n\nThe Front Porch Alcoholic\n\nWhy don’t you ever go to work sir? ",
"This guy is harmless, but his life pretty much consists of chillin’ on the porch with a brown bag surrounded by other brown bags, and then walking to the corner store when he needs a refill. ",
"Every now and then he mumbles something at you which can feel troublesome, but no one ever really know what he is saying so its cool. ",
"We aren’t really sure how he pays his rent.",
"\n\nAnd then there’s you. ",
"Just out here trying to live your best life. ",
"Up next: the worst of New Orleans landlords. ",
"We know you have stories.",
"😂\n\nWe are excited to embark on a new journey, one that we think you will want to join. ",
"Today, a chapter is not ending, but a vision expanding.",
"\n\nBabes & Beignets began in 2014 when we were introduced by a mutual friend before a night out in a town that was still relatively new to us both, our beloved New Orleans. ",
"We were both young, curious, embarking on new careers as “adults” and always ready to explore all that our city had to offer. ",
"With that said, we connected immediately. ",
"It wasn’t long before we did everything together: met up after work to try the newest happy hours and chatted endlessly about the restaurants, events and festivals we should be going to on the weekends. ",
"Babes & Beignets was formed to document our adventures in the best city in the world and to fill a void by doing so from the perspective of two young women.",
"\n\nSince then, we have met so many inspiring people from restaurateurs to business owners to foodies and generally passionate New Orleanians. ",
"We have had the best meals of our lives, drank a lot and attended more events than we can count. ",
"All the while, learning the ins and outs of running a business and developing a brand together. ",
"We’ve been shocked, humbled and felt honored to have people rely on us and ask us to be a part of amazing things.",
"\n\nMost of all, we have fallen even more in love with New Orleans.",
"\n\nWhen we learned one of us would be moving to Houston, we were both sad yet excited for a new chapter. ",
"We talked about how our brand has evolved and especially how we have evolved as women in the past five and a half years. ",
"With the realization that our interests and goals had matured as well as the world around us, we thought it was a fitting time for our brand to do the same.",
"\n\nbabes xo is a new and improved Babes & Beignets. ",
"It will be a place to support modern women of all ages in being babes, which to us means living life to the fullest, working hard and being true to oneself.",
"\n\nWe won’t ever stop talking about or loving New Orleans. ",
"We are simply evolving beyond that. ",
"From eating and drinking to travel, relationships and fashion, we are creating a place where you can have it all online. ",
"We will interview even more Badass Babes so you can continue to be inspired by women from all over.",
"\n\nWe are so grateful to everyone who has believed in us or followed along on our journey. ",
"And we hope you will continue with us along this road.",
"\n\nAs always, thank you for your support. ",
"We love you!",
"\n\nXO,\n\nAlessandra & Maggie\n\nTo guarantee the best possible user experience, our site uses technical and third parties cookies. ",
"By continuing to browse this website, you accept the use of cookies."
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"Grigsby [2 takcles and 2 assists] looked good, but (and I hate to say this with all the pub Boomer gets, and he sounds like a great kid) Liam Ezekial was a MONSTER [6 tackles and 1 assist]. ",
"The only question with Ezekial is can he drop into coverage, cause Grigsby certainly looked like he had the speed to do it. ",
"Plus Grigsby was a beast on special teams.",
"\n\nDarrel Reid, DT form Minnesota, looked good. ",
"Drew the occasional double team, and free-ed up which ever backer played behind him pretty constantly. ",
"2 tackles 0 assists and a sack (only player to register a sack).",
"\n\nBrandon Jacobs, RB form Southern Illionis, also looked good. ",
"May have changed teh minds of some teams about changing him into a TE. ",
"Kid ran over a few players today. ",
"11 rushes for 102 yrds 2 tds with a long of 52 yrds.",
"\n\nThe biggest disappionment was Scott Mruczkowski, C from Bowling Green. ",
"Kid looked strong, but I never recalled him getting to the second level of the defense."
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[
"5 Unbelievable Health Benefits of Fish Oil\n\nIf we were to take cartoons or sitcoms at face value, we’d be stuck believing that fish oil is another unpleasant supplement mother forcibly feed their children by the spoonful. ",
"The reality couldn’t be more different. ",
"It is in fact a naturally nutritious substance that can improve both health and appearance, and it comes in capsule form as well.",
"\n\nWhat Is Fish Oil?",
"\n\nObviously enough, the term ‘fish oil’ refers to the oil extracted from the tissue of fish. ",
"Due to obvious reason, fatty species are the best source of it. ",
"These include salmon, trout, herring, mackerel, and sardines. ",
"It is full of omega-3 fatty acids, which are beneficial for our wellbeing in many ways. ",
"And if you’re not a fan of the food, you can consume it in the form of supplements.",
"\n\nThe positive results obtained by the daily consumption of fish oil have been discussed by nutritionists and health buffs alike. ",
"In fact, according to DefendYourHealthcare, it is highly recommended on the list of permitted ingredients of the world famous leptin diet. ",
"So, why does everyone love fish oil? ",
"Let’s find out.",
"\n\nThe Major Benefits\n\nAs mentioned above, the omega-3 fatty acids are an essential nutrient for a happy, healthy life. ",
"Their effects have been noticed and analyze carefully by the medical community for years now. ",
"Here are five of the most important benefits of this substance, be it in supplement form, or directly from the meat of the marine creature.",
"\n\n1. ",
"Promotes Cardiovascular Health\n\nOne of the most notable benefits of consuming fish oil has to do with its positive effect on cardiovascular health. ",
"Seeing as how heart disease is the number one cause of death on a global scale, it’s important to find ways in which to avoid it. ",
"And fortunately, this widely available natural supplement is one of them.",
"\n\nFirst of all, the omega-3 fatty acids in it are effective in raising the levels of HDL cholesterol, which is the good kind. ",
"While they don’t reduce LDL, or the bad kind, it is still a notable enhancement that will deliver visible results as far as your wellbeing is concerned. ",
"What is more, it lowers triglycerides by as much as 30%, and it decreases blood pressure as well.",
"\n\n2. ",
"Improves Mental Wellness\n\nOmega-3 fatty acids are compounds known for their ability to improve mental health. ",
"In countries whose diet is rich in fish and seafood, it has been noticed that the incidence of depression is lower than in other areas of the globe. ",
"And when you’re not a fan of the meat itself, you can still consume these essential nutrients in the form of a supplement.",
"\n\nThis is good news, seeing as how they provide relief in the case of anxiety and depression, as well as stress, restlessness, and loss of libido. ",
"Furthermore, studies have shown that adding fish oil to one’s diet leads to the amelioration of symptoms associated with schizophrenia and bipolar disorder. ",
"Just remember that it is not supposed to eliminate the need for medication.",
"\n\n3. ",
"Prevents Cognitive Decay\n\nThe mental health benefits of fish oil don’t end with mood disorders. ",
"Because omega-3 fatty acids are vital for brain function, this supplement can slow down cognitive decay and prevent the atrophy of the organ. ",
"This makes it an amazing ally in the battle against neurodegenerative disorders such as Alzheimer’s or dementia.",
"\n\n4. ",
"Alleviates Arthritic Pain\n\nFish oil has incredible anti-inflammatory properties. ",
"This natural remedy can sometimes deliver considerably valuable results even in areas where non-steroidal tablets fail. ",
"For this reason, it is used in the treatment of pain and stiffness caused by arthritis, an inflammation of the tissue cushioning the joints. ",
"This reduces the need for medication among patients.",
"\n\n5. ",
"Enriches Skin and Hair\n\nFinally, it’s worth mentioning that fish oil and its active compounds work wonders in terms of beauty as well. ",
"The supplement is an incredible source of vigor and plumpness for the skin, leaving it smooth and elastic. ",
"In this way, it also combats the aging process, which means that you will maintain your youthful glow for longer.",
"\n\nAs far as the hair is concerned, fatty acids prevent thinning and dandruff. ",
"Your locks will thus look clean, shiny, and playful all day long, and you won’t have to worry about any problems as far as their wellbeing is concerned. ",
"Thus, not only does fish oil help you feel like a million bucks, it prompts you to look the part as well.",
"\n\nFinal Thoughts\n\nExtracted straight from the tissue of fish, fish oil is ripe with beneficial omega-3 fatty acids. ",
"These nutrients promote heart health, alleviate the aches caused by arthritis, and enhance the function of your mind as well. ",
"And to top it all off, they give you beautiful skin and hair too, which is a great bonus for anyone.",
"\n\nLuke S. Mitchell is an MS Undergraduate in Sports Journalism and manager of Defend Your Healthcare. ",
"He is interested not only in the mind-body relationship and how motivation shapes our bodies but also in how we draw energy just from one simple yet powerful thought.",
"You can find him on Facebook and Twitter.",
"\n\nDisclaimer:\n\nWe work diligently to bring quality material to our readers and provide credit and/or links to sources. ",
"As much of our website is made of user-generated content, we can't always verify sources. ",
"If you believe we have used your copyrighted content without permission, send us an email at admin@zestythings.com and we will remove it immediately or provide proper attribution to the material (your preference).",
"\n\nCopyright Zesty Things. ",
"All right reserved.",
"\nSite maintenance provided by CGS Computers"
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"1. ",
"Field of the Invention\nThe present invention relates to an Wireless tag system having a plurality of antenna feeding points.",
"\n2. ",
"Description of the Related Art\nRecently, radio data carriers that have set up therein an inherent radio frequency ID (RFID), have a memory area in which information can be rewritten, and read out their RFID or write and read out arbitrary information in a noncontact manner using radio are coming into use.",
"\nFor example, mail articles and deliveries are made to have attached thereto or built therein respectively a radio data carrier having an inherent RFID so as to identify such mobile objects.",
"\nMeanwhile, when mail articles and deliveries are piled up, the resonance frequency of Wireless tag fluctuates from a designed resonance frequency due to the mutual inductance of respective Wireless tag. ",
"Thus, the resonance syntony state with a reader/writer comes to be out of alignment, and an electric power that is fed from a reader/writer to Wireless tag is lowered, which deteriorates the communication performance.",
"\nAccordingly, there is suggested a technique in which, even if the resonance frequency of Wireless tag comes to be out of alignment, the transmission frequency is changed in turn by a frequency scan circuit so that the effective frequency band within which communication with an Wireless tag is sufficiently possible in the transmission frequency sent from a transmitting antenna covers resonance frequency fluctuation ranges of all Wireless tag (Jpn. ",
"Pat. ",
"Appln. ",
"Laid-Open Publication No. ",
"2000-227947).",
"\nAs a frequency band used in a Wireless tag for managing an object, a microwave band of 2.45 GHz is used in Japan. ",
"In many cases, the transmission frequency band cannot be easily changed. ",
"There is raised a problem that, in case Wireless tag are densely located, or sparsely located, a reading system is not realistic since the transmission frequency band is changed."
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"tive of t(x). ",
"Let f(w) = 4*w - 2. ",
"What is -4*f(y) + 3*n(y)?",
"\n-y + 2\nLet b(m) = -3*m**2 - m. Let c(h) = -2*h**2 - 1. ",
"Give -3*b(d) + 4*c(d).",
"\nd**2 + 3*d - 4\nLet x(y) = 10*y - 7. ",
"Let s(o) = 7*o**2 + 2*o. ",
"Let n be s(-1). ",
"Let d(j) = 7*j - 5. ",
"Calculate n*x(v) - 7*d(v).",
"\nv\nLet v(j) = 8*j. ",
"Let x(h) = 12*h. ",
"Suppose -4*d - 48 = 2*d. ",
"Give d*v(y) + 5*x(y).",
"\n-4*y\nLet y(q) = q**3 - 2*q**2 + q + 11. ",
"Let p(f) = -f. ",
"Calculate p(s) - y(s).",
"\n-s**3 + 2*s**2 - 2*s - 11\nLet w(h) = -2*h - 3. ",
"Let j(c) = -c - 6. ",
"Let d be j(-9). ",
"Let v = d - 4. ",
"Let s(n) = 3*n - 5*n + 3*n + 1. ",
"Calculate v*w(u) - s(u).",
"\nu + 2\nSuppose -r = -0*r + 4*d - 12, 4*d - 60 = 5*r. ",
"Let m(q) = q. Let j(z) = 10*z + 2. ",
"What is r*m(v) + j(v)?",
"\n2*v + 2\nLet f(q) = 4*q**3 - 4*q + 7. ",
"Let j(i) be the second derivative of -i**5/20 + i**3/6 - i**2 - 9*i. ",
"Give -6*f(l) - 22*j(l).",
"\n-2*l**3 + 2*l + 2\nLet d(v) be the third derivative of 13*v**6/120 - 4*v**5/15 - 7*v**4/24 + 2*v**2 - 12. ",
"Let o(t) = 4*t**3 - 5*t**2 - 2*t. ",
"What is -2*d(h) + 7*o(h)?",
"\n2*h**3 - 3*h**2\nLet z(g) = g**3 - g**2 - g - 1. ",
"Let j(y) be the first derivative of y**3/3 + y**2/2 - y + 46. ",
"Calculate -j(k) - z(k).",
"\n-k**3 + 2\nLet i(d) = 6*d**2 - 5*d. ",
"Let n(r) = 18*r**2 - 16*r. ",
"Determine 10*i(p) - 3*n(p).",
"\n6*p**2 - 2*p\nLet v(j) = 5*j**2 + 3*j. ",
"Let s(u) = -u**3 - 11*u**2 - 7*u. ",
"Calculate 4*s(t) + 9*v(t).",
"\n-4*t**3 + t**2 - t\nLet b(a) = -6*a - 13. ",
"Suppose -4*x = 3*r - 20, -3*r + 1 = -11. ",
"Let d(c) = -x*c + 2 + 3*c + 0*c. ",
"Determine -6*b(i) - 39*d(i).",
"\n-3*i\nLet c(q) = -q**2 + 1. ",
"Let u = 3 - -2. ",
"Let f(o) = 2*o**2 + o - 2*o**2 - 6*o**2 + 6 - 2*o. ",
"Give u*c(r) - f(r).",
"\nr**2 + r - 1\nLet z(u) = 1. ",
"Let q(n) = 7*n + 1. ",
"What is -q(p) + 4*z(p)?",
"\n-7*p + 3\nLet f(t) = 2*t + 21. ",
"Let n(y) = 6*y + 64. ",
"Calculate -10*f(r) + 3*n(r).",
"\n-2*r - 18\nLet u(z) = -z**2 - 2. ",
"Let h(l) = -l**3 - l**2 + 2*l - 3. ",
"Let t(d) = -d. ",
"Let s(k) = -h(k) - 2*t(k). ",
"Determine 2*s(i) + 3*u(i).",
"\n2*i**3 - i**2\nLet j(l) = -l - 5. ",
"Let o(f) = -2*f - 6. ",
"What is -5*j(p) + 4*o(p)?",
"\n-3*p + 1\nLet j(q) = 9*q**2 + 5*q - 1. ",
"Let z(r) = 13*r**2 + 7*r - 2. ",
"Give -7*j(a) + 5*z(a).",
"\n2*a**2 - 3\nLet b(j) = -2*j**3 - 6*j**2 - 3*j - 4. ",
"Let y(z) = 6*z**3 + 17*z**2 + 9*z + 11. ",
"What is 17*b(o) + 6*y(o)?",
"\n2*o**3 + 3*o - 2\nLet v(w) = -11*w**2 + 7*w - 7. ",
"Let j(q) = -6*q**2 + 4*q - 4. ",
"Give 7*j(r) - 4*v(r).",
"\n2*r**2\nLet y(h) = h - 1. ",
"Let f(r) = r + 1. ",
"Let i(k) = -k + 4. ",
"Let q(x) = 4*f(x) - 2*i(x). ",
"Suppose -5 + 0 = 5*c. ",
"Determine c*q(t) + 5*y(t).",
"\n-t - 1\nSuppose 20 = -0*j + 4*j. ",
"Suppose 3*n - j*l = 4, -4*n + 16 = l + 3*l. ",
"Let f(u) = 9*u**2 - 1. ",
"Let y(p) = 5*p**2 - 1. ",
"What is n*f(b) - 5*y(b)?",
"\n2*b**2 + 2\nLet x(l) = -5*l**2 + 49*l - 16. ",
"Let a(n) = n**2 - 10*n + 3. ",
"Determine -11*a(q) - 2*x(q).",
"\n-q**2 + 12*q - 1\nSuppose -3*i - 14 = -13*h + 14*h, 5*h + 10 = 5*i. ",
"Let s(y) = -8*y**2 + 3. ",
"Let m(b) = 105*b**2 - 40. ",
"What is i*m(o) - 40*s(o)?",
"\n5*o**2\nLet m(s) = 3 + s + 5*s**2 + 2 - 2*s + 2*s**2. ",
"Let u(r) = 8*r**2 - r + 6. ",
"Let w(y) = 2*y - 8. ",
"Let a be w(7). ",
"Determine a*m(g) - 5*u(g).",
"\n2*g**2 - g\nLet z(d) = -d**3 - 7*d**2 + 5*d - 6. ",
"Let b = -48 + 42. ",
"Let f(k) = 2*k**3 + 15*k**2 - 11*k + 13. ",
"What is b*f(t) - 13*z(t)?",
"\nt**3 + t**2 + t\nLet p(r) = -13*r**2 - 4*r + 7. ",
"Let i(s) = -6*s**2 - 2*s + 3. ",
"Calculate 7*i(y) - 3*p(y).",
"\n-3*y**2 - 2*y\nLet g(d) = 5*d**2 - 5*d**3 - 4*d**2 + 6*d + 4*d**2 - d**3. ",
"Let l(j) = 5*j**3 - 4*j**2 - 5*j. ",
"Give -6*g(m) - 7*l(m).",
"\nm**3 - 2*m**2 - m\nLet p(v) = v**3 + 5*v**2 + 4*v + 1. ",
"Let c(g) = g**3 + 3*g**2 + 2*g + 1. ",
"Suppose 0 = -3*u + 7*u + 20. ",
"Determine u*c(k) + 3*p(k).",
"\n-2*k**3 + 2*k - 2\nLet c = -11 + 11. ",
"Suppose 4*g + 9 + 3 = c. Let q(r) = r. Let d(t) = -t - 1. ",
"Let p(f) = -d(f) + 4*q(f). ",
"Let a(o) = -9*o - 1. ",
"Give g*a(b) - 5*p(b).",
"\n2*b - 2\nLet y(d) = -3*d**3 - 3*d**2 - 5*d - 5. ",
"Let o(w) = 12 + 7 - w - 20 - w**3. ",
"Determine 5*o(x) - y(x).",
"\n-2*x**3 + 3*x**2\nLet j(i) = i**3 + 3*i**2 - 3*i + 2. ",
"Suppose -4*m - 22 = -5*h - m, -4*h = -4*m - 24. ",
"Let n(a) = 10 + a**3 - 4 - 2*a - 4 + 2*a**h. ",
"What is 2*j(l) - 3*n(l)?",
"\n-l**3 - 2\nLet o = 0 + 2. ",
"Suppose -4*h = o*r - 18, 3*r - 5*h - 3 = 2. ",
"Let l(z) = -4*z**3 - 3*z**2 - 5*z - 1. ",
"Let m(g) = 5*g**3 + 4*g**2 + 6*g + 1. ",
"What is r*m(x) + 6*l(x)?",
"\nx**3 + 2*x**2 - 1\nLet a(q) = q**2 - 3*q**3 + 2*q**3 + 4*q - 4*q. ",
"Let x(d) = d + 1. ",
"Let y(c) = -c**3 + c**2 - 3*c - 4. ",
"Let g(m) = -4*x(m) - y(m). ",
"Give -2*a(i) - g(i).",
"\ni**3 - i**2 + i\nLet w(i) = 8*i**2 - 2*i - 5. ",
"Let z be 6/(-36) - 123/18. ",
"Let x(j) = -7*j**2 + 2*j + 4. ",
"Calculate z*x(v) - 6*w(v).",
"\nv**2 - 2*v + 2\nLet z(r) be the third derivative of 0*r - 5/24*r**4 - 11/6*r**3 - 3*r**2 + 0. ",
"Let f(g) = 2*g + 4. ",
"Calculate 11*f(d) + 4*z(d).",
"\n2*d\nLet m(i) = -5*i**3 + 2*i**2 + 6. ",
"Let q = 0 + 3. ",
"Let g(p) = 0*p**3 + q + 3*p**3 - 2*p**3 - 4. ",
"What is -6*g(b) - m(b)?",
"\n-b**3 - 2*b**2\nLet n(g) = 3*g**2 + 3. ",
"Let t(f) = -7*f**2 - f - 7. ",
"Suppose 3*i = -8*i + 22. ",
"Calculate i*t(a) + 5*n(a).",
"\na**2 - 2*a + 1\nLet p(k) be the second derivative of -7*k**3/6 - k**2 + 3*k. ",
"Let s(y) = -6*y - 2. ",
"Determine -5*p(w) + 6*s(w).",
"\n-w - 2\nLet t(j) be the second derivative of j**3/6 + 13*j**2/2 + 33*j. ",
"Let l(w) = 3. ",
"Calculate 9*l(x) - 2*t(x).",
"\n-2*x + 1\nLet g(v) = -22*v**2 + 3*v + 3. ",
"Let z(i) = 67*i**2 - 8*i - 8. ",
"Give -8*g(m) - 3*z(m).",
"\n-25*m**2\nLet f be -1 + 27 + (3 - 1). ",
"Let o be 8/f + 5/7. ",
"Let h(z) = -2 + o + 0. ",
"Let a(y) = -y**2 - y - 3. ",
"What is -2*a(x) + 6*h(x)?",
"\n2*x**2 + 2*x\nLet b(s) = -5*s**3 - 3*s**2 - 4*s. ",
"Let p = 1379/831 + 2/277. ",
"Let x(j) be the first derivative of 0*j + 9/4*j**4 + 1 + p*j**3 + 7/2*j**2. ",
"Give -5*b(t) - 3*x(t).",
"\n-2*t**3 - t\nLet l(a) = 2*a + 5. ",
"Let k(y) = -11. ",
"Let t(p) = -9. ",
"Let m(j) = 4*k(j) - 5*t(j). ",
"Calculate l(o) - 6*m(o).",
"\n2*o - 1\nLet q(p) be the second derivative of -p**3/6 - p**2/2 + p. Let b(v) = -2*v + 1711 + 0*v - 1714. ",
"What is b(o) - 3*q(o)?",
"\no\nLet s(h) = -7*h**2 - 7*h - 4. ",
"Let o(x) = -3*x**2 - 3*x - 2. ",
"Let v(y) = y + 14. ",
"Let c be v(-16). ",
"Let g be (4 - 3)/(c/4). ",
"What is g*s(w) + 5*o(w)?",
"\n-w**2 - w - 2\nLet b(m) = 9*m + 2. ",
"Let u(i) = -i**2 + 3*i + 15. ",
"Let z be u(5). ",
"Let s(a) = 5*a + 1. ",
"Give z*s(h) - 3*b(h).",
"\n-2*h - 1\nSuppose -2*v - 2 = 4. ",
"Let a(z) = 11*z**3 - 11*z**2 - 8*z. ",
"Let b(r) = -4*r**3 + 4*r**2 + 3*r. ",
"Calculate v*a(k) - 8*b(k).",
"\n-k**3 + k**2\nLet c(b) = -b**2 - 7*b - 2. ",
"Suppose -k - k = -2. ",
"Let z = 6 - k. Let v = -3 + 4. ",
"Let q(n) = n. Give v*c(t) + z*q(t).",
"\n-t**2 - 2*t - 2\nLet t be ((-8)/3 - -2)/(22/495). ",
"Let l(p) = -12*p - 15. ",
"Let b(m) = -4*m - 4 - 3*m + 4*m. ",
"What is t*b(i) + 4*l(i)?",
"\n-3*i\nLet g(y) be the third derivative of y**6/120 - y**5/30 + y**4/8 - y**3/6 + 5*y**2 + 8*y. ",
"Let d(s) = -s**2 + s. Calculate -2*d(n) + g(n).",
"\nn**3 + n - 1\nLet c(r) = -r**3 - 7*r**2 + 2*r + 10. ",
"Let u be c(-7). ",
"Let q(a) = 2*a + 4. ",
"Let o(m) = -1. ",
"Let g(y) = 1. ",
"Let n(s) = 2*g(s) + 3*o(s). ",
"Calculate u*n(f) - q(f).",
"\n-2*f\nLet f(j) = -5*j - 2. ",
"Suppose 9 = z - 4*z. ",
"Let q(v) be the first derivative of -v**2 - v + 5. ",
"Determine z*f(s) + 8*q(s).",
"\n-s - 2\nLet p(j) = j**3 - j. Let b = 15 - 14. ",
"Let y(r) = -8*r**3 + 4*r. ",
"Give b*y(t) + 4*p(t).",
"\n-4*t**3\nLet s(y) = -8 + 1 + 10*y - 5. ",
"Let i(h) = -h + 1. ",
"Suppose -3*x + 2 = -1. ",
"Determine x*s(t) + 12*i(t).",
"\n-2*t\nLet v(r) = 0 - 4*r + 10*r**2 - 2*r**2 + 3 + 5*r. ",
"Let a(n) = 17*n**2 + 2*n + 6. ",
"Give 6*a(b) - 13*v(b).",
"\n-2*b**2 - b - 3\nSuppose -6 = 2*a, 9 = 4*k - 0*k + a. Let s(c) = 3*c**2 + 4 + 1 - 10 + 7. ",
"Let j(h) = 4*h**2 + h + 2. ",
"Calculate k*s(q) - 2*j(q).",
"\nq**2 - 2*q + 2\nLet l(s) = -28*s**2 - 3*s. ",
"Let b(r) = -28*r**2 - 2*r. ",
"What is 4*b(t) - 3*l(t)?",
"\n-28*t**2 + t\nLet n(s) = 3*s**2 + 2*s. ",
"Let h(r) = 6*r**2 + 5*r. ",
"Let j be 4/8*(-12)/(-3). ",
"Determine j*h(z) - 5*n(z).",
"\n-3*z**2\nLet u = 208 - 202. ",
"Let m(y) = 5*y**2 - 8*y - 4. ",
"Let g(c) = 6*c**2 - 9*c - 5. ",
"What is u*g(x) - 7*m(x)?",
"\nx**2 + 2*x - 2\nLet d(x) = -x**2 - 6*x - 7. ",
"Let k(f) = f**2. ",
"Let q(s) = -d(s) + 3*k(s). ",
"Let c(a) = -2*a**2 - 3*a - 4. ",
"Determine 7*c(b) + 4*q(b).",
"\n2*b**2 + 3*b\nSuppose -68*n + 32 = -60*n. ",
"Let j(d) = -5*d**2 + 5*d - 1. ",
"Let i(o) = -9*o**2 + 9*o - 2. ",
"Let a = -18 - -11. ",
"Calculate a*j(l) + n*i(l).",
"\n-l**2 + l - 1\nLet k(u) = -26*u**2 - 3*u. ",
"Let c(i) = -261*i**2 - 29*i. ",
"Calculate -3*c(v) + 29*k(v).",
"\n29*v**2\nLet r(s) = -60*s**3 + 4*s - 4. ",
"Let f(u) = u**3 + u - 1. ",
"Calculate -4*f(z) + r(z).",
"\n-64*z**3\nLet u(w) = 2*w. ",
"Let s(o) = o - 2 + 2. ",
"Let b = -25 - -9. ",
"Let m = b + 13. ",
"Determine m*s(a) + 2*u(a).",
"\na\nLet z(c) = -c. ",
"Suppose 0 = -s - 0 - 1. ",
"Let a(b) = 1 - b + 5*b - 3*b. ",
"What is s*a(p) - 2*z(p)?",
"\np - 1\nLet c(r) be the first derivative of r**4/4 + 5*r**3/3 - r"
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"Behenna — whose mother helped prosecute Oklahoma City bomber Timothy McVeigh and whose father worked as an FBI analyst and Oklahoma state investigator — has always insisted he acted in self-defense when he killed Ali Mansur. ",
"Behenna was paroled five years ago after an effort to overturn his conviction failed. ",
"Now he is seeking a different path to vindication: a presidential pardon.",
"\n\nAD\n\nIt is a bold ask for someone who stripped a prisoner naked, interrogated him without authorization, shot him twice and then claimed at his court-martial that he was protecting himself. ",
"But Behenna, now a 35-year-old ranch hand, is pitching his case at an opportune moment for U.S. military members accused of war crimes.",
"\n\nAD\n\nPrompted by adulatory profiles on “Fox & Friends,” President Trump has voiced support for a Navy SEAL and a Green Beret who have been charged with murder but haven’t faced trial yet.",
"\n\nTrump tweeted last month that Eddie Gallagher — a Navy SEAL charged with fatally stabbing a teenage ISIS prisoner under his care in Iraq and then holding his reenlistment ceremony with his corpse — would be moved to a less restrictive confinement before his trial.",
"\n\nAD\n\n“In honor of his past service to our Country,” Trump said on Twitter. ",
"And in December, Trump tweeted that he would review the case of Matt Golsteyn, an Army major and former Green Beret who admitted to killing a suspected Taliban bombmaker in Afghanistan and burning his body. “",
"A U.S. Military hero,” Trump declared on Twitter.",
"\n\nThere are signs that lobbying efforts on Behenna’s behalf may be having an impact, too.",
"\n\nAD\n\n“We know we have a president who is very sympathetic to the very difficult situation that soldiers, sailors and Marines were put in during the Iraq and Afghanistan wars,” said John Richter, a former head of the Justice Department’s criminal division who is representing Behenna and has spoken to White House officials about his case.",
"\n\nAD\n\nBehenna was told that he had to complete his parole in 2024 before he could seek a pardon recommendation from the Justice Department. ",
"But two months ago, he said he learned that the agency might be considering his application, after all.",
"\n\nRosalind Sargent-Burns, the agency’s deputy pardon attorney, had called with questions about his case.",
"\n\n“Sounds like your pardon request and paperwork [are] being processed,” Behenna’s parole officer texted him.",
"\n\nSargent-Burns declined a request to discuss Behenna’s case.",
"\n\nAD\n\nSince Behenna’s conviction, elected officials and retired military brass have come forward to support his unsuccessful court appeals and, more recently, his pardon request. ",
"Among his backers: former Oklahoma governor Mary Fallin (R) and state Attorney General Mike Hunter (R), as well as more than three dozen retired generals and admirals, including retired Gen. Anthony Zinni, who, up until January, served as a Trump administration special envoy for the Persian Gulf.",
"\n\nBehenna is hopeful but cautious. ",
"A presidential pardon, which can be granted only to people convicted of federal crimes, can restore the right to vote, the ability to hold political office, obtain certain jobs, or, as Behenna would like to do, adopt a child.",
"\n\nAD\n\n“I am not expecting anything, because this is a lesson I learned years ago in my trial,” said Behenna, who called the pardon “a way to ask for forgiveness from the country I fought for . . . ",
"and a way for me to get justice for a trial that was unjust.”",
"\n\nCarnage from a roadside bomb\n\nOn April 21, 2008, Behenna was riding in the second of a three-truck convoy north of Baghdad when the third vehicle behind him suddenly shot into the sky, blown up by a roadside bomb.",
"\n\nAD\n\nAs Behenna, then a 24-year-old lieutenant, looked out the side mirror, whorls of smoke and dust blanketed the air. ",
"He made his way to the third truck and immediately performed CPR on his dying comrade, Sgt. ",
"Adam Kohlhaas, 26. ",
"A second friend, Sgt. ",
"Steven Christofferson, 20, who lost a coin toss and had to man the gun turret, was also killed — his body cut in half after the truck landed 75 feet from the blast site on its top.",
"\n\nAD\n\nBehenna was familiar with explosions and death before he arrived in Iraq. ",
"He was in fifth grade in 1995 when McVeigh, a Persian Gulf War veteran, detonated a truck bomb at the Alfred P. Murrah Federal Building in downtown Oklahoma City, killing 168 people. ",
"One of the victims was Behenna’s middle-school baseball coach.",
"\n\nBut he’d never seen carnage up close. ",
"Shortly after his friends’ deaths in the roadside bombing, Behenna seized on a new intelligence report revealing the identity of an al-Qaeda operative who possibly helped orchestrate the explosion: Ali Mansur. ",
"Mansur was found at his house with several weapons, including an illegal machine gun and an Iraqi passport indicating that he’d made two trips to Syria, “a known training ground for terrorists,” according to Behenna’s pardon application.",
"\n\nAD\n\nMansur was interrogated but freed because the military could not find conclusive evidence linking him to the explosion, the pardon application says.",
"\n\nAD\n\nBehenna was livid when Mansur was released. ",
"And it was Behenna who was ordered to transport him back to his village and set him free.",
"\n\n'In total shock'\n\nLess than a month later, out by a secluded railroad culvert near the northern town of Baiji, Behenna decided to interrogate Mansur again on his own.",
"\n\n“I stripped him naked to intimidate him,” Behenna said. “",
"I told him I wanted more intel on local leaders of al-Qaeda, and that I wanted him to tell me about his stops in Saudi Arabia and Syria, and the [roadside] bomb explosion. ",
"But he kept saying, ‘I don’t know. ",
"I don’t know.’ ”",
"\n\nFinally, Mansur said something in Arabic that Behenna didn’t understand. ",
"He turned to his interpreter for help, he said, but that’s when he heard something hit the culvert’s wall. ",
"When he pivoted back to the prisoner, Behenna claims Mansur had just thrown a rock and was now standing up and reaching for his Glock. ",
"Behenna stepped to the side and fired one shot into Mansur’s chest, the next to his head.",
"\n\nAD\n\nAD\n\n“I was in disbelief that just happened,” Behenna recalled. “",
"I just remember this numb feeling. ",
"I was in total shock.”",
"\n\nBehenna left the body in the culvert and didn’t tell anyone.",
"\n\nThen he went back to his base.",
"\n\nThe next day, Iraqi police found Mansur’s body, and Behenna was later charged with premeditated murder. ",
"He faced life in prison without parole. ",
"The whole time their son was under investigation, his parents, lifelong criminal investigators, said they never felt disappointed or embarrassed.",
"\n\n“When he got charged, I wasn’t like, ‘Dear God, what are people going to think?’ ",
"I was thinking, ‘What can I do to help my son?’ ” ",
"recalled Vicki Behenna, who is now the executive director of the Oklahoma Innocence Project, which works to exonerate the wrongfully convicted.",
"\n\nAt his 2009 court-martial, military prosecutors at Fort Campbell, Ky., told the jury they believed Behenna executed Mansur to avenge the loss of his two soldiers in the roadside explosion, according to press coverage of the trial.",
"\n\nAD\n\nAD\n\nProsecutors said Behenna ordered another soldier, Staff Sgt. ",
"Hal Warner, to burn the body with an “incendiary” grenade — an accusation that Behenna denies and that two other witnesses said they never heard. (",
"Warner pleaded guilty to assault, maltreatment of a subordinate and making a false statement and was sentenced to 17 months in prison.)",
"\n\nAs his trial wound down, Behenna took the stand.",
"\n\n“I was scared Ali Mansur was going to take my weapon and use it against me,” he told the military panel of seven officers. “",
"This happened very fast.” ",
"He described the sequence of his movements — turning to face Mansur and stepping to the left before firing two shots.",
"\n\nIn the gallery, the government’s own forensics expert was listening, concerned, according to Behenna’s pardon application. ",
"The expert had outlined a similar scenario to prosecutors the day before.",
"\n\nAD\n\nBut the prosecution rested its case without calling the expert, portraying Behenna as an unprovoked aggressor.",
"\n\nThe next day, Behenna was found guilty of unpremeditated murder, a result that spared him of the possibility of life in prison without parole. ",
"Still, he filed for a mistrial, contending prosecutors hid exculpatory evidence — a constitutional violation — and should have immediately alerted him since the expert’s testimony could have benefited his case. ",
"The judge denied the motion. ",
"Behenna appealed to a higher court for a new trial, but lost, 3 to 2. ",
"Ultimately, Behenna’s 25-year sentence was reduced to 15.",
"\n\nIn early 2013, four years into his prison term at Fort Leavenworth, Kan., 37 retired military generals and admirals, including five four-stars, sent an amicus brief to the Supreme Court urging the justices to review his case and clarify the self-defense rights of service members in combat zones. “[",
"A]ssuming that Lieutenant Behenna’s claim that Mansur lunged for his weapon is truthful — an assumption made by [the country’s highest military appeals court] that this Court must also make — it would have been a dereliction of duty for Lt. ",
"Behenna not to fire in self-defense,” they wrote.",
"\n\nThough the Supreme Court didn’t accept his case, Behenna was released a year later on March 14, 2014, a moment the Oklahoman featured on its front page with an enormous photo of Behenna hugging his sister-in-law, while his mother was bawling off to the side.",
"\n\nNow, as Behenna seeks another front-page headline, he spends his days near his hometown north of Oklahoma City, feeding cattle on a commercial ranch and raising goats and chickens at his own 20-acre property.",
"\n\nIn his living room, he displays a framed photograph of Kohlhaas and Christofferson with the son of an Iraqi villager sitting between them. ",
"His comrades are smiling and laughing, and the boy is wearing one of their helmets and a pair of their sunglasses. ",
"It was taken the day before they were killed — and the day before Behenna’s life changed.",
"\n\n“I was responsible for them,” Behenna said. “",
"I’m afraid if I didn’t see the picture as often as I do, then maybe I would forget them.”"
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"Hello, gardening friends, welcome to\nBackyard Basics, I'm John Dromgoole.",
"\nWell we end up with problems in our\ngarden sometimes as the weather changes\nmaybe from winter to spring, or goes in\nand out of winter and spring. ",
"The plants\nget distressed because they think it's\none season or the other, and when they're\ndistressed, here come the insects that do\nthe damage. ",
"It's their job to get rid of\nthe weaker species. ",
"So as we see that, we\nneed to do something about managing\nthose. ",
"So there's some homemade remedies.",
"\nThere's always something around the\nhouse that will manage some of the insects.",
"\nHere's one of the traditional ones, and\nit's the blending of peppers. ",
"I'm not\ngoing to touch those because they're\nvery hot. ",
"And then some garlic. ",
"You blend\nthe two of those together, and there's\ndifferent formulas. ",
"There's books out\nthere that will show you how to make\nthese blends. ",
"And so you blend the garlic\nand the pepper, and it's one of the best\ninsecticides that you're going to find\nfor homemade remedies. ",
"Another one is eggshells.",
"\nNow eggshells are very sharp, and so\nwhen slugs or snails are coming over to the\nplants, you put it around the base of the\nplant. ",
"Last year we had a big snail\nproblem, so we put those around the base\nin containers. ",
"Or out in the garden, you\ncan put quite a few out there also,\naround the seedlings, that is, and that\nwill keep them off of there, and that's\nreally important when they're getting\nstarted because they will nibble the\nedge of it and then the plant will fall over.",
"\nSo that's a good way to keep them away.",
"\nAnother thing that we use is coffee\ngrounds. ",
"They have a tendency to slow\ndown the slugs and snails also. ",
"Some\nother insects it keeps awake longer and\nthey do a lot more damage, but it's a\ngreat mulch and a great nutrient also. ",
"So\ncoffee grounds are another thing not to waste,\ngo ahead and use them in the garden, and\nit doesn't make any difference whether\nit's a plant in the ground or a\ncontainer plant. ",
"They're all going to\nbenefit from the coffee grounds. ",
"Now some\nflying insects are managed easily, too,\nwith these types of traps.",
"\nThese are called sticky traps, and they\nhave a nice gluing substance on them, and\nall the insects are attracted to the\nyellow color, and so this is placed in\nthe ground\nnext to the insects in the garden or in\ncontainer plants indoors. ",
"A lot of us have\nflying insects like gnats, fungus gnats.",
"\nThey get caught on this rather easily.",
"\nThese are replaceable. ",
"I think that this\nis a very good alternative: non-toxic and\nit will do the job for you to manage\nflying insects. ",
"Now there's one here\nthat's white and that is for the\nwhitefly. ",
"The whitefly will go to that\none every time, and you'll get management\nof those. ",
"Also, there are many products on\nthe market that if you don't have time\nto make your own, you can select from\nsome of these and they're all safe to\nuse: certified organic. ",
"It's the kind of\nthing you want to use around your home.",
"\nNow the books, they're great, and they're\nmany of them. ",
"Actually there are some\nbooks that the city of Austin produces\ntoo about well-adapted plants. ",
"It's not so\nmuch about in and out of the season but\nabout plants that are well-adapted, and\nwhen they're well adapted, they're much\nmore resistant to insects. ",
"And one thing\nyou need to remember is that if you\nkill a beneficial insect, you inherit their job.",
"\nSkip Victor always mentioned that one.",
"\nFor Backyard Basics, I'm John Dromgoole.",
"\nSee you next time.",
"\n"
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"Q:\n\nInside a trigger create database is not working\n\nI tried using trigger concept. ",
"If any new row inserts in the user table my trigger will fire and try to create a dynamic database for that new user. ",
"But Inside a trigger create database is not working. ",
"How can get it done. ",
"Can anybody have any suggestions.",
"Please help of on this\n\nA:\n\nAs the documentation says (under Trigger Limitations):\n\nThe following Transact-SQL statements are not allowed in a DML trigger: \n (...) CREATE DATABASE (...)\n\nThere are further limitations, consulting the documentation is highly recommended.",
"\nThis all means, to my understanding, that you should solve this problem at application level*: check whether your user is successfully created, and then issue a CREATE DATABASE statement. ",
" Preferably these two are enclosed in a transaction, so that either both succeed or both fail - this way you can be sure that your data are in a consistent state. ",
" You may have to issue a DROP DATABASE if the transaction fails. ",
" (Thanks to Jon Seigel for pointing this out.)",
"\n*: From this point of view, the 'application' may be a T-SQL block of code (or a stored procedure) as well.",
"\n\nA:\n\nYou do not want to provision a seat for your new tenant in a DML trigger. ",
"For one, provisioning is usually lengthy and you do not want to block the DML, and likely the HTTP request for new user onboarding behind it. ",
"You want to be able to create and onboard the user, then start the provisioning process. ",
"If the provisioning goes awry, you can resolve the problem with offline intervention later, and you have the user record to contact and to notify that the account is now properly provisioned (ie. ",
"you do not lose a customer). ",
"The second reason is the problem you run into, namely that some provisioning operations are simply not do-able from a DML trigger, and as things will evolve and your architecture becomes more complex the provisioning will become more complex (eg. ",
"load balance the account resources).",
"\nThe universal solution is to decouple the two and use queues. ",
"The account creation drops a message into a queue to request provisioning the account. ",
"A queue monitoring process kicks off (after the user account is committed!) ",
"and does the necessary work, asynchronously. ",
"SQL Server can handle this via Service Broker and internal activation. ",
"This is much better than doing it at application level because is reliable (ie. ",
"if the user stops the page load during the account creation, which in turn triggers a cancellation of the application processing the HTTP request, it does not leave an orphaned non-provisioned account).",
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"Q:\n\nConvert to date using formatters parameter in pandas to_string\n\nI know there are several ways to convert a column to a date object, but what I am looking for is a way to do so while simultaneously formatting other columns. ",
"Say I have the following data frame:\nimport pandas as pd\n\nurl = \"https://raw.github.com/pandas-dev/pandas/master/pandas/tests/data/tips.csv\"\ndf = pd.read_csv(url)\ndf[\"date\"] = list(range(42005, 42005+len(df)))\n\nWhat I'm trying to achieve is the ability to print these data using some formatting, so I might do something like the following:\nprint(\n df\n .head(10)\n .to_string(\n formatters={\"total_bill\": \"${:,.2f}\".format, \n \"tip\": \"${:,.2f}\".format\n }\n )\n)\n\nBut I also want to format the date in this step as well. ",
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"\nIs it possible to do this in a simple manner?",
"\n\nA:\n\nTurns out this is incredibly easy once you realize how the function actually works... \nprint(\n df\n .head(10)\n .to_string(\n formatters={\"total_bill\": \"${:,.2f}\".format, \n \"tip\": \"${:,.2f}\".format,\n \"date\": lambda x: \"{:%m/%d/%Y}\".format(pd.to_datetime(x, unit=\"D\"))\n }\n )\n)\n\n total_bill tip sex smoker day time size date\n0 $16.99 $1.01 Female No Sun Dinner 2 02/08/2017\n1 $10.34 $1.66 Male No Sun Dinner 3 02/09/2017\n2 $21.01 $3.50 Male No Sun Dinner 3 02/10/2017\n3 $23.68 $3.31 Male No Sun Dinner 2 02/11/2017\n4 $24.59 $3.61 Female No Sun Dinner 4 02/12/2017\n5 $25.29 $4.71 Male No Sun Dinner 4 02/13/2017\n6 $8.77 $2.00 Male No Sun Dinner 2 02/14/2017\n7 $26.88 $3.12 Male No Sun Dinner 4 02/15/2017\n8 $15.04 $1.96 Male No Sun Dinner 2 02/16/2017\n9 $14.78 $3.23 Male No Sun Dinner 2 02/17/2017\n\n"
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"Evaluation of dyspnoea in a sample of elderly subjects recruited from general practice.",
"\nThe objectives of this study were to investigate the cause of dyspnoea in a sample of elderly individuals and to assess the diagnostic yield of a three-step examination algorithm for the evaluation of dyspnoea paired with a cost analysis. ",
"A total of 152 subjects were examined. ",
"A predefined diagnostic approach in three steps was carried out to find the cause of dyspnoea. ",
"Step 1 included lung spirometry and ECG; step 2 included lung diffusion capacity, echocardiography, haemoglobin and thyroid function; and step 3 included cardiac magnetic resonance imaging, chest radiography and exercise test. ",
"Of 129 subjects with dyspnoea, 68 (53%) had signs of lung disease, 27 (21%) had heart disease, a total of 43 (33%) were obese, 20 (16%) were obese without other causes of dyspnoea and five (4%) had general physical deconditioning. ",
"Twelve per cent had none of the above-mentioned potential causes of dyspnoea. ",
"Steps 1, 1 + 2 and 1 + 2 + 3 revealed a cause of dyspnoea in 39%, 63%, and 73% of subjects respectively. ",
"The cost per diagnosed case at steps 2 and 3 was twice and 3.5 times the cost per diagnosed case at step 1. ",
"In this sample of elderly subjects, a potential cause of dyspnoea was identified in most cases, the most frequent being lung disease followed by heart disease and obesity. ",
"These data shed light on the diagnostic yield that can be expected from a relatively simple diagnostic approach, including the most frequent recommended initial screening tests. ",
"As expected, the incremental nature of this algorithm translated into incremental costs per diagnosis achieved."
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"In situ integration of graphene foam-titanium nitride based bio-scaffolds and microfluidic structures for soil nutrient sensors.",
"\nIt is challenging to integrate porous graphene foam (GF) and GF-based nanocomposites into microfluidic channels and even create microfluidic structures within these materials. ",
"This is because their irregular interior pore shape and geometry, rough exterior surface, and relatively large material thickness make it difficult to perform conventional photolithography and etching. ",
"This challenge has largely hindered the potential of using GF-based materials in microfluidics-based sensors. ",
"Here we present a simple approach to create well-defined flow-through channels within or across the GF-based materials, using a liquid-phase photopolymerization method. ",
"This method allows embedding of a nanocomposite-based scaffold of GF and titanium nitride nanofibers (GF-TiN NFs) into a channel structure, to realize flow-through microfluidic electrochemical sensors for detecting nitrate ions in agricultural soils. ",
"The unique GF-TiN nanocomposite provides high electrochemical reactivity, high electron transfer rate, improved loading capacity of receptor biomolecules, and large surface area, serving as an efficient electrochemical sensing interface with the help of immobilized specific enzyme molecules. ",
"The microfluidic sensor provides an ultralow limit of detection of 0.01 mg L-1, a wide dynamic range from 0.01 to 442 mg L-1, and a high sensitivity of 683.3 μA mg-1 L cm-2 for nitrate ions in real soil solution samples. ",
"The advantageous features of the GF-TiN nanocomposite, in conjunction with the in situ integration approach, will enable a promising microfluidic sensor platform to monitor soil ions for nutrient management towards sustainable agriculture."
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"Y = midpoint.",
"Y - value.",
"Height\nDim rc As New Rectangle(New Point(point.",
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"Y), value.",
"Size)\nrects1.Add(rc)\ntemp.",
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"Width\nPictureBox1.Invalidate()\nno_clicks += 1\n\nnPoint(i) = New Point() {point.",
"X, point.",
"Y}\nnSize(i) = New Size(value.",
"Width, value.",
"Height)\ni += 1\n\nI have a point variable which is a drawing point of a rectangle.",
"I have declared another array of point as DIM npoint() as point so that i can assign the rectangles value to it.",
"Problem is it is saying value of integer cannot be converted into system.srawing.point.",
"\n\nA:\n\nThis:\nnPoint(i) = New Point() {point.",
"X, point.",
"Y}\n\nis obviously wrong. ",
"Judging by this:\nnSize(i) = New Size(value.",
"Width, value.",
"Height)\n\nwhich is the line that follows it, that first line should be this:\nnPoint(i) = New Point(point.",
"X, point.",
"Y)\n\nThat said, Point and Size are both value types, so there's no point to creating new instances from parts of existing instances. ",
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"1. ",
"Field of the Invention\nThe present invention relates to an information processing system, an information processing device, and an authentication information management method.",
"\n2. ",
"Description of the Related Art\nIn recent years, network systems (web systems) such as a LAN (local area network) and an intranet are set up to enable a client terminal connected to a network to use various web services that are provided by a web server using a web browser.",
"\nIn such a web system, a user typically inputs a login name (account) and a password upon requesting authentication by an authentication service from a web browser. ",
"The web system passes the login name and password input by the user to the authentication service and provides an authentication ticket to a web service so that the user may be allowed to use a function of the web service (see e.g., Japanese Laid-Open Patent Publication No. ",
"2007-53580).",
"\nA client making a request to a server for a service such as data conversion or data processing provides the server with a valid authentication ticket issued in response to a user login operation and uses a function of the server under the authority of the logged in user. ",
"Note that some processes such as data conversion or data processing that are executed by the server may require a long process time.",
"\nHowever, the client using a function of the server under the authority of the logged in user may be unable to continue using the function of the server once the user logs out.",
"\nAccordingly, a technique is desired for enabling continuation of a process executed under the authority of a user even after the user logs out."
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"Friday, January 30, 2009\n\nI've posted about epigrams before, including this one by Samuel Taylor Coleridge:\n\nWhat is an Epigram? ",
"A dwarfish whole;Its body brevity, and wit its soul.",
"\n\nBut I was feeling the need for something short and to the point today, so epigrams seem just the thing. ",
"An epigram need not rhyme, although poetic ones often do. ",
"An epigram is a short, clever, usually witty statement that is memorable. ",
"Here is an epigram from Benjamin Franklin, writing as Poor Richard:\n\nEarly to bed and early to risemakes a man healthy, wealthy and wise\n\nFranklin used repetition (early to), internal rhyme (healthy and wealthy) and alliteration (wealthy and wise) as well as end-rhyme, virtually guaranteeing that this would be memorable after only one hearing.",
"\n\nThis poem is a parody poem of a well-known \"instructional\" poem at that time, \"Against Idleness and Mischief\" from a book called Divine Songs for Children by Isaac Watts, a noted Nonconformist hymnwriter. ",
"It begins \"How doth the little busy bee/Improve each shining hour,/And gather honey all the day/From every opening flower!\", ",
"and ends with exhortations to be diligent and avoid Satan. ",
"You can read the full text here. ",
"Carroll's poem is still widely anthologized for children these days, whereas the Watts poem has fallen out of favor.",
"\n\nI've been hearing the Overture from Le Nozze di Figaro (The Marriage of Figaro) in my head, and think I will go put on the CD of one of my favorite of his symphonies, \"The Haffner\", Symphony No. ",
"35, K. 385, and then spend some quality time with the copy of The Annotated Alice: The Definitive Edition that hubby got me for the holidays.",
"\n\nTuesday, January 27, 2009\n\nBelieve me when I say that the title of this post is 100% accurate as to content today. ",
"The long story version is that I've just read an interview (from which I will be quoteskimming come Sunday), in which Keats was mentioned. ",
"And I thought (as I so often do) \"F*ck me, I love Keats\", which happens to be a line spoken by Hugh Grant's character in the first Bridget Jones's Diary movie, but also happens to be true. ",
"My next thought was \"Why not share some Keats today?\", ",
"and I fumbled about and came up with this sonnet, usually referred to as \"Bright Star\". ",
"But during the reading of it, the little man in the upstairs fileroom started to clamor about Robert Frost and choral music since, twice upon a time, I've been in choruses that have performed choral settings of certain Frost poems (known as Frostiana, music by Randall Thompson, by the by).",
"\n\nToday, you get Keats's Shakespearian sonnet - a love poem that is entirely swoon-worthy on its own, plus Frost's \"Choose Something Like a Star\" in both poem and choral form. ",
"Plus discussion in analysis. ",
"I am totally the Crazy Eddie of poetry this morning: Everything must go!",
"\n\nBright Starby John Keats\n\nBright star, would I were stedfast as thou art—Not in lone splendour hung aloft the nightAnd watching, with eternal lids apart,Like nature's patient, sleepless Eremite*,The moving waters at their priestlike taskOf pure ablution round earth's human shores,Or gazing on the new soft-fallen maskOf snow upon the mountains and the moors—No— yet still stedfast, still unchangeable,Pillow'd upon my fair love's ripening breast,To feel for ever its soft fall and swell,Awake for ever in a sweet unrest,Still, still to hear her tender-taken breath,And so live ever— or else swoon to death.",
"\n\n*Eremite: a hermit, particularly a religious recluse who lives alone in the wilderness\n\nDiscussion of the Keats poem\n\nIn this single-sentence sonnet, Keats fixes his attention on a star in the sky, wishing that he had the ability to be as steadfast and watchful of the woman he loved (almost certainly Fanny Brawn at this point in his life). ",
"Keats is such a drama queen that he wants to lie with her forever, \"pillowed on her breast\" or else swoon to death. ",
"The poem, which is written using the Shakespearian sonnet form, uses iambic pentameter and the ABABCDCDEFEFGG rhyme scheme. ",
"The volta or \"turn\" in the poem comes at the start of line 9, when Keats turns his attention from the star up in the sky to describing how he wishes to be able to have that star's immortal constancy in order to stay with his beloved.",
"\n\nNote how Keats begins the poem by addressing the star in the sky, but when he reaches the volta, he pretty much ceases to address the star, and talks to himself. ",
"Were this a performance on the stage, the actor might start his recitation by looking up to the star and gesturing, but he would almost undoubtedly turn his attention from the star to a more inward performance by the start of the 9th line, or else he might have a conveniently placed woman on a chaise lying about with whom to conclude the recitation.",
"\n\nI should note that this poem is sometimes referred to as \"the last sonnet\", because it was for quite a long time believed to be the final sonnet Keats wrote before his death. ",
"Some dispute as to whether that is correct exists, but it does appear to be one of the last poems he completed before his death, even if it was drafted earlier than first believed.",
"\n\nChoose Something Like a Starby Robert Frost\n\nO Star (the fairest one in sight),We grant your loftiness the rightTo some obscurity of cloud—It will not do to say of night,Since dark is what brings out your light.",
"Some mystery becomes the proud.",
"But to the wholly taciturnIn your reserve is not allowed.",
"Say something to us we can learnBy heart and when alone repeat.",
"Say something! ",
"And it says, 'I burn.",
"'But say with what degree of heat.",
"Talk Fahrenheit, talk Centigrade.",
"Use language we can comprehend.",
"Tell us what elements you blend.",
"It gives us strangely little aid,But does tell something in the end.",
"And steadfast as Keats' Eremite,Not even stooping from its sphere,It asks a little of us here.",
"It asks of us a certain height,So when at times the mob is swayedTo carry praise or blame too far,We may choose something like a starTo stay our minds on and be staid.",
"\n\nThis 25-line poem is written in iambic tetrameter (four iambic feet per line), and uses a complicated nested rhyme scheme (AABAABCBDEDEFGGFGHIIHJKKJ), although one could fairly characterize the final eight lines as stanzas set in envelope rhyme. ",
"The starting 17 lines use a nested rhyme technique that is quite similar to what T.S. Eliot used in \"The Love Song of J. Alfred Prufrock\", posted here the other day, and rest assured, that is no coincidence.",
"\n\nDiscussion of Frost's poem\n\nFrost specifically references Keats's poem within his, both by addressing a star in the first place and by specifically talking about the steadfastness of the star and \"Keats' Eremite\". ",
"Were this poem to be performed on the stage, there'd be no fainting couch around, and the speaker would essentially be arguing with the star for a good 17 lines. ",
"Because it's not until the final 8 lines of this poem that Frost stops addressing the star directly. ",
"Yet there, at the start of the 18th line - \"And steadfast as Keats' Eremite\" - is a volta, where the poet stops hollering at the star and turns to his audience to address them directly.",
"\n\nNow, Frost's poem is actually quite lovely on its surface. ",
"It purports to be about a star in the night sky, and the speaker asks it questions, seeking answers, and the star tells us precious little about itself. \"",
"It says 'I burn'./But say with what degree of heat./Talk Fahrenheit. ",
"Talk Centigrade. ",
"Tell us what elements you blend.\" ",
"The speaker wants facts and specifics, something he can wrap his head around.",
"\n\nBut, to quote Eliot from the other day, \"That is not it at all,/That is not what [he] meant, at all.\" ",
"Frost, you see, told at least one of his classes that the \"star\" to which the poem is addressed was a contemporary star in the world of poetry: T.S. Eliot. ",
"Frost is being his usual cantankerous self, criticizing Eliot for his highbrow ways and for combining elements (Sanskrit, Hebrew, mythology, pop culture and more) in his poems in a way that many average folk found incomprehensible. ",
"In \"On Extravagance: A Talk\", Frost doesn't straightaway acknowledge Eliot, but gets around to it. ",
"In discussing this poem, he says\n\nBy that star I mean the Arabian Nights or Catullus or something in the Bible or something way off or something way off in the woods, and when I've made a mistake in my vote. (",
"We were talking about that today. ",
"How many times we voted this way and that by mistake.)",
"\n\nAnd then see little personal things like this.... (Do you know the real motivation probably of it all...? ",
"Take the one line in that, \"Some mystery becomes the proud.\" ",
"Do you know where I got that? ",
"Out of long efforts to understand contemporary poets. ",
"You see, let them be a mystery. ",
"And that's my generosity--call it that! ",
"If I was sure they meant anything to themselves it would be all right.)",
"\n\nOn the one hand, Frost is challenging other contemporary poets to speak more plainly in their writing. ",
"On the other, if one wants to be \"generous\" (and I sincerely doubt that Frost was being generous, but let's play along): he's saying at the start of the poem that Eliot is pretentious and proud and obscure. ",
"That's Eliot's choice and he's welcome to it. ",
"But Frost goes on to challenge Eliot (and possibly other contemporary writers) to \"say something to us we can learn/by heart and when alone repeat.\" ",
"After flinging out the challenge a second time, the star says only \"I burn\", and doesn't elucidate further. ",
"Frost says, in essence, that that is a telling response. ",
"Dispensing with the clever use of intertwined rhyme that Eliot frequently uses, Frost turns to the reader for the last eight, settled lines. ",
"Referencing Keats, a master of the sonnet form, and using what is essentially two quatrains of envelope rhyme, Frost settles into a more staid form himself to present the argument that choosing his sort of poetry (more formal in the traditional sense than that of some of his contemporaries), one can stay their minds \"and be staid.\"",
"\n\nHe's a curmudgeon, but man can he turn a phrase.",
"\n\nTo hear a lovely choral setting of the Frost poem sung by the Mormon Tabernacle Choir, and with amazing images from the Hubble space telescope, check out this video on YouTube. (",
"Sorry - the owner disabled embedding so I couldn't make this easier on you.",
"\n\nFriday, January 23, 2009\n\nLast week's Poetry Friday host, Karen Edmisten, posted the poem \"The Writer\" by Richard Wilbur, noting that she usually posts it about every four months. ",
"Today, I'm reposting one of my all-time favorite poems (I have many, so don't ask me to rank it), \"The Love Song of J. Alfred Prufrock\" by T.S. Eliot.",
"\n\nI posted the poem once before, back in September of 2006, and I think it's worth posting again, particularly because it ties in with Praise Song for the Day from the inauguration the other day. ",
"How so, you ask? ",
"Well, on Wednesday evening, Elizabeth Alexander was a guest on The Colbert Report, and he quoted from Prufrock: \"I hear the mermaids singing, each to each,/I do not think they will sing to me.\" ",
"You can check out the whole interview below, and, geek that I am, I transcribed the whole thing. ",
"Read the transcript, watch the interview, and read the analysis of the poem below.",
"\n\nLet us go then, you and I,When the evening is spread out against the skyLike a patient etherised upon a table;Let us go, through certain half-deserted streets,The muttering retreatsOf restless nights in one-night cheap hotelsAnd sawdust restaurants with oyster-shells:Streets that follow like a tedious argumentOf insidious intentTo lead you to an overwhelming question . . ",
".Oh, do not ask, \"What is it?\"Let us go and make our visit.",
"\n\nIn the room the women come and goTalking of Michelangelo.",
"\n\nThe yellow fog that rubs its back upon the window-panes,The yellow smoke that rubs its muzzle on the window-panesLicked its tongue into the corners of the evening,Lingered upon the pools that stand in drains,Let fall upon its back the soot that falls from chimneys,Slipped by the terrace, made a sudden leap,And seeing that it was a soft October night,Curled once about the house, and fell asleep.",
"\n\nAnd indeed there will be timeFor the yellow smoke that slides along the street,Rubbing its back upon the window-panes;There will be time, there will be timeTo prepare a face to meet the faces that you meet;There will be time to murder and create,And time for all the works and days of handsThat lift and drop a question on your plate;Time for you and time for me,And time yet for a hundred indecisions,And for a hundred visions and revisions,Before the taking of a toast and tea.",
"\n\nIn the room the women come and goTalking of Michelangelo.",
"\n\nAnd indeed there will be timeTo wonder, \"Do I dare?\" ",
"and, \"Do I dare?\"Time to turn back and descend the stair,With a bald spot in the middle of my hair —[They will say: \"How his hair is growing thin!\"]My morning coat, my collar mounting firmly to the chin,My necktie rich and modest, but asserted by a simple pin —[They will say: \"But how his arms and legs are thin!\"]Do I dareDisturb the universe?In a minute there is timeFor decisions and revisions which a minute will reverse.",
"\n\nFor I have known them all already, known them all: —Have known the evenings, mornings, afternoons,I have measured out my life with coffee spoons;I know the voices dying with a dying fallBeneath the music from a farther room.",
"So how should I presume?",
"\n\nAnd I have known the eyes already, known them all—The eyes that fix you in a formulated phrase,And when I am formulated, sprawling on a pin,When I am pinned and wriggling on the wall,Then how should I beginTo spit out all the butt-ends of my days and ways?And how should I presume?",
"\n\nAnd I have known the arms already, known them all—Arms that are braceleted and white and bare[But in the lamplight, downed with light brown hair!]It is perfume from a dressThat makes me so digress?Arms that lie along a table, or wrap about a shawl.",
"And should I then presume?And how should I begin?",
"\n\n. . . . .",
"\n\nShall I say, I have gone at dusk through narrow streetsAnd watched the smoke that rises from the pipesOf lonely men in shirt-sleeves, leaning out of windows? . . .",
"\n\nI should have been a pair of ragged clawsScuttling across the floors of silent seas.",
"\n\n. . . . .",
"\n\nAnd the afternoon, the evening, sleeps so peacefully!Smoothed by long fingers,Asleep . . . ",
"tired . . . ",
"or it malingers,Stretched on the floor, here beside you and me.",
"Should I, after tea and cakes and ices,Have the strength to force the moment to its crisis?But though I have wept and fasted, wept and prayed,Though I have seen my head [grown slightly bald] brought in upon a platter,I am no prophet—and here's no great matter;I have seen the moment of my greatness flicker,And I have seen the eternal Footman hold my coat, and snicker,And in short, I was afraid.",
"\n\nAnd would it have been worth it, after all,After the cups, the marmalade, the tea,Among the porcelain, among some talk of you and me,Would it have been worth while,To have bitten off the matter with a smile,To have squeezed the universe into a ballTo roll it toward some overwhelming question,To say: \"I am Lazarus, come from the dead,Come back to tell you all, I shall tell you all\"—If one, settling a pillow by her head,Should say: \"That is not what I meant at all.",
"That is not it, at all.\"",
"\n\nAnd would it have been worth it, after all,Would it have been worth while,After the sunsets and the dooryards and the sprinkled streets,After the novels, after the teacups, after the skirts that trail along the floor—And this, and so much more?—It is impossible to say just what I mean!But as if a magic lantern threw the nerves in patterns on a screen:Would it have been worth whileIf one, settling a pillow or throwing off a shawl,And turning toward the window, should say:\"That is not it at all,That is not what I meant, at all.\"",
"\n\n. . . . .",
"\n\nNo! ",
"I am not Prince Hamlet, nor was meant to be;Am an attendant lord, one that will doTo swell a progress, start a scene or two,Advise the prince; no doubt, an easy tool,Deferential, glad to be of use,Politic, cautious, and meticulous;Full of high sentence, but a bit obtuse;At times, indeed, almost ridiculous—Almost, at times, the Fool.",
"\n\nI grow old . . . ",
"I grow old . . ",
".I shall wear the bottoms of my trousers rolled.",
"\n\nShall I part my hair behind? ",
"Do I dare to eat a peach?I shall wear white flannel trousers, and walk upon the beach.",
"I have heard the mermaids singing, each to each.",
"\n\nI do not think that they will sing to me.",
"\n\nI have seen them riding seaward on the wavesCombing the white hair of the waves blown backWhen the wind blows the water white and black.",
"\n\nWe have lingered in the chambers of the seaBy sea-girls wreathed with seaweed red and brownTill human voices wake us, and we drown.",
"\n\n*If I thought my answer were givento anyone who would ever return to the world,this flame would stand still without moving any further.",
"But since never from this abysshas anyone ever returned alive, if what I hear is true,without fear of infamy I answer you. ",
"from Inferno by DanteStephen Colbert interview with Elizabeth Alexander, Wednesday, January 21, 2009\n\nSC: I have a general question about poetry. ",
"It's something that worries me. ",
"Poems aren't true, are they?",
"\n\nEA: Well. . .",
"\n\nSC: They're made up, right? ",
"Like, they're made up. ",
"'Cause I recently read this thing called \"The Love Song of J. Alfred Prufrock\", which is about a guy in his mid-40s, like I am, and he's facing his mortality and he's got a sense that no matter what he's achieved in his life, he's never really gonna be great.",
"\n\nEA: Do you like that poem?",
"\n\nSC: That's not true, right? ",
"That's not true?",
"\n\nEA: It's not true in the sense that the newspaper is true, it's not true in the sense that journalism is . . .",
"\n\nSC: Well the newspaper is never true. ",
"Have you read the New York Times?",
"\n\nEA: Well, uh, meant to be true. ",
"It's not true in the sense of the strictly factual, but a poem should be in some way emotionally true, true to the language that it has, so that's why, for example, \"J. Alfred Prufrock\" might speak to you because there's something in the poem that resonates, that feels true to you, and that's how people connect with poems.",
"\n\nSC: He says that \"I have heard the mermaids singing each to each, I do not think they sing to me\". ",
"They're, are - They're still singing to me though, aren't they?",
"\n\nEA: (Laughs) They are, if you want them to be, but . . .",
"\n\nSC: Desperately.",
"\n\nEA: I know, I know, I know. ",
"But that's your experience with the poem, so the point is that phrase, that language, is . . . ",
"off enough, it is decentered enough from the way that you hear language every day that it makes you stop and think about what that could mean, so . . .",
"\n\nSC: Okay, let's talk about meaning for a second, let's talk about meaning for a second, okay? ",
"Okay. ",
"Metaphors, okay? ",
"What's the difference between a metaphor and a lie?",
"\n\nEA: Laughs\n\nSC: Okay, because, you know, 'I AM THE SUN, YOU ARE THE MOON!' ",
"That's a lie, you're not the moon. ",
"I'm not the sun, okay? ",
"What's the difference between a metaphor and a lie?",
"\n\nEA: Well, that was both a metaphor and a lie. ",
"So, the two are not necessarily exclusive. ",
"A metaphor is a way of using language where you make a comparison to let people understand something as it relates to something else, and that's how we use the language to increase meaning.",
"\n\nSC: Well, why not just go, say, say what you mean, instead of dressing things up in all this flowery language like, you know, the great romantic poets, you know 'Shall I compare thee to a summer's day?' ",
"Why not just say 'You are hot, let's do it'?",
"\n\nEA: Laughs\n\nSC: 'Let's get it on! ",
"Okay? ",
"I got a mountain in my pants', and that is not a metaphor. ",
"That is not a lie!",
"\n\nEA: But that's a metaphor.",
"\n\nSC: No it's not. ",
"It's not, lady. ",
"It is not at all.",
"\n\nEA: (Laughs and fans herself)\n\nSC: Now listen, listen. ",
"Let's go to your poem\n\nEA: Line by line\n\nSC: Beautiful, beautiful. ",
"But no 'explicate it line by line', no. ",
"The poem is \"Praise Song for the Day\". ",
"Let's talk about it. ",
"Is this a \"praise song\" for the day, or are we praising song for the day? ",
"Is praise there a command?",
"\n\nEA: No. ",
"It is a \"praise song\", which is a form of poetry, an ode if you will, that is often written in various West African countries, it's made its way here. ",
"It's a way of praising, not to say \"gold star\", \"good job\", but rather to name something that we take joy from, that we rejoice in, so that's what a praise song is. ",
"It's a form.",
"\n\nSC: And you called it \"an occasional poem\"\n\nEA: Yes\n\nSC: What is an occasional poem, is it a \"sometimes\" poem?",
"\n\nEA: An occasional poem is a poem written for an occasion. ",
"So obviously yesterday's occasion was the inaugural. ",
"It could be for a graduation, it could be for a wedding, any occasion.",
"\n\nSC: So its context is associated specifically to that event.",
"EA: Exactly.",
"\n\nSC: Like the lyrics to \"Mad Max Beyond Thunderdome\"\n\nEA: Well there you go.",
"\n\nSC: We don't need another hero. ",
"We don't need to know the way home. ",
"All we want is life beyond Thunderdome. ",
"It really only makes sense when you're at Thunderdome.",
"\n\nEA: And you see, that's a very good example of the occasional poem that actually doesn't resonate beyond the occasion. ",
"So if you look at those words, what you want to do is mark the occasion, but write words that will last afterwards, and be useful afterwards in some kind of way.",
"\n\nSC: Your poem was marked by, I would say, the commonality of experience that you were naming in it. ",
"Why not soaring rhetoric? ",
"Why not just light up the crowd with, you know, with one of your metaphor lies, like \"Barack Obama is a blazing star of hope\"? ",
"Why not that? ",
"Why not really, really gild him with these words?",
"\n\nEA: Well, I don't think he needs gilding, and I think that actually what's been so powerful about his campaign and now, hopefully, his presidency, is that even though he is the leader, it's about the people. ",
"It's about many people feeling invested in changing this country and looking to something better, and working for it. ",
"So, he said, \"it's not about me, it's about us.\"",
"\n\nSC: If poetry is in some way about economy of language, could I suggest a poem that might have gotten to the heart of it a little quicker?",
"\n\nEA: (Nodding) Please, please do.",
"\n\nSC: Hickory dickory dock, we elected a guy named Barack. ",
"Think about it?",
"\n\nThe short-form explanation of this poem is that the speaker, Mr. J. Alfred Prufrock, who lives in a London of the early 20th century polluted with sulfurous smog, is getting dressed to go to a party, where he will see a lady to whom he would like to declare his love. ",
"He talks to his reflection as he gets ready to go, projecting what his evening will be like in the rooms where \"women come and go,/talking of Michelangelo.\" ",
"And he worries. ",
"What if the woman turns him down, or, worse yet, mocks him? ",
"Rather than face the possibility of rejection, he opts not to venture out at all. ",
"He stays in his rooms, facing a future full of regret wondering whether he dares to eat a peach, growing old and rolling his trousers at the bottoms. ",
"Poor guy.",
"\n\nPrufrock misses his chance to declare his feelings, and perhaps find real love, because he cannot bring himself to put himself out there. ",
"As a good friend once pointed out, he's worried about both his emotional and perhaps also his literal impotence, as when he asks, \"will I have the strength to force the moment to its crisis?\" ",
"He's a very careful man. ",
"You can tell this from the line, \"I have measured out my life with coffee spoons.\" ",
"A coffee spoon, for those of you who do not carefully pay attention to silverware, is significantly smaller than a teaspoon. ",
"He is a man who worries. ",
"What will people think of his appearance? ",
"The insect metaphor - that if he declares himself, he will be an insect, formulated (anesthetized) and pinned to the wall on display - is brilliant. ",
"How clinical, how horrible, to think of being a specimen pinned up for all to see and discuss.",
"\n\nThe poem is beautifully unified by quite a lot of end rhyme. ",
"It doesn't follow any obvious, fixed pattern, yet it is there, and by being there, the poem feels much like music when read aloud. ",
"In the end, I find myself feeling sorry for Prufrock and his missed opportunities. ",
"As I said in 2006, \"the pity one feels for Prufrock is tempered by disdain for his decision not to act. ",
"Because it becomes clear that when we don't act, it's not just inertia (an object at rest remaining at rest). ",
"Because we are not objects, we are subjects -- we act (or choose not to). ",
"And so, for today, I will not measure out my day with a coffee spoon. ",
"I will not roll my trousers. ",
"I will go out into the day, and greet it. ",
"And I will hope it greets me back.\"",
"\n\nFriday, January 16, 2009\n\nToday, another poem by Puritan poet, Anne Bradstreet, who emigrated to Ipswich, Massachusetts in the 1630s with her husband, Simon Bradstreet, a Cambridge graduate who eventually became Governor of the Colony of Massachusetts. ",
"Anne and her husband had eight children, despite Anne suffering from some paralysis, possibly as a lasting effect of smallpox. ",
"She wrote in the Elizabethan tradition, as her mastery of rhymed couplets and iambic pentameter show. ",
"A volume of poems was published in England during her lifetime. ",
"Shortly after her death, a volume of her work (containing today's poem) was published in America. ",
"A third collection of religious poems was published in the 19th century.",
"\n\nI started reading more Bradstreet after a dear friend confessed a massive crush on Anne, and I completely see why. ",
"Happy birthday, B!",
"\n\nToday's poem is a beautiful love poem, methinks. ",
"I don't believe I know too many women who are as content in their marriages as Anne seems to have been, gauging from this poem, but it is decidedly something to aspire to.",
"\n\nTo My Dear and Loving Husbandby Anne Bradstreet\n\nIf ever two were one, then surely we.",
"If ever man were loved by wife, then thee;If ever wife was happy in a man,Compare with me ye women if you can.",
"I prize thy love more than whole mines of gold,Or all the riches that the East doth hold.",
"My love is such that rivers cannot quench,Nor ought but love from thee give recompense.",
"Thy love is such I can no way repay; The heavens reward thee manifold, I pray.",
"Then while we live, in love let's so persever,That when we live no more we may live ever.",
"\n\nThursday, January 15, 2009\n\nOn one of my trips to the marvelous Children's Book World with in the fall, I purchased a small novel in verse. ",
"And I do mean small: it's approximately 5 x 7 inches, and it's very slim at 112 pages. ",
"Each page contains a lot of white space due to the nature of the poems contained thereon, making it a quick read for the adept reader, and exceedingly appealing to the reluctant one.",
"\n\nT4 was the name of the program that existed in Nazi Germany from about 1939 to 1943 that was designed to eliminate disabled individuals from German society. ",
"It took its name from the address of its headquarters, Tiergartenstrasse 4. ",
"Using a series of poems attributed to a fictional deaf Catholic girl named Paula Becker, LeZotte explains not only the threat of the T4 program and the terror felt by its intended victims, but also, on a more personal level, some of the challenges of being deaf, particularly in the first half of the 20th century, when some people still thought it could be cured by filling a person's ears with hot wax.",
"\n\nI found the book utterly engrossing and quite well done. ",
"The T4 program is one aspect of the Holocaust that has not been given nearly enough space in books, and this novel does much to rectify that situation (as well as mentioning the efforts to eradicate the gypsy population, as well as the Jews).",
"\n\nThe author is herself deaf, which doubtless informs her writing. ",
"The first poem in the book is not so much a part of the text, but is, instead, an introduction (although it was not labelled as such, and may therefore confuse some younger readers).",
"\n\nHere's that opening poem, which gives you an idea of the masterful poems found in the book, and establishes what is going on, while referencing its place among other poems and poets, from William Blake to William Carlos Williams.",
"\n\nHear the Voice of the Poetby Ann Clare LeZotte\n\nHere the voice of the poet!I see the past, future, and present.",
"I am Deaf, but I have heardThe beauty of song\n\nAnd I wish to share it withYoung readers.",
"A poem can be simple,About a cat or a redWheelbarrow.",
"\n\nOr it can illuminate the livesOf people who lived, loved,And died. ",
"You can makePeople think or feel\n\nFor other people, if youWrite poetry. ",
"In T4, the factsAbout history are true, andMy characters tell the story.",
"\n\nThe simple, spare poems convey so much, without infusing too much emotion into the text itself; that part is left to the reader to infer. ",
"Here, for example, is a poem entitled \"Doctor Bouhler\", about one of the directors of T4:\n\nDoctor Bouhler\n\nInsisted the deathsShould bePainless.",
"\n\nHe didn't wantThe patientsTo know what wasGoing to happen.",
"\n\nBut they died ofLethal injectionAnd starvation.",
"\n\nSimple, clear expression of the facts is, in this instance, more upsetting than a poem describing a reaction to that news, I think.",
"\n\nA well-done verse novel, recommended for middle grade readers, elementary and middle school libraries, and public libraries, as well as for anyone with a Holocaust collection.",
"\n\nFriday, January 09, 2009\n\nWhether true or not, it's been reported that Alfred, Lord Tennyson once said that others may have written better poetry than he, but none had written poems that sounded better. ",
"He may be on to something there, because he was certainly (as I have commented before) a master of assonance (repetition of vowel sounds), alliteration (repetition of consonants) and, well, repetition, as well as onomatopoeia (use of a word that imitates the sound it describes). ",
"And he crafted poems that were designed for performance, or at the very least, to be read aloud. ",
"All those devices he uses make his lines easier to memorize, and enable a speaker to sound like a true orator.",
"\n\nToday's poem is actually a small section of a long work entitled The Princess. ",
"The premise of The Princess is that a group of seven college students, on holiday, get involved in a discussion with a few females about issues such as women's rights - and, more specifically, the education of women - as well as the need for education of the poor. ",
"Tennyson (quintessential Victorian that he was) felt the need to make clear that women should not be treated the same as men, nor should they aspire to the same careers or positions as men, and his poem was criticized as anti-feminist. ",
"That said, The Princess does champion the notion of teaching women history, science (as then understood, which included phrenology, which is, if memory serves, how to understand human nature by reading the bumps on one's skull), socialism (then a word for learning how to understand society and the roles within it), math, geography and languages. ",
"And if one reads some of it, one gets the idea that Tennyson realized that women were largely treated like cattle - and may not have approved. ",
"But truly, I digress.",
"\n\nToday's excerpt from The Princess is usually referred to by its first line. ",
"You should know that its my opinion that this is an erotic, sexual poem, a position which I'll set forth in full. ",
"But first the poem, and then the conversation.",
"\n\nNow Sleeps the Crimson Petal, Now the Whiteby Alfred, Lord Tennyson\n\nNow sleeps the crimson petal, now the white;Nor waves the cypress in the palace walk;Nor winks the gold fin in the porphyry* font:The fire-fly wakens: waken thou with me.",
"\n\nNow droops the milkwhite peacock like a ghost,And like a ghost she glimmers on to me.",
"\n\nNow lies the Earth all Danaë** to the stars,And all thy heart lies open unto me.",
"\n\nNow slides the silent meteor on, and leavesA shining furrow, as thy thoughts in me.",
"\n\nNow folds the lily all her sweetness up,And slips into the bosom of the lake:So fold thyself, my dearest, thou, and slipInto my bosom and be lost in me.",
"\n\n*porphyry: a purple-red stone containing crystals throughout, coming from a quarry in the eastern desert of Egypt that was rediscovered in 1823 (during Tennyson's lifetime).",
"\n\n**Danaë: a character in Greek mythology, who, after being locked up in a subterranean chamber made of bronze by her father, was euphemistically \"visited\" by Zeus in a shower of gold rain (no - I do not want to call it a \"golden shower\"), thereby conceiving Perseus. ",
"Those of you interested in astronomy are likely aware that the Perseid meteor showers derive their name from Perseus, because they appeared to come from the constellation of Perseus.",
"\n\nLet's look first at form\n\nA first reading of this poem gives you the impression that it is written in rhyme, but that is not actually correct. ",
"In fact, the end rhyme that you see is actually just the word \"me\", repeated. ",
"The poem is written in blank verse (unrhymed iambic pentameter), but the frequent use of the end-word \"me\" makes it seem like it rhymes, as does the near-rhyme in the final four-line stanza of the words \"up\" and \"slip\".",
"\n\nThe poem is broken into five stanzas: the first and fifth contain four lines, and the middle three contain two lines each.",
"\n\nAnalysis of the poem's structure and meaning\n\nWhy the five stanzas? ",
"Well, the first and fifth stanzas are bookends, I think. ",
"The first sets the scene (it's evening! ",
"in a garden!), ",
"but it also makes clear that this poem is addressed to someone (whether male or female is unclear, but the notion that it is a lover, or a person that the speaker would like to have as a lover, comes clear as we go along).",
"\n\nThe second stanza talks about two things: seeing a white peacock in the night garden, and how the woman he sees glimmers in the moonlight/starlight like a ghost. ",
"It's an interesting construction, because he is talking to a person, but in this stanza, it sounds as if he's talking about her instead.",
"\n\nThe third makes another comparison, and this is where things get really, really interesting. ",
"In the first line, he says \"Now lies the Earth all Danaë to the stars\". ",
"Taking into account the story of Danaë that I summarized above, he means that the Earth is open to being (for want of a better term) fertilized by the stars in the night sky. ",
"He quickly follows that with \"And all thy heart lies open unto me.\" ",
"Romantic, definitely. ",
"Erotic, too, if you ask me.",
"\n\nThe fourth makes both the romantic and the sexual intent clear in my books (not that I've seen a lot of discussion of this poem as having anything to do with sex, but that's only because historically, folks liked to give Victorians credit for not talking about sex. ",
"As we know, however, that's not really the case, and many \"repressed Victorians\" were horndogs, or at least keen enough observers of human nature to know what was what.) ",
"That fourth stanza again? \"",
"Now slides the silent meteor on, and leaves /A shining furrow, as thy thoughts in me.\" ",
"The sliding meteor and shining furrow, in conjunction with the earlier reference to Danaë is not, I would argue, accidental, but is again a sexual reference. ",
"That said, it's also a reference to the Perseid meteor showers, which rain down on the earth from \"the stars\", with tails that could appear to leave (for the moment) a furrow in the sky. ",
"Going back to the reference to Danaë, however, those meteors come streaking toward earth like golden rain, fertilizing their target. ",
"In this stanza, it's the poets thoughts that have been fertilized. (",
"Again, this works on two levels - he's thinking about the unknown female subject of the poem, but he's also thinking fertile thoughts, if you catch my drift.) ",
"Also? ",
"Traditionally the word \"furrow\" goes with \"plowing\" and \"plowing\" is a euphemism for intercourse. ",
"So.",
"\n\nThe fifth stanza is the closing bookend - four lines for additional weight, to balance the four that started the poem and set the scene. ",
"Here's the payoff, as it were (pun intended). ",
"In the first two lines, he talks of how the water lily folds \"all her sweetness up/And slips into the bosom of the lake\". ",
"First off, water lilies don't really submerge, to my knowledge, and given the keen interest in botany that prevailed during Victorian times, Tennyson probably knew that, which means that he was taking license for a reason. ",
"That reason? ",
"To show a complete yielding of the female, who is not merely submerged, but completely possessed by the the lake. ",
"The final two lines are spoken directly to the person who was addressed at the close of the first stanza:\n\nSo fold thyself, my dearest, thou, and slip Into my bosom and be lost in me.",
"\n\nWow. ",
"So lovely. ",
"And still, erotic. ",
"And quite possibly a reference to the notion of two becoming one, although with the phrase \"be lost in me\", it seems to intimate a sense of sexual abandon, I think.",
"\n\nI love this poem for its surface beauty, for its gorgeous word choices, and for the way it sounds when you read it aloud. ",
"And I love it even more for its erotically charged imagery and secret meaning. ",
"How 'bout you?",
"\n\n`Twas brillig, and the slithy tovesDid gyre and gimble in the wabe;All mimsy were the borogoves,And the mome raths outgrabe.",
"\n\nI did not learn this poem as a child, which is a shame. ",
"Also, a bit of a shock, since as an adult I learned that it is one of my mother's all-time favorite poems. ",
"I never even knew she had favorite poems!",
"\n\nThe poem opens and closes with the same stanza: what starts as an introduction forms almost a benediction at the end. ",
"Each stanza is cross-rhymed (ABAB). ",
"The first three lines of each stanza have 8 syllables (4 roughly iambic feet), and the fourth line has 6 (3 feet). ",
"The poem was intended by its author to be a condemnation/satire of pretentious poetry, a poke at literary critics, and a sort of warning along the lines of \"how not to write a poem\", but it quickly earned its own reputation, and is one of the best-known and widely-referenced poems in the English language (to the extent that all those portmanteau and nonsense words count as English).",
"\n\nWhy this poem rocks for kids\n\nDuring school visits where I've come in to teach kids about poetry, I frequently open my presentation by reciting (nay, performing) this poem from memory. ",
"I do this for several reasons: 1) it's fun; 2) I love this poem; 3) nothing grabs kids' attention more quickly than launching into something that makes them pause long enough to say (or think) \"what the hell?\", ",
"because when they pause, they fall silent and start listening and paying attention, trying to sort things out - it's so much more fun (and interesting) than opening with \"now boys and girls, please quiet down so I can lecture you\"; and 4) this poem makes kids feel really smart.",
"\n\nBut wait, Kelly, you say - what's with that last one? ",
"Didn't you just acknowledge how confusing and strange this poem is? ",
"You betcha - and yet, if you sit and listen to it all the way through, you can sort out the story. ",
"I have confidence in you, you see, because kids as young as first grade have sorted this one out for me. ",
"Now, not all first and second graders get it all the way right, but as a whole, a class of kids will totally get this poem, which has lots of action (some of it mildly horrifying, which is better still). ",
"And they adore the wordplay in it. ",
"In some cases, the younger kids are better with the portmanteau words and nonsense words in the poem than older kids or grownups are. ",
"My theory is that this is so because there are just so many, many words that they don't yet know the meaning of that they don't spend as much time thinking \"hey! ",
"that's not a real word!\" ",
"Also? ",
"Because I actually do perform the poem as opposed to simply reciting it, I incorporate \"standing and thinking\" poses and swashbuckling and head-carrying actions into what I'm doing, so the visual cues probably help, too.",
"\n\nWhile I do not have a large cast, and I generally pantomime carrying a severed head, this skit from The Muppet Show does a pretty good job at reflecting the actions described in the poem:\n\nAbout Me\n\nI'm a poet and children's book author. ",
"My current projects are a biography of Jane Austen written in verse using period forms and an early middle grade novel involving garden gnomes. (",
"And yes, these are separate projects. ",
"Thus far, I've found no evidence of interaction between Jane Austen and any gnomes.)"
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"\nVietnamese: \nXin mến chào đại gia đình mình chúc cho cả nhà một ngày mới được nhiều an vui\nHôm nay mình có làm cái món nấm\nnhân thì mình làm bằng tàu hũ trắng\nmón này cũng rất là dễ làm\nMời cả nhà cùng làm cho vui\nỪ thì đây là cái loại nấm mình nói là cả nhà ạ\nNhà mình có bốn cái tay nắm khoảng 300 rưỡi ra má thì mình đã rửa steroid thì cái loại nấm này nó\nngười thì gọi là cái nấm nắp Tại nó giống như cái nắp như thế này nữa ah\nVụ tai nạn\nGửi đây mình có một miếng tàu hũ\n400 rưỡi ram cái loại này thì nó sẽ vừa nó không có mềm\n\nEnglish: \n Dear my family, I wish the whole family a new day of happiness \n Today I made a mushroom dish \n In the case, we are made of white tofu \n This dish is also very easy to make \n Invite the whole family to make fun \n Well, this is the mushroom I said to be the whole family \n My family has four handles that hold about 300 and a half on my cheeks, then I washed the steroids, this fungus is it \n People call it the lid mushroom Because it's like a lid like this too \n The accident \n Send here I have a piece of tofu \n 400 and a half ram of this type will fit it without being soft \n\nVietnamese: \nừ ừ\nCó khoảng nửa chén Vũng Tàu đường sắt lại như vậy\nrồi khoảng nửa chén nấm mèo mình cũng sắp sợ hả Ừ\nỪ mùng 4 cà phê đường một chút xíu tiêu chút xíu muối và 1 muỗng cà phê bột nêm em\nừ ừ - 1 muỗng cà phê bột bắp Để một lát mình để vào trong cá nhân\nRồi bây giờ mình sẽ đi\nluộc tàu hũ nha cả nhà mình sẽ sát nhỏ cho nó dễ chút\nrồi mình đem đi luộc luộc sơ thôi\nKhi còn ấm này á thì mình lấy cái chân ở bên trong nó ra như thế này\nđể lát mình để cá nhân nó vô à\n\nEnglish: \n OK OK \n There are about half a cup of Vung Tau railway like that \n Then about half a cup of cat mushrooms, we're about to be scared, huh \n Yes 4 coffee sugar a little bit of pepper a little salt and 1 teaspoon of powdered seasoning baby \n Yes yeah - 1 teaspoon cornstarch Leave a moment to personalize \n Now I will go \n Boiling the tofu for the whole family will make it easy for them to be small \n Then we bring it to boil only \n When I was this warm, I took out the leg inside it like this \n For a moment to personalize it \n\nEnglish: \n pear \n Minh told a little bit of salt \n Yes \n I was sleeping for about 2 minutes \n I slime aside this bowl. ",
"Let me wait a while to wait for it to be warm, then I walk the skin \n eat \n My tofu has been drained to drain the whole family and now I mix the spices \n Love to put cornstarch with sugar and then deliver it all \n\nVietnamese: \nLê\nMinh dặn vô chút xíu muối\nỪ\nMình trụng sơ khoảng 2 phút thôi\nmình nhớt qua bên cái tô này Để một lát nữa mình đợi nó hơi ấm ấm thì mình dắt cái nước da\năn\nTàu hũ mình đã giắt nước ra nó ráo gì là cả nhà rồi bây giờ mình trộn gia vị vô\nTình để bột bắp muối đường rồi giao hết luôn à\n\nVietnamese: \nÊ mèo đều cho đến khi mà nó mình mới dẻo là được à\nVì mình thấy tàu hũ nó đã dẻo rồi bây giờ mình đi ngồi do nấm được rồi cả nhà\nLắm mình đã làm xong rồi Bây giờ mà đi qua mình sẽ chuyên xơ nó kho lại\na cho vào khoảng 2 muỗng canh dầu bị ra cả nhà ạ\n\nEnglish: \n Hey cat, until it is new to be flexible \n Because I saw that the tofu was already flexible, now I can go sit due to the fungus \n I have done it already. ",
"Now that I go through, I will specialize in making it stock again \n I put in about 2 tablespoons of oil, the whole family \n\nEnglish: \n Brother \n listening through the other side, then here we do a good job our whole family put in a tablespoon of oil \n Add to 1 tablespoon ginger 1 tablespoon tilapia for fragrant \n For the sauce, I have 1 tablespoon soy sauce 1 tablespoon oyster oil 1 tablespoon vegetarian seasoning powder \n 1 tbsp sugar \n half a teaspoon of pepper \n 1 cup of water \n Viet due to a standing posture \n It's okay to top it \n Now when I come over and watch it hot, I will leave the nick Face through the photo-shaped mushroom \n Because \n\nVietnamese: \nAnh\nđang nghe xuyên nắm bên kia thế thì bên đây mình làm tốt nha cả nhà mình cho vào một muỗng canh dầu\nđể vào một muỗng canh gừng 1 muỗng canh bơ rô phi cho thơm\nphần nước sốt thì mình có một muỗng canh nước tương 1 muỗng canh dầu hào 1 muỗng canh bột nêm chay\n1 muỗng canh đường\nnửa muỗng cà phê tiêu\n1 chén nước\nViệt do một dáng đứng\nCái nó top nó được rồi\nBây giờ mình qua mình coi nóng nó ràng thì mình sẽ để cái nick Face qua nấm hình kho ạ\nVì\n\nEnglish: \n The reason I let the top country come here is to give mother to each other \n Back then, I still have the root of the mushroom that I used to let it go \n I reduce my low heat a little \n After that, I filled the lid and cooked for about 5 more minutes \n for \n When it is ripe, I stirred 1 teaspoon of cornstarch and 2 tablespoons of water I prepared \n it's clean and bad for it to taste good \n Let the cilantro be in, then the whole family \n Leave a little bit I sprinkle it on my face \n For a little more pepper it will taste good and \n Because \n So the mushroom is sitting, but the tofu is done \n\nVietnamese: \nLý do mình để cho nước top mà qua đây để mẹ cho nhau cả mà\nHồi nãy mình còn cái gốc cái nấm mà nãy mình lấy cho rằng để vô luôn\nMình bớt lửa nhỏ tôi một chút\nXong mình đầy nắp lại nấu thêm khoảng 5 phút nữa là được\ncho\nnấm nó chín rồi mình mới quậy thêm 1 muỗng cà phê bột bắp với 2 muỗng canh nước mình chế vô\nvào cái nước nó sạch tệ cho nó ngon\nđể hành ngò vô được rồi cả nhà\nĐể lại chút xíu một tí mình rắc lên mặt\nCho thêm tí tiêu nó sẽ ngon và\nVì\nVậy là món nấm ngồi nhưng tàu hũ mình đã làm xong\n\nVietnamese: \nnhững gia vị mình chia sẻ Cả nhà có thể ra giảm theo khẩu vị riêng\nnhìn cũng rất là ngon cho nên hôm nào có dịp cả nhà làm cho gia đình mình dùng thử\nEm\ncảm ơn cả nhà đã dành thời gian tem video\nMến chúc cả nhà được nhiều sức khỏe hạnh phúc và an dư trong cuộc sống mến chào và hẹn gặp lại lần sau\n\nEnglish: \n the spices we share The whole family can reduce according to their own taste \n It also looks delicious, so every day we have the opportunity to let our family try it \n You \n Thank you all for taking the time to stamp the video \n I wish the whole family happy health and well-being and well-being in life and welcome you and see you next time \n"
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[
"[](https://travis-ci.org/WebAssembly/wabt) [](https://ci.appveyor.com/project/WebAssembly/wabt/branch/master)\n\n# WABT: The WebAssembly Binary Toolkit\n\nWABT (we pronounce it \"wabbit\") is a suite of tools for WebAssembly, including:\n\n - **wat2wasm**: translate from [WebAssembly text format](https://webassembly.github.io/spec/core/text/index.html) to the [WebAssembly binary format](https://webassembly.github.io/spec/core/binary/index.html)\n - **wasm2wat**: the inverse of wat2wasm, translate from the binary format back to the text format (also known as a .wat)\n - **wasm-objdump**: print information about a wasm binary. ",
"Similiar to objdump.",
"\n - **wasm-interp**: decode and run a WebAssembly binary file using a stack-based interpreter\n - **wat-desugar**: parse .wat text form as supported by the spec interpreter (s-expressions, flat syntax, or mixed) and print \"canonical\" flat format\n - **wasm2c**: convert a WebAssembly binary file to a C source and header\n\nThese tools are intended for use in (or for development of) toolchains or other\nsystems that want to manipulate WebAssembly files. ",
"Unlike the WebAssembly spec\ninterpreter (which is written to be as simple, declarative and \"speccy\" as\npossible), they are written in C/C++ and designed for easier integration into\nother systems. ",
"Unlike [Binaryen](https://github.com/WebAssembly/binaryen) these\ntools do not aim to provide an optimization platform or a higher-level compiler\ntarget; instead they aim for full fidelity and compliance with the spec (e.g.\n1:1 round-trips with no changes to instructions).",
"\n\n## Online Demos\n\nWabt has been compiled to JavaScript via emscripten. ",
"Some of the functionality is available in the following demos:\n\n- [index](https://cdn.rawgit.com/WebAssembly/wabt/aae5a4b7/demo/index.html)\n- [wat2wasm](https://cdn.rawgit.com/WebAssembly/wabt/aae5a4b7/demo/wat2wasm/)\n- [wasm2wat](https://cdn.rawgit.com/WebAssembly/wabt/aae5a4b7/demo/wasm2wat/)\n\n## Cloning\n\nClone as normal, but don't forget to get the submodules as well:\n\n```console\n$ git clone --recursive https://github.com/WebAssembly/wabt\n$ cd wabt\n```\n\nThis will fetch the testsuite and gtest repos, which are needed for some tests.",
"\n\n## Building (macOS and Linux)\n\nYou'll need [CMake](https://cmake.org). ",
"If you just run `make`, it will run\nCMake for you, and put the result in `out/clang/Debug/` by default:\n\n> Note: If you are on macOS, you will need to use CMake version 3.2 or higher\n\n```console\n$ make\n```\n\nThis will build the default version of the tools: a debug build using the Clang\ncompiler.",
"\n\nThere are many make targets available for other configurations as well. ",
"They\nare generated from every combination of a compiler, build type and\nconfiguration.",
"\n\n - compilers: `gcc`, `clang`, `gcc-i686`, `gcc-fuzz`\n - build types: `debug`, `release`\n - configurations: empty, `asan`, `msan`, `lsan`, `ubsan`, `no-re2c`,\n `no-tests`\n\nThey are combined with dashes, for example:\n\n```console\n$ make clang-debug\n$ make gcc-i686-release\n$ make clang-debug-lsan\n$ make gcc-debug-no-re2c\n```\n\nYou can also run CMake yourself, the normal way:\n\n```console\n$ mkdir build\n$ cd build\n$ cmake ..\n...\n```\n\n## Building (Windows)\n\nYou'll need [CMake](https://cmake.org). ",
"You'll also need\n[Visual Studio](https://www.visualstudio.com/) (2015 or newer) or \n[MinGW](http://www.mingw.org/).",
"\n\nYou can run CMake from the command prompt, or use the CMake GUI tool. ",
"See\n[Running CMake](https://cmake.org/runningcmake/) for more information.",
"\n\nWhen running from the commandline, create a new directory for the build\nartifacts, then run cmake from this directory:\n\n```console\n> cd [build dir]\n> cmake [wabt project root] -DCMAKE_BUILD_TYPE=[config] -DCMAKE_INSTALL_PREFIX=[install directory] -G [generator]\n```\n\nThe `[config]` parameter should be a CMake build type, typically `DEBUG` or `RELEASE`.",
"\n\nThe `[generator]` parameter should be the type of project you want to generate,\nfor example `\"Visual Studio 14 2015\"`. ",
"You can see the list of available\ngenerators by running `cmake --help`.",
"\n\nTo build the project, you can use Visual Studio, or you can tell CMake to do it:\n\n```console\n> cmake --build [wabt project root] --config [config] --target install\n```\n\nThis will build and install to the installation directory you provided above.",
"\n\nSo, for example, if you want to build the debug configuration on Visual Studio 2015:\n\n```console\n> mkdir build\n> cd build\n> cmake .. -DCMAKE_BUILD_TYPE=DEBUG -DCMAKE_INSTALL_PREFIX=..\\bin -G \"Visual Studio 14 2015\"\n> cmake --build .. --config DEBUG --target install\n```\n\n## Changing the lexer\n\nIf you make changes to `src/wast-lexer.cc`, you'll need to install\n[re2c](http://re2c.org). ",
"Before you upload your PR, please run `make\nupdate-re2c` to update the prebuilt C sources in `src/prebuilt/`.",
"\n\nCMake will detect if you don't have re2c installed and use the prebuilt source\nfiles instead.",
"\n\n## Running wat2wasm and wast2json\n\nSome examples:\n\n```sh\n# parse and typecheck test.wat\n$ out/wat2wasm test.wat\n\n# parse test.wat and write to binary file test.wasm\n$ out/wat2wasm test.wat -o test.wasm\n\n# parse spec-test.wast, and write verbose output to stdout (including the\n# meaning of every byte)\n$ out/wat2wasm spec-test.wast -v\n\n# parse spec-test.wast, and write files to spec-test.json. ",
"Modules are written\n# to spec-test.0.wasm, spec-test.1.wasm, etc.",
"\n$ out/wast2json spec-test.wast -o spec-test.json\n```\n\nYou can use `-h` to get additional help:\n\n```console\n$ out/wat2wasm -h\n```\n\nOr try the [online demo](https://cdn.rawgit.com/WebAssembly/wabt/aae5a4b7/demo/wat2wasm/).",
"\n\n## Running wasm2wat\n\nSome examples:\n\n```sh\n# parse binary file test.wasm and write text file test.wat\n$ out/wasm2wat test.wasm -o test.wat\n\n# parse test.wasm and write test.wat\n$ out/wasm2wat test.wasm -o test.wat\n```\n\nYou can use `-h` to get additional help:\n\n```console\n$ out/wasm2wat -h\n```\n\nOr try the [online demo](https://cdn.rawgit.com/WebAssembly/wabt/aae5a4b7/demo/wasm2wat/).",
"\n\n## Running wasm-interp\n\nSome examples:\n\n```sh\n# parse binary file test.wasm, and type-check it\n$ out/wasm-interp test.wasm\n\n# parse test.wasm and run all its exported functions\n$ out/wasm-interp test.wasm --run-all-exports\n\n# parse test.wasm, run the exported functions and trace the output\n$ out/wasm-interp test.wasm --run-all-exports --trace\n\n# parse test.json and run the spec tests\n$ out/wasm-interp test.json --spec\n\n# parse test.wasm and run all its exported functions, setting the value stack\n# size to 100 elements\n$ out/wasm-interp test.wasm -V 100 --run-all-exports\n```\n\nYou can use `-h` to get additional help:\n\n```console\n$ out/wasm-interp -h\n```\n\n## Running wasm2c\n\nSee [wasm2c.md](wasm2c/README.md)\n\n## Running the test suite\n\nSee [test/README.md](test/README.md).",
"\n\n## Sanitizers\n\nTo build with the [LLVM sanitizers](https://github.com/google/sanitizers),\nappend the sanitizer name to the target:\n\n```console\n$ make clang-debug-asan\n$ make clang-debug-msan\n$ make clang-debug-lsan\n$ make clang-debug-ubsan\n```\n\nThere are configurations for the Address Sanitizer (ASAN), Memory Sanitizer\n(MSAN), Leak Sanitizer (LSAN) and Undefine Behavior Sanitizer (UBSAN). ",
"You can\nread about the behaviors of the sanitizers in the link above, but essentially\nthe Address Sanitizer finds invalid memory accesses (use after free, access\nout-of-bounds, etc.), ",
"Memory Sanitizer finds uses of uninitialized memory, \nthe Leak Sanitizer finds memory leaks, and the Undefined Behavior Sanitizer\nfinds undefined behavior (surprise!).",
"\n\nTypically, you'll just want to run all the tests for a given sanitizer:\n\n```console\n$ make test-asan\n```\n\nYou can also run the tests for a release build:\n\n```console\n$ make test-clang-release-asan\n...\n```\n\nThe Travis bots run all of these tests (and more). ",
"Before you land a change,\nyou should run them too. ",
"One easy way is to use the `test-everything` target:\n\n```console\n$ make test-everything\n```\n\nTo run everything the Travis bots do, you can use the following scripts:\n\n```console\n$ CC=gcc scripts/travis-build.sh\n$ CC=gcc scripts/travis-test.sh\n$ CC=clang scripts/travis-build.sh\n$ CC=clang scripts/travis-test.sh\n```\n"
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"Interactions at the mild steel acid solution interface in the presence of O-fumaryl-chitosan: Electrochemical and surface studies.",
"\nThe performance of synthesised O-fumaryl-chitosan (OFC) as corrosion inhibitor for mild steel in 1M HCl has been evaluated through various studies. ",
"The initial screening by weight loss method revealed the good inhibition efficiency by the inhibitor. ",
"Thermodynamic and kinetic parameters have been calculated and discussed. ",
"The mode of adsorption is physical in nature and it follows Langmuir adsorption isotherm. ",
"Electrochemical measurements supported the inhibition of mild steel by the fumaryl derivative of chitosan. ",
"Polarisation studies provided the information that the inhibition is of mixed type. ",
"The formation of inhibitor film is assured by surface morphological studies with Scanning electron microscopy (SEM) and Atomic force microscopy (AFM). ",
"The mechanism of inhibition is derived from the Fourier-transform infrared (FTIR) spectroscopy and zero charge potential measurement. ",
"The adsorbed film is characterised using FTIR and X-ray diffraction studies (XRD)."
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