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"The National Enquirer has his sexts and, it appears, saucy photos. ",
"His imminent divorce is the talk of Wall Street.",
"\n\nSo Jeff Bezos, the Amazon boss, announced on Friday that he had turned to a singular figure for help – a “bodyguard to the stars” and Los Angeles-based security consultant named Gavin de Becker.",
"\n\nBezos and his longtime wife, MacKenzie, are divorcing and the National Enquirer has revealed his extramarital affair with the television personality Lauren Sánchez. ",
"Writing on the website Medium, Bezos said the publisher of the National Enquirer had threatened to publish sexual photos he and Sanchez shared if he did not call off De Becker, whom Bezos described as an acquaintance of 20 years. ",
"De Becker, Bezos said, is still on the job with instructions “to proceed with whatever budget he needed to pursue the facts” related to the Enquirer’s possession of his communications.",
"\n\nBy declaring his troubles so publicly, Bezos was operating straight out of De Becker’s playbook.",
"\n\nIn his 1997 book The Gift of Fear, De Becker offered a strategy for dealing with extortionists. ",
"De Becker described the case of a client, a young actress, whose ex-boyfriend was demanding $50,000 for his silence about a private matter. ",
"He suggested she “kill the threat” by disclosing the matter to her parents so they wouldn’t learn it in “a tabloid’s way”.",
"\n\n“Disclosing harmful information oneself is so radical an idea that most victims of extortion never even consider it,” De Becker wrote, noting that his firm has “a few cases” like that each year.",
"\n\nDe Becker & Associates, the 64-year-old’s firm, provides security and, as in Bezos’s case, conducts private investigations for the rich, the powerful and the famous. ",
"He has been romantically linked to stars – he dated Geena Davis and Alanis Morissette – and was a guest on Oprah. ",
"He and his employees are frequent foes of the tabloid press.",
"\n\nIn The Gift of Fear, De Becker recounts, in jarring detail, a childhood defined by brutality. ",
"He describes picking up a pistol his mother used to shoot his stepfather, gripping it by the hot barrel as his mother went to tend the wounded man she’d just shot. ",
"As he tells it, those experiences drove him to understand violence and fear.",
"\n\nDe Becker’s destiny was set at least in part by geography. ",
"Attending Beverly Hills high school, he befriended the actor Carrie Fisher, at whose funeral he spoke, and Miguel Ferrer, son of Rosemary Clooney. ",
"De Becker went to work for Elizabeth Taylor at 19 as an assistant, took over security for Shaun Cassidy, then a teen star, at 23, and launched his security firm in 1978.",
"\n\nBy ever-increasing degrees, has been a Hollywood fixture since, and has become known for injecting analytical processes into the security business.",
"\n\nDr Park Dietz, a forensic psychiatrist called as an expert to testify against Jeffrey Dahmer, Ted Kaczynski and other prominent killers, began working with De Becker in 1983. ",
"De Becker’s firm had collected thousands of threatening communications sent to stars, and built dossiers on their authors to gauge those threats. ",
"Those records, and a similar collection held by the US Capitol police, formed the basis for their joint research: a National Institute of Justice-supported study of threats against public figures.",
"\n\nThey shared an interest, Dietz said, in changing the way society understood stalking, which was then viewed as something experienced only by a handful of famous women. ",
"Dietz, founder of the Threat Assessment Group, believes their work made stalking “a household word” and a prosecutable crime.",
"\n\nDe Becker, Dietz said, was “on the vanguard of recognizing the similarity of paparazzi and other stalkers”. ",
"De Becker also wanted to revolutionize his industry.",
"\n\nJeff Bezos said he told De Becker ‘to proceed with whatever budget he needed to pursue the facts’. ",
"Photograph: Cliff Owen/AP\n\n“His concern was that Hollywood was full of thug-like bodyguards, with various backgrounds in the martial arts or the military or professional wrestling,” Dietz told the Guardian. “",
"He wanted to try to professionalize an industry that was not well-regarded. ",
"And he was a leader in that.”",
"\n\nDe Becker’s approach hinges on the proposition that violence can be anticipated, and that would-be subjects of violent crime can be prepared. ",
"He has developed, and aggressively defended from critics, a suite of threat assessment tools known as Mosaic. ",
"Users answer a series of questions about a threatening person or occurrence using an online interface; the system analyses the answers and gauges the level of threat. ",
"His efforts landed De Becker a seat across from Oprah in 2008, an appearance that solidified his status as a security guru.",
"\n\nDe Becker is an unabashed protector of the famous and powerful. ",
"He launched a private terminal at LAX, the Los Angeles airport, catering to that cohort. ",
"He took out an advertisement in the Hollywood Reporter defending Mel Gibson after the actor’s drunken antisemitic tirade was caught on police video.",
"\n\nHis other clients have been reported to include Madonna, Cher, Michelle Pfeiffer, Michael J Fox and John Travolta. ",
"He provided security to Planned Parenthood, was hired by the Cosbys to investigate threats against the family after the only son of Bill and Camille Cosby was killed, and has consulted on blockbuster films, including The Bodyguard.",
"\n\nDuring the years Dietz worked with De Becker, he visited the homes of De Becker’s clients, touring guardhouses and safe rooms De Becker’s team had installed.",
"\n\nDe Becker, Dietz said, moved with the glad-handing charm of a successful sports agent. “",
"He knew every aspect of the world of celebrity, and had a real commitment to trying to protect them against people who would do them harm,” Dietz said.",
"\n\nDietz described his time in De Becker’s celebrity-stuffed world eye-opening “and, frankly, fun”.",
"\n\nOthers have made a similar appraisal, though considering his travails, Bezos may not be enjoying his time in De Becker’s orbit quite so much.",
"\n\n“He’s the funniest civilian I know,” the comedian Harry Shearer told the Los Angeles Times in 2002 of De Becker. “",
"He’s very bright and disciplined, and he’d probably be good at whatever he wanted to do. ",
"But if you do what he does for a living, having a light side would be a necessity.”"
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"Moreover, the lens length of the rod lens became long, so that the material cost for the lens increased.",
"\nAn object of the present invention is to provide a light source-optical fiber coupler which can couple a semiconductor laser to a single-mode optical fiber with low loss and which can be produced as a small-size coupler inexpensively and easily.",
"\nA subject of the present invention is a light source-optical fiber coupler using a gradient index rod lens for coupling a diffused luminous flux emitted from a light source onto an end surface of an optical fiber. ",
"Here, the gradient index rod lens has a planar end surface on the light source side, and a convex spherical end surface on the optical fiber side. ",
"The gradient index rod lens has a light source side numerical aperture NA2 in a range of from 0.40 to 0.75, an effective lens radius r0 in a range of from 0.3 to 1.0 mm, and a spherical curvature radius R1 in a range of from 1.2 to 2.0 mm.",
"\nIn the present invention, a semiconductor laser is generally used as the light source. ",
"When a single-mode optical fiber having a small diameter is used as the optical fiber which is a partner of optical coupling, the highly efficient optical coupling effect is particularly remarkable. ",
"In the present invention, the planar surface is used on the semiconductor laser side and the convex spherical surface is used on the optical fiber, so that high numerical aperture is attained and the aberration of light can be corrected by the refractive-index distribution shape of the rod lens. ",
"Hence, reduction in size of an optical module can be achieved when such a gradient index rod lens having a small lens diameter is used.",
"\nPreferably, the gradient index rod lens used in the present invention has a center axis refractive index n0 in a range of from 1.5 to 1.8, a value of n0xc2x7gxc2x7r0 in a range of from 0.40 to 0.65 (in which a is a secondary refractive-index distribution coefficient), and a value of Z/R1 in a range of from 1.1 to 2.5 (in which Z is the lens length of the rod lens) . ",
"Preferably, the gradient index rod lens has a value of g in a range of from 0.38 to 0.50, a value of h4 in a range of from xe2x88x920.2 to 1.8, a value of h6 in a range of from xe2x88x920.5 to 10, and a value of h8 in a range of from xe2x88x9225 to 45 (in which h4, h6 and h8 are higher-order refractive-index distribution coefficients respectively).",
"\nA practical optical module structure may be preferably configured as follows. ",
"A semiconductor laser is used as the light source. ",
"A chip of the semiconductor laser and the gradient index rod lens are retained in a housing in the condition that they are arranged closely to each other. ",
"The housing is provided so that a ferrule of an optical plug which is used as a mating partner and in which a single-mode optical fiber is retained can be fitted and retained in the housing.",
"\nThe present disclosure relates to the subject matter contained in Japanese patent application No. ",
"2000-326471 (filed on Oct. 26, 2000), which is expressly incorporated herein by reference in its entirety."
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"Q:\n\nWhy does timezone info from Ruby sequel and Postgres psql differ?",
"\n\nHere is a query result from Postgres:\n$ psql ... -c 'select the_date from foo where foo_id in (998,999)'\n the_date \n------------------------\n 2012-03-07 09:34:47.98\n 2012-03-16 11:31:25.336\n\nthe_date is \"timestamp without time zone\".",
"\nHere is a Ruby program:\n#!",
"/usr/bin/env ruby\n\nrequire 'sequel'\n\n@DB = Sequel.connect({...})\nquery = \"select the_date from foo where foo_id in (998,999)\"\n@DB[query].each do |row|\n warn row\nend\n\nand some output:\n{:the_date=>2012-03-07 09:34:47 -0600}\n{:the_date=>2012-03-16 11:31:25 -0500}\n\nWhere does the -0500 and -0600 come from? ",
" That is the \"Olson timezone\" of the server and the client machines (US/Central), but why does Ruby add it and psql does not?",
"\nI've been reading the docs, and I'm thoroughly confused.",
"\nThe server is Postgres 9.0.4, the client is psql 9.1.4, sequel is 3.33.0.",
"\n\nA:\n\nThe column is of type 'timestamp without timezone'. ",
"Thus when Postgres displays a value in this column it just displays the timestamp with no timezone. ",
"However, Sequel wants to convert a Postgres timestamp to an instance of the Ruby Time class, and an instance of the Time class must have a timezone specified - either it's a time in the local timezone or it's a time in UTC. ",
"Thus Sequel must choose one. ",
"By default, it's choosing your local timezone.",
"\nYou may configure the database and application timezone in Sequel. ",
"See http://sequel.rubyforge.org/rdoc/classes/Sequel/Timezones.html\nFor example, here's the default Sequel behavior with a database I had handy:\n> c['select * from actors'].each do |row|; puts row[:created_at]; end\nThu Jul 12 20:33:17 -0400 2012\n\nHere the timestamp is assumed to be in my local timezone (EDT).",
"\nHowever, if I do this:\n> Sequel.database_timezone = :utc\n => :utc \n> c['select * from actors'].each do |row|; puts row[:created_at]; end\nThu Jul 12 20:33:17 UTC 2012\n\nThen the timestamp is assumed to be in UTC.",
"\n\n"
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"Q:\n\nAS3 9 slice scaling\n\ni'm making window component on Flash Professional CS4 and now i have 1 problem. ",
"At the top of window is dynamic textfield for title. ",
"When i scale window text is also scaling. ",
"So is there any way to make textfield fixed in same position and same size?",
"\nP.S. 9-slice scaling is enabled.",
"\n\nA:\n\nYou should separate your window's background from the rest of it:\nwindow > title\n > background\n\nYour background would be the only clip to be 9-sliced and you will scale your window by doing:\nwindow.getChildByName(\"background\").scaleX = newScaleX;\nwindow.getChildByName(\"background\").scaleY = newScaleY;\n\n"
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"second derivative of -5363321*i**2 - 751678*i.",
"\n-10726642\nWhat is the third derivative of 13603*i**3*j**2 - 2*i**3 + 6438*i**2*j - 167*i**2 + i*j**2 wrt i?",
"\n81618*j**2 - 12\nWhat is the second derivative of -907*b**5 - b**3 - 66513*b**2 - 368735490*b?",
"\n-18140*b**3 - 6*b - 133026\nWhat is the second derivative of 23483466*n**3 - 51370098*n wrt n?",
"\n140900796*n\nWhat is the first derivative of -19458115*n + 35688118 wrt n?",
"\n-19458115\nWhat is the second derivative of -65440406*v**5 - 2*v - 14726294?",
"\n-1308808120*v**3\nWhat is the derivative of -11059824*h - 9600016?",
"\n-11059824\nWhat is the second derivative of 2*d**3*h*z + 4749*d**3 + 2479*d**2*h**2 - 5721*d*h**2*z + 2*d*h**2 - 35*d*h*z + 2*d*h + 2*d*z wrt d?",
"\n12*d*h*z + 28494*d + 4958*h**2\nWhat is the second derivative of -13*k**4 - 2266*k**3 - 399*k**2 - 183*k + 9580 wrt k?",
"\n-156*k**2 - 13596*k - 798\nWhat is the third derivative of 1913*d**3*f**3 - 93*d**3*f**2 - 37*d**3*f + 2*d**2*f**2 - 5*d**2*f + 517*f**4 + f wrt f?",
"\n11478*d**3 + 12408*f\nWhat is the second derivative of -f**5 - 8932800*f**3 + 4845947*f?",
"\n-20*f**3 - 53596800*f\nFind the first derivative of -20405*b**4 + b**2 - 7*b - 5803067 wrt b.\n-81620*b**3 + 2*b - 7\nFind the third derivative of -19*b**4*c - 202833*b**3*c + 222*b**2*c - 3*b**2 - 14*c - 1 wrt b.\n-456*b*c - 1216998*c\nDifferentiate -516*h**3 + 2513*h**2 + 34256237.",
"\n-1548*h**2 + 5026*h\nDifferentiate 612*x**4 + 299*x - 1712925 wrt x.\n2448*x**3 + 299\nWhat is the second derivative of -12126166*v**4 + 4851272*v - 1 wrt v?",
"\n-145513992*v**2\nWhat is the first derivative of 334954118*d**4 - 650794861 wrt d?",
"\n1339816472*d**3\nWhat is the second derivative of -2*z**3 - 802408*z**2 + 5440950*z?",
"\n-12*z - 1604816\nFind the third derivative of -13*o**4 - 59972*o**3 + o**2 - 11*o - 28150 wrt o.\n-312*o - 359832\nWhat is the third derivative of -1176300*r**5 - 41*r**3 + 365*r**2 + 916*r + 22 wrt r?",
"\n-70578000*r**2 - 246\nFind the first derivative of -2308*l*w**3 - 3458*l + 2*w**3 - 72879544*w wrt l.\n-2308*w**3 - 3458\nFind the third derivative of k**5 - k**4 + 1872860*k**3 - 83217*k**2 + k + 352.",
"\n60*k**2 - 24*k + 11237160\nFind the third derivative of -7438835*b**4 - 16*b**2 + b + 61539.",
"\n-178532040*b\nFind the first derivative of 1873*y**4 - 161*y**3 + 2*y**2 - 11379107.",
"\n7492*y**3 - 483*y**2 + 4*y\nWhat is the first derivative of -285*c**3*r - 222247078*c**3 + 370534*c*r wrt r?",
"\n-285*c**3 + 370534*c\nWhat is the second derivative of -760440033*g**2*j - 2*g*j + g - 7*j - 5018597 wrt g?",
"\n-1520880066*j\nWhat is the first derivative of -g*r*x + g*r - 2*g*x - 32*g - 113*r*x - 2076*r + 2270*x - 2 wrt x?",
"\n-g*r - 2*g - 113*r + 2270\nFind the second derivative of -107697*b*q**3 + 2*b*q**2 + 2*b*q - 308*b + 7*q**3 - 4*q**2 - 23764*q - 1 wrt q.\n-646182*b*q + 4*b + 42*q - 8\nFind the third derivative of 4*a*i*o**3 + a*i*o**2 - 2*a*i*o - 299*a*i - 585*a*o**3 - 991*a*o**2 + 2*i*o**3 + 2042*o**3 - o - 2 wrt o.\n24*a*i - 3510*a + 12*i + 12252\nFind the first derivative of -10495006*a**3*h + 48*a**3 + 25*a**2 - 29*a wrt h.\n-10495006*a**3\nDifferentiate -3966*v**4 + 232*v**3 + 2885416 with respect to v.\n-15864*v**3 + 696*v**2\nFind the second derivative of -915942710*r**4 + 64457162*r.",
"\n-10991312520*r**2\nWhat is the derivative of 6*r*t + 2625916*r + 3980*t**4 - 20*t**3 - 1 wrt t?",
"\n6*r + 15920*t**3 - 60*t**2\nWhat is the derivative of -1354280*u**2 - 34*u - 289184622 wrt u?",
"\n-2708560*u - 34\nWhat is the third derivative of -20*f**5 - 66383*f**3 + 12409720*f**2?",
"\n-1200*f**2 - 398298\nDifferentiate 130151431*g*l**2 - l**2 + 11014810 wrt g.\n130151431*l**2\nWhat is the second derivative of -a**3*f - 1485*a**3 - 33*a**2*f + 97*a**2 + 404*a*f - a + 242*f + 84 wrt a?",
"\n-6*a*f - 8910*a - 66*f + 194\nWhat is the second derivative of 408356257*o**2*w**2 - 5*o**2 + 25906872*w wrt w?",
"\n816712514*o**2\nFind the third derivative of 54416176*s**3 + 1424*s**2 + 1281*s - 5 wrt s.\n326497056\nFind the first derivative of -73340749*k + 206637680 wrt k.\n-73340749\nWhat is the derivative of 3*c**2*q**2 - 1904*c**2 - 249*c*q - 40210558*q**2 wrt c?",
"\n6*c*q**2 - 3808*c - 249*q\nWhat is the second derivative of -176*b**4 - 1650650*b**2 - 221127488*b wrt b?",
"\n-2112*b**2 - 3301300\nDifferentiate -134821636*a*p**3 + 2*p**3 - 17821995*p**2 - p wrt a.\n-134821636*p**3\nFind the third derivative of -2732577*b**4*d**3 - 239*b**2*d + 6634*b*d**3 wrt b.\n-65581848*b*d**3\nDifferentiate -2308357*l**3 + 15*l**2 - l + 254569136.",
"\n-6925071*l**2 + 30*l - 1\nFind the third derivative of -18715*m**2*v**4 - m**2*v**2 - 1344*m**2 + 590*m*v**4 - 29423*v**2 wrt v.\n-449160*m**2*v + 14160*m*v\nDifferentiate 144510*a**4 + 98*a**3 + 68283580 with respect to a.\n578040*a**3 + 294*a**2\nFind the first derivative of 12293606*y - 4576216.",
"\n12293606\nFind the second derivative of o**4 + 15128295*o**3 + 2*o - 3372779.",
"\n12*o**2 + 90769770*o\nFind the second derivative of 75961*w**4 + 10*w**3 + 42*w**2 - 41495653*w - 1 wrt w.\n911532*w**2 + 60*w + 84\nWhat is the derivative of -147128178*i**2 - 2*i - 3763263?",
"\n-294256356*i - 2\nWhat is the second derivative of -30671*a**3*h**2 + 2*a**3*h + a**3 + 310*a**2*h + a*h**4 + 2*a*h**2 - 2*a - 41*h + 2 wrt h?",
"\n-61342*a**3 + 12*a*h**2 + 4*a\nWhat is the third derivative of 34*g**3*j*q - 2*g**3*j - 46*g**3*q + 85*g**3*z - 2*g**2*q*z - 12*g**2 - 3*g*j*z + 251*q*z wrt g?",
"\n204*j*q - 12*j - 276*q + 510*z\nWhat is the second derivative of 2*g**2*i*l**2 + 303*g**2*i - 3*g**2*l - 2*g**2 + 3154*g*i*l**2 - 1518*g*l**2 + 38*g*l - 27*l wrt l?",
"\n4*g**2*i + 6308*g*i - 3036*g\nWhat is the second derivative of -191213*b**2*l - 245*b**2*y - 431*b*l*y - 2*b*y + 14*b + 4*l*y - 5*l + 2*y wrt b?",
"\n-382426*l - 490*y\nWhat is the first derivative of 199537485*k**3 + 56836601 wrt k?",
"\n598612455*k**2\nFind the third derivative of o**4*w + 3*o**4 + 79*o**3*w + 482*o**3 + 8*o**2 + 80158*w - 1 wrt o.\n24*o*w + 72*o + 474*w + 2892\nWhat is the second derivative of -60468016*u**5 - 51917*u + 506 wrt u?",
"\n-1209360320*u**3\nWhat is the second derivative of 22233*k*s**2*u**3 + 114*k*s*u**3 - k*s*u + 10*s**2*u**3 - 4*s**2*u - 119*s*u + 40*u wrt s?",
"\n44466*k*u**3 + 20*u**3 - 8*u\nWhat is the first derivative of -48157963*q**3 - 21415248?",
"\n-144473889*q**2\nFind the third derivative of -2004161*q**4 + 25145071*q**2.",
"\n-48099864*q\nFind the second derivative of 60803733*f**2 + 28499239*f.",
"\n121607466\nDifferentiate 1285317*f**2 - f*o + 11024250*o with respect to f.\n2570634*f - o\nWhat is the derivative of -9285692*b**2 + 12990848?",
"\n-18571384*b\nWhat is the third derivative of -39065816*t**3 + 36037591*t**2?",
"\n-234394896\nDifferentiate -71462*n**3 - 66*n**2 - n - 154279025 with respect to n.\n-214386*n**2 - 132*n - 1\nFind the third derivative of -37*f*k**3 - 4*f*k*v + 27*f + 9250*k**3*v**2 + 2*k**3 - 2791*k**2*v**2 - k**2*v wrt k.\n-222*f + 55500*v**2 + 12\nDifferentiate -236221*b*i*v**2 + b*v**3 - b*v + 2*b - 20*i*v**3 + 3*i - 747*v**3 - 2*v - 135 with respect to i.\n-236221*b*v**2 - 20*v**3 + 3\nFind the third derivative of 186135357*f**3 - 81851438*f**2.",
"\n1116812142\nWhat is the third derivative of -481966*m**2*o**3 - 17923*m**2*o**2 - 22*o**4 - 11*o**2 wrt o?",
"\n-2891796*m**2 - 528*o\nDifferentiate 2299*b*f*x + 76*b - 4312*f*x**2 - 78515*x**2 wrt f.\n2299*b*x - 4312*x**2\nWhat is the third derivative of 94824*n*p**4 - n*p**2 + 31*n*p - 3*n + 30*p**6 - 258*p**2 + 2 wrt p?",
"\n2275776*n*p + 3600*p**3\nDifferentiate 519182500*q - 257264126.",
"\n519182500\nDifferentiate -4*b*f*k - 7*b*f + 173068*b - 62659750*f*k with respect to b.\n-4*f*k - 7*f + 173068\nWhat is the first derivative of -63569974*z**3 + 2*z**2 - 294252969?",
"\n-190709922*z**2 + 4*z\nWhat is the second derivative of 560251832*c**2 + 675356680*c?",
"\n1120503664\nWhat is the third derivative of 164*i*j**3 + 2483*i*j**2 - 1023*i + 40323*j**3 - 2*j**2 wrt j?",
"\n984*i + 241938\nWhat is the derivative of -9*p**3 - 84488*p - 5319634 wrt p?",
"\n-27*p**2 - 84488\nWhat is the third derivative of -213002649*p**3 - 9*p**2 + 212010*p - 1?",
"\n-1278015894\nWhat is the second derivative of 124*a**2*p**3 + 1453*a**2*p**2 + 122*a**2 - a*p**3 + a*p**2 + 2*a*p + 370*p**3 + 8*p - 1381 wrt a?",
"\n248*p**3 + 2906*p**2 + 244\nWhat is the third derivative of 20168499*o**3 - 412*o**2 - 6695 wrt o?",
"\n121010994\nFind the first derivative of 837*o**4 + 2*o**3 + 13*o**2 + 50291 wrt o.\n3348*o**3 + 6*o**2 + 26*o\nWhat is the third derivative of -53*v**5 - 1420574*v**4 - 1159*v**2 + 17162?",
"\n-3180*v**"
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"Alzheimer's disease damages the nervous system in many different ways. ",
"This is because the disease affects not only neurons but also other brain cells, such as the astrocytes. ",
"These support the normal function of neurons and are involved in the regulation of cerebral blood flow. ",
"Through experimental studies scientists of the German Center for Neurodegenerative Diseases (DZNE) at the Bonn and Berlin sites have now gained new insights into how Alzheimer's interferes with the metabolism of astrocytes. ",
"The research team also showed that the pathological changes of the astrocytes can be mitigated by pharmacological treatment. ",
"The triggering molecules turned out to be energy carriers of the cell such as ATP: These molecules can induce the astrocytes to switch into a hyperactive state, which is characterized by sudden fluctuations in the concentration of calcium. ",
"As the researchers describe in the scientific journal \"Nature Communications\", their study suggests a novel potential approach for the treatment of Alzheimer's disease.",
"\n\nShare on Pinterest Besides neurons, the brain harbors a variety of other cells with very specific functions. ",
"This image (tissue sample of a mouse with hallmarks of Alzheimer's disease) shows some of them. ",
"The projections of so-called astrocytes are colored white. ",
"Astrocytes support the function of neurons and are involved in the regulation of cerebral blood flow. ",
"In Alzheimer's disease they alter their shapes and activity. ",
"This image also depicts the nuclei (blue) of astrocytes, neurons and other cells. ",
"The green structures stem from a protein associated with Alzheimer's.",
"\n\nCredit: DZNE / A. Delekate, T. Schumacher, G. Petzold\n\nIn a way, the brain resembles a large symphonic orchestra, whereby although the various instruments play together, each assumes a special part. ",
"Accordingly, the brain consists of nerve cells, also called \"neurons\", that are woven into a network in which they relay signals to one another. ",
"On the other hand, so-called glial cells are also equally important for brain function. ",
"These cells were once regarded as mere connective tissue of the brain. ",
"However, it is now known that they assume tasks that are far more complex than previously thought. ",
"One prominent member of this versatile family of glial cells are the astrocytes.",
"\n\n\"Astrocytes have various functions in the brain. ",
"For example, they supply the neurons with nutrients, but they also dispose waste products of metabolism\", explains Professor Gabor Petzold, who leads a research group at the Bonn site of the DZNE and also supervises the Neurovascular Unit at the University Hospital Bonn. „",
"In addition, they influence the communication of neurons with one another, and are involved in the control of cerebral blood flow.\"",
"\n\nAlzheimer's disease alters the astrocytes\n\nIt has long been known that astrocytes change their shapes as a consequence of Alzheimer's. ",
"Cells located near the \"plaques\", as the protein deposits typical for this disease are called, grow in size and form additional extensions. ",
"However, until know it was largely unclear how these changes affect the function of astrocytes.",
"\n\nThus, Petzold and his colleagues studied mice whose brains exhibited the typical protein deposits of Alzheimer's. ",
"They discovered that the calcium metabolism of astrocytes in the vicinity of plaques was disturbed. ",
"Calcium plays an important role as a regulator of cellular function and metabolism. \"",
"The astrocytes were hyperactive. ",
"This means that calcium levels in these cells could suddenly rise. ",
"We also noted that this effect often travelled to neighbouring astrocytes, causing so-called calcium waves. ",
"The effect is quite similar to throwing a stone into water,\" Petzold notes. \"",
"Normal astrocytes, in contrast, only rarely exhibit these variations in the concentration of calcium.\"",
"\n\nEnergy carriers with a signal effect\n\nThese fluctuations were caused by the actions of a cellular molecule named ATP. ",
"When the researchers blocked its release with the help of drugs, the activity of the astrocytes normalized. ",
"The same effect was achieved when the scientists disabled a specific receptor for these molecules. ",
"As Petzold's team determined, this receptor was present in unusually high numbers on the surface of astrocytes in the vicinity of plaques. ",
"This circumstance made the cells particularly susceptive.",
"\n\n\"ATP and similar molecules normally supply the cells with energy. ",
"However, it has already been known that they can also act as messenger molecules that can trigger specific reactions\", Petzold explains. \"",
"Although these molecules occur in most cell tissues, it is assumed that their release is increased in the vicinity of the plaques. ",
"We could show that this causes the astrocytes to switch into hyperactivity. ",
"The signalling pathway is mediated by a special receptor on the cell surface of astrocytes.\"",
"\n\nAn influence on blood flow\n\nIt is still uncertain whether the astrocytes' hyperactivity constitutes a protective defence reaction or whether it is associated with negative consequences. ",
"However, the current study shows that the calcium waves may in some cases be associated with local changes in brain perfusion. \"",
"This is interesting, because there have long been indications that Alzheimer's has a vascular component. ",
"Alterations of blood vessels and blood flow appear to play an important role\", says Petzold.",
"\n\nAccording to the Bonn scientist the current study could open up new avenues for therapy: \"Our investigations demonstrate that it is possible to mitigate the hyperactivity of these cells. ",
"This could point to a novel approach for treatment. ",
"It might perhaps also be possible to modify the course of the disease with the help of suitable pharmaceuticals.\"",
"\n\nSo far, the scientist clarifies, the signalling pathways were studied at the level of the cellular network in the brain. ",
"In future studies, Petzold and his colleagues intend to investigate what effect the inhibition of hyperactivity has on disease symptoms."
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"JONESBORO,\nArk. (3/6/13) – Arkansas State University’s Sharika Nelvis will be on the national stage this weekend (March\n8-9) as a result of qualifying for the 2013 NCAA Indoor Track and Field Championships.",
"\n\nThe\n2013 NCAA-I Championships, which are being hosted by the University of\nArkansas, will be held at the Randal Tyson Track Center on the UA campus in Fayetteville, Ark.\n\nNelvis,\nthe 2013 Sun Belt Conference’s (SBC) Most Outstanding Women’s Indoor Track Performer, will be competing in the\nlong jump and 200-meter dash at nationals.",
"\nThe ASU junior is presently seeded 13th in the long jump at 6.32 meters\n(20-7.35) and clocked a 23.38 time in the 200-meters to claim a present 11th\nseeding.",
"\n\n“We\nare certainly pleased with Sharika qualifying for the NCAA indoor meet,” said\nASU head coach Jim Patchell. “",
"For the second year in a row, Sharika has\nqualified in both the long jump and 200-meter events. ",
"Last year, she qualified in the 60-meter\nhurdles and although the hurdles is probably her main event, we have changed a\nfew things about her hurdles race. ",
"She\nhasn’t quite got it yet, but she’s very close much like a golfer making a swing\nchange.\"",
"\n\nNelvis,\na three-time (1/9, 1/30 and 2/13) SBC Indoor Track Athlete of the Week pick this season,\nwas the 2013 SBC Women’s Indoor Track\nChampionships’ top scorer with 40.5 points and led the Red Wolves to the 2013\nSun Belt Indoor Championship title. ",
"She was\ncrowned the meet champion in four events – long jump, 55-meter hurdles,\n55-meter dash, and 200-meter dash to claim first-team All-Sun Belt recognition.",
"\n\n“Sometimes\nyou have to take a step back in order to make a leap forward,” Patchell\nsaid. “",
"However, qualifying in events\nthat aren’t necessarily her focus does say a lot about her talent and work\nethic. ",
"We’re excited to see how things\npan out this weekend.”",
"\n\nA\nseason ago, Nelvis garnered U.S. Track and Field/Cross Country Coaches\nAssociation (USTFCCCA) Second-Team All-America honors as a result of her\nperformance and participation in the 2012 NCAA Indoor Championships in Nampa,\nIdaho. ",
"Nelvis competed in the 60-meter\nhurdles, placing sixth in the first heat and then claiming ninth overall in the\nqualifiers to miss the finals. ",
"Nelvis\nran a time of 8.18 in the event which was only .02 seconds in back of the\neighth-place and final qualifier.",
"\n\nThe 2013 NCAA-I Indoor Track and Field Championships will be streamed live by ESPN3.com this Friday (March 8), beginning\nat 5:25 p.m. (CT) and this Saturday (March 9), starting at 5:50 p.m. (CT).",
"\nEarly portions of the meet not covered by ESPN3 will be streamed at NCAA.com\nand ArkansasRazorbacks.com, beginning with the start of competition each day. ",
"A\ntape-delay broadcast of the national meet will air Sunday, March 17, at 6 p.m. (CT)\non ESPNU.",
"\n\nFor tickets to the 2013 NCAA-I Indoor Track Championships, fans can log on\nto the online ticket center at ArkansasRazorbacks.com or call 800-982-HOGS\n(4647)"
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"The exhaust manifold is connected to an exhaust pipe which communicates exhaust gases into the open air. ",
"Internal combustion engines suffer from common drawbacks associated with the use of exhaust systems. ",
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"Therefore, reducing back pressures within the exhaust system causes the engine to operate more efficiently, provides enhanced performance, and increases the gas mileage of an internal combustion engine.",
"\nPrevious attempts to reduce back pressure within exhaust systems include removing or modifying the exhaust muffler. ",
"However, such attempts are replete with drawbacks. ",
"For example, removing the exhaust muffler eliminates one source of the back pressure, but also results in non-desirous engine noise. ",
"Other attempts to reduce back pressure within exhaust systems include positioning of a turbine within the exhaust pipe to draw the exhaust gases from the exhaust pipe. ",
"The internal turbine may be powered by the motion of the exhaust gases or be powered by a power source, such as an electric motor within the exhaust system. ",
"However, internal turbines powered by exhaust gases alone produce little effect. ",
"The efficiency of an internal turbine may be increased if the internal turbine is powered by an electric motor; however, electric motors operating within an exhaust pipe are subjected to high temperatures and corrosive effects from exhaust gases passing through the exhaust pipe. ",
"Electric motors operating in high temperature environments suffer from increased wear and tear and therefore require more frequent maintenance.",
"\nIn other attempts to reduce the back pressure of an exhaust system, an external turbine is positioned such that the blades of the external turbine are disposed about the exhaust pipe near the discharge end of the exhaust pipe. ",
"While use of an external turbine has met with success, this design can limit the air flow that reaches the external turbine due to the obstruction of air flow by the exhaust pipe itself. ",
"Further, the amount of surface area of the blades available for receiving air flow is reduced for a turbine having a selected diameter due to the space filled by the exhaust pipe, thereby lowering efficacy of the system.",
"\nTherefore, a need exists for an exhaust evacuation apparatus that reduces the back pressure of an exhaust system, and improves the efficiency and performance of an associated internal combustion engine while avoiding the drawbacks of the previous systems. ",
"It is to such an exhaust evacuation apparatus that the present invention is directed."
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"The observations of the\ncluster known as MACS J0025.4-1222 indicate that a titanic collision has\nseparated the dark from ordinary matter and provide an independent\nconfirmation of a similar effect detected previously in a target dubbed\nthe Bullet Cluster. ",
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"---\nauthor:\n- Richard Kowar\n- Otmar Scherzer\ntitle: Photoacoustic Imaging Taking into Account Attenuation\n---\n\nIntroduction\n============\n\n*Photoacoustic Imaging* is one of the recent hybrid imaging techniques, which attempts to visualize the distribution of the *electromagnetic absorption coefficient* inside a biological object. ",
"In photoacoustic experiments, the medium is exposed to a short pulse of a relatively low frequency electromagnetic (EM) wave. ",
"The exposed medium absorbs a fraction of the EM energy, heats up, and reacts with thermoelastic expansion. ",
"This induces acoustic waves, which can be recorded outside the object and used to determine the electromagnetic absorption coefficient. ",
"The combination of EM and ultrasound waves (which explains the usage of the term *hybrid*) allows one to combine high contrast in the EM absorption coefficient with high resolution of ultrasound. ",
"The method has demonstrated great potential for biomedical applications, including functional brain imaging of animals [@WanPanKuXieSto03], soft-tissue characterization, and early stage cancer diagnostics [@KruKisMilRey00], as well as imaging of vasculature [@ZhaLauBea07]. ",
"For a general survey on biomedical applications see [@XuWan06]. ",
"In comparison with the X-Ray CT, photoacoustics is non-ionizing. ",
"Its further advantage is that soft biological tissues display high contrasts in their ability to absorb frequency electromagnetic waves. ",
"For instance, for radiation in the near infrared domain, as produced by a Nd:YAG laser, the absorption coefficient in human soft tissues varies in the range of $0.1/\\textrm{cm}$–$0.5/\\textrm{cm}$ [@ChePraWel90]. ",
"The contrast is also known to be high between healthy and cancerous cells, which makes photoacoustics a promising early cancer detection technique. ",
"Another application arises in biology: Multispectral optoacoustic tomography technique is capable of high-resolution visualization of fluorescent proteins deep within highly light-scattering living organisms [@RazDisVinMaPerKoeNtz09]. ",
"In contrast, the current fluorescence microscopy techniques are limited to the depth of several hundred micrometers, due to intense light scattering.",
"\n\nDifferent terms are often used to indicate different excitation sources: *Optoacoustics* refers to illumination in the visible light spectrum, *Photoacoustics* is associated with excitations in the visible and infrared range, and *Thermoacoustics* corresponds to excitations in the microwave or radio-frequency range. ",
"In fact, the carrier frequency of the illuminating pulse is varying, which is usually not taken into account in mathematical modeling. ",
"Since the corresponding mathematical models are equivalent, in the mathematics literature, the terms opto-, photo-, and thermoacoustics are used interchangeably. ",
"In this article, we are addressing only the *photoacoustic tomographic technique* PAT (which is mathematically equivalent to the thermoacoustic tomography TAT).",
"\n\nVarious kinds of photoacoustic imaging techniques have been implemented. ",
"One should distinguish between photoacoustic *microscopy* (PAM) and *tomography* (PAT). ",
"In microscopy, the object is scanned pixel by pixel (or voxel by voxel). ",
"The measured pressure data provides an image of the electromagnetic absorption coefficient [@ZhaMasStoWan06]. ",
"Tomography, on the other hand, measures pressure waves with detectors surrounding completely or partially the object. ",
"Then the internal distribution of the absorption coefficients is reconstructed using mathematical inversion techniques (see the sections below).",
"\n\nThe common underlying mathematical equation of PAT is the *wave equation* for the pressure $$\\label{eq:ex:wave3d}\n\\boxed{\n\\frac{1}{c_0^2} \\frac{\\partial^2 p}{\\partial t^2}({{\\vec{x}}},t ) - \\nabla^2 p({{\\vec{x}}},t) = \\frac{d j}{d t}(t)\n \\left( \\frac{\\mu_{\\rm abs}({{\\vec{x}}}) \\beta ({{\\vec{x}}}) J({{{\\vec{x}}}})}{c_p({{\\vec{x}}})} \\right)\\,,\\;{{\\vec{x}}}\\in {{\\mathbb{R}}}^3,\\,t > 0 \\;.}$$ Here $c_p$ denotes the specific heat capacity, $J$ is the spatial intensity distribution, $\\mu_{\\rm abs}$ denotes the absorption coefficient, $\\beta$ denotes the thermal expansion coefficient and $c_0$ denotes the speed of sound, which is commonly assumed to be constant. ",
"The assumption that there is no acoustic pressure before the object is illuminated at time $t = 0$ is expressed by $$\\label{eq:ex:ini3d}\n\\boxed{\n p({{\\vec{x}}},t) = 0 \\,, \\qquad {{\\vec{x}}}\\in {{\\mathbb{R}}}^3, t < 0\\;.}$$ In PAT, $j(t)$ approximates a pulse, and can be considered as a $\\delta$-impulse $\\delta(t)$. Introducing the shorthand notations $$\\label{eq:u(x)}\n\\begin{aligned}\n\\rho({{\\vec{x}}}) := \\frac{\\mu_{\\rm abs}({{\\vec{x}}}) \\beta ({{\\vec{x}}}) J({{\\vec{x}}})}{c_p({{\\vec{x}}})}\\,,\n\\end{aligned}$$ one reduces (\\[eq:ex:wave3d\\]) and (\\[eq:ex:ini3d\\]) to $$\\label{eq:ex:ivp3d}\n\\boxed{\n\\frac{1}{c_0^2} \\frac{\\partial^2 p}{\\partial t^2} ({{\\vec{x}}},t) - \\nabla^2 p({{\\vec{x}}},t)= 0\\,,\\quad {{\\vec{x}}}\\in {{\\mathbb{R}}}^3, t > 0\\,,}$$ with initial values $$\\label{eq:init_values}\n\\boxed{\n p({{\\vec{x}}},0) = \\rho({{\\vec{x}}})\\,, \\quad \\frac{\\partial p}{\\partial t}({{\\vec{x}}},0) = 0 \\qquad {{\\vec{x}}}\\in {{\\mathbb{R}}}^3\\;.}$$ The quantity $\\rho$ in (\\[eq:ex:wave3d\\]) and (\\[eq:u(x)\\]) is a combination of several physical parameters. ",
"All along this paper $\\rho$ should not be confused with the source term $$\\label{eq:source}\n\\boxed{f({{\\vec{x}}},t) = \\frac{d j}{d t}(t) \\rho({{\\vec{x}}})\\,,\\quad {{\\vec{x}}}\\in {{\\mathbb{R}}}^3, t > 0 \\;.}$$ In PAT, some data about the pressure $p({{\\vec{x}}},t)$ are measured and the main task is to reconstruct the initial pressure $\\rho$ from these data. ",
"While the excitation principle is always as described above and thus (\\[eq:ex:ivp3d\\]) holds, the specific type of data measured depends on the type of transducers used, and thus influences the mathematical model.",
"\n\nNowadays there is a trend to incorporate more and more modeling into photoacoustic. ",
"In particular, taking into account locally varying *wave speed* and *attenuation*. ",
"Even more there is a novel trend to *qualitative photoacoustics*, which is concerned with estimating physical parameters from the imaging parameter of standard photoacoustics. ",
"In this paper we focus on attenuation correction, where we survey some recent progress. ",
"Inversion with varying wave speed has been considered for instance in [@AgrKuc07; @HriKucNgu08], and is not further discussed here.",
"\n\nThe outline of this paper is as follows: First, we review existing attenuation models and discuss their causality properties, which we believe to be essential for algorithms for inversion with attenuated data. ",
"Then, we survey causality properties of common attenuation models. ",
"We also derive integro-differential equations which the attenuated waves are satisfying. ",
"In addition we discuss the ill–conditionness of the inverse problem for calculating the unattenuated wave from the attenuated one.",
"\n\nAttenuation {#sec:attenuation}\n===========\n\nThe difficult issue of effects of and corrections for the attenuation of acoustic waves in PAT has been studied[@RivZhaAna06; @BurGruHalNusPal07; @PatGre06; @KowSchBon10], although no complete conclusion on the feasibility of these models has been reached.",
"\n\nMathematical models for describing attenuation are formulated in the frequency domain, taking into account that attenuation disperses high frequency components more rapidly over traveled distance. ",
"Let ${\\mathcal{G}}({{\\vec{x}}},t)$ denote the attenuated wave which originates from an impulse ($\\delta_{{{\\vec{x}}},t}$-distribution) at ${{\\vec{x}}}=0$ at time $t=0$. In mathematical terms ${\\mathcal{G}}$ is the Green-function of attenuated wave equation. ",
"Moreover, we denote by $$\\boxed{\n\\label{eq:g0} {\\mathcal{G}}_0({{\\vec{x}}},t) =\n\\frac{\\delta\\left(t-\\frac{{\\left|{{\\vec{x}}}\\right|}}{c_0}\\right)}{4\\,\\pi\\,{\\left|{{\\vec{x}}}\\right|}}}\\index{Green function, standard wave equation}$$ the Green function of the unattenuated wave equation; that is, it is the solution of , with constant sound speed $c(x) \\equiv c_0$ and initial conditions $${\\mathcal{G}}_0({{\\vec{x}}},0)=0 \\quad\\mbox{ and}\\quad\n \\frac{\\partial {\\mathcal{G}}_0}{\\partial t}({{\\vec{x}}},0)=\\delta_{{{\\vec{x}}},t}\\,.$$ Common mathematical formulations of *attenuation* assume that $$\\label{eq:GG0}\n\\boxed{\n{\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}({{\\vec{x}}},\\omega) = {\\exp \\left( -\\beta^*\n({\\left|{{\\vec{x}}}\\right|},\\omega) \\right)}\\, {\\mathcal{F} \\left\\{ {\\mathcal{G}}_0 \\right\\}}({{\\vec{x}}},\\omega)\\,, \\quad {{\\vec{x}}}\\in {{\\mathbb{R}}}^3,\\, \\omega \\in {{\\mathbb{R}}}\\;.}$$ Here ${\\mathcal{F} \\left\\{ \\cdot \\right\\}}$ denotes the Fourier transform with respect to time $t$ (cf. ",
"Appendix \\[sec:app\\]). ",
"Applying the inverse Fourier transform ${\\mathcal{F}^{-1} \\left\\{ \\cdot \\right\\}}$ to [(\\[eq:GG0\\])]{} gives $$\\label{eq:FGG0}\n\\boxed{{\\mathcal{G}}({{\\vec{x}}},t) = K({{\\vec{x}}},t) *_t {\\mathcal{G}}_0({{\\vec{x}}},t)}\n\\qquad \\mbox{($*_t$ time convolution)}$$ where $$\\label{eq:kernel}\nK({{\\vec{x}}},t) := \\frac{1}{\\sqrt{2\\,\\pi}} {\\mathcal{F}^{-1} \\left\\{ {\\exp \\left( -\\beta^* ({\\left|{{\\vec{x}}}\\right|},\\cdot) \\right)} \\right\\}}(t)\\;.$$ From (\\[eq:FGG0\\]) and (\\[eq:g0\\]) it follows that $$\\begin{aligned}\n{\\mathcal{G}}({{\\vec{x}}},t) &= K({{\\vec{x}}},t) *_t {\\mathcal{G}}_0({{\\vec{x}}},t)\\\\\n&= \\int_{{\\mathbb{R}}}K({{\\vec{x}}},t-\\tau) \\frac{\\delta (\\tau - \\frac{{\\left|{{\\vec{x}}}\\right|}}{c_0})}{4 \\pi {\\left|{{\\vec{x}}}\\right|}}\\,d\\tau \\\\\n&= \\frac{K\\left({{\\vec{x}}},t-\\frac{{\\left|{{\\vec{x}}}\\right|}}{c_0}\\right)}{4\\pi {\\left|{{\\vec{x}}}\\right|}}\\;.",
"\n\\end{aligned}$$ Consequently, $$\\label{eq:Kandgreen}\n\\boxed{\n{\\mathcal{G}}({{\\vec{x}}},t+{\\left|{{\\vec{x}}}\\right|}/c_0) = K({{\\vec{x}}},t)/(4\\pi {\\left|{{\\vec{x}}}\\right|})\\;.}$$ Moreover, we emphasize that the Fourier transform of a real and even (real and odd) function is real and even (imaginary and odd). ",
"Since ${\\mathcal{G}}$ and ${\\mathcal{G}}_0$ are real valued, $K$ must be real valued and consequently the real part $\\Re(\\beta^*)$ of $\\beta^*$ has to be even with respect to the frequency $\\omega$ and $\\Im(\\beta^*)$ has to be odd with respect to $\\omega$. Attenuation is caused if $\\Re(\\beta^*)$ is positive and since then $\\beta^*$ has a nonzero imaginary part due to the Kramers-Kronig relation, attenuation causes dispersion. ",
"In the literature the following product ansatz is commonly used $$\\label{eq:attenuation_law}\n\\boxed{\n\\beta^* ({\\left|{{\\vec{x}}}\\right|},\\omega) = \\alpha^*(\\omega)\\,{\\left|{{\\vec{x}}}\\right|} \\qquad \\omega\\in{{\\mathbb{R}}},\\,{{\\vec{x}}}\\in{{\\mathbb{R}}}^3\\;.}$$ In the sequel we concentrate on these models and use the following terminology:\n\nWe call $\\beta^*$ of standard form if (\\[eq:attenuation\\_law\\]) holds. ",
"Then the function $$\\label{eq:attenuation_coefficient}\n\\alpha^*: {{\\mathbb{R}}}\\to {{\\mathbb{C}}}$$ is called *standard attenuation coefficient* and $\\alpha = \\Re(\\alpha^*)$ is called the *attenuation law*. ",
"We also call $\\beta^*$ the *attenuation coefficient*.",
"\n\nFrom the relation (\\[eq:attenuation\\_law\\]), it follows that $\\Re (\\alpha^*)$ is even, $\\Im (\\alpha^*)$ is odd, and $\\Re (\\alpha^*)>0$ (the last inequality guarantees attenuation).",
"\n\nIn the following we summarize common attenuation coefficients and laws: In what follows $\\alpha_0$ denotes a positive parameter and $$\\label{ta0} {\\tilde{\\alpha}}_0 = \\frac{\\alpha_0}{\\cos\n\\left(\\frac{\\pi}{2}\\gamma\\right)} \\qquad\\quad (0 < \\gamma \\not\\in {{\\mathbb{N}}})\\,$$ is a possibly non-positive coefficient.",
"\n\n- [**Frequency Power Laws:**]{}\n\n - Let $0 < \\gamma \\not\\in {{\\mathbb{N}}}$. The frequency power law *attenuation coefficient* is defined by $$\\label{eq:powlaw1}\n \\alpha_{pl}^*(\\omega) = {\\tilde{\\alpha}}_0 (-{{\\mathrm{i}}\\,}\\omega)^\\gamma = {\\tilde{\\alpha}}_0 {\\left|\\omega\\right|}^\\gamma \\left(\n \\cos \\left(\\frac{\\pi}{2} \\gamma \\right) - {{\\mathrm{i}}\\,}{\\mbox{sgn}}(\\omega) \\sin \\left(\\frac{\\pi}{2} \\gamma \\right)\\right)$$ for $\\omega \\in {{\\mathbb{R}}}$. Therefore, the *attenuation law* is given by $$\\label{eq:powlaw1b}\n \\boxed{\n \\alpha_{pl}(\\omega) = \\alpha_0\\,{\\left|\\omega\\right|}^\\gamma\\;.}$$ These models have been considered for instance in [@Sza94; @Sza95; @WatHugBraMil00; @WaMoMi05].",
"\n\n - Let $\\gamma=1$ and $\\omega_0\\neq0$, the attenuation coefficient is defined by $$\\label{alphagamma1}\n \\begin{aligned}\n \\alpha_{pl}^*(\\omega)\n := \\alpha_0\\,{\\left|\\omega\\right|}\n + {{\\mathrm{i}}\\,}\\frac{2}{\\pi}\\,\\alpha_0\\,\\omega\\,\\log\\left|\\frac{\\omega}{\\omega_0}\\right|\n \\qquad \\omega\\in{{\\mathbb{R}}}\\,.",
"\n \\end{aligned}$$ The attenuation law is $$\\label{alphagamma1a}\n \\boxed{\n \\alpha_{pl}(\\omega):=\\alpha_0\\,{\\left|\\omega\\right|}\\;.}$$ This model has been considered in [@Sza95; @WaMoMi05].",
"\n\n- [[**Szabo:**]{}]{} Let $0 < \\gamma\\not\\in{{\\mathbb{N}}}$. The attenuation coefficient [^1] of Szabo’s law is defined by $$\\label{alpha*szabo}\n \\begin{aligned}\n \\alpha_{sz}^*(\\omega) &= \\frac{1}{c_0}\\,\\sqrt{ (-{{\\mathrm{i}}\\,}\\omega)^2\n + 2 {\\tilde{\\alpha}}_0 c_0 (-{{\\mathrm{i}}\\,}\\omega)^{\\gamma+1} }\n + {{\\mathrm{i}}\\,}\\frac{\\omega}{c_0}\\;.",
"\n \\end{aligned}$$ We denote Szabo’s attenuation law by $$\\boxed{\n \\alpha_{sz}(\\omega) := \\Re (\\alpha_{sz}^*(\\omega))\\;.}$$ For small frequencies $\\alpha_{sz}(\\omega)$ behaves like $\\alpha_0\\,|\\omega|^\\gamma$. This model has been considered in [@Sza94; @Sza95] where, in addition, also a model for $\\gamma \\in {{\\mathbb{N}}}$ has been introduced.",
"\n\n- [[**Thermo-Viscous Attenuation Law:**]{} (see e.g. [@KinFreCopSan00; @Sza94]):]{} Here, for $\\tau_0>0$, the attenuation coefficient is defined by $$\\label{alpha*th}\n \\begin{aligned}\n \\alpha_{tv}^*(\\omega)\n = \\frac{-{{\\mathrm{i}}\\,}\\omega}{c_0\\,\\sqrt{1-{{\\mathrm{i}}\\,}\\tau_0\\,\\omega}}\n + \\frac{{{\\mathrm{i}}\\,}\\omega}{c_0}\\;\n \\end{aligned}$$ with attenuation law $$\\label{alpha*threal}\n \\begin{aligned}\n \\boxed{\n \\alpha_{tv}(\\omega)\n = \\frac{\\tau_0\\, \\omega^2}{ \\sqrt{2}\\,c_0\\,\\sqrt{ (1+\\sqrt{1+(\\tau_0\\,\\omega)^2})\\,(1+(\\tau_0\\,\\omega)^2)} } \\;.}",
"\n \\end{aligned}$$ For small frequencies $\\alpha_{tv}(\\omega)$ behaves like $\\frac{\\tau_0\\,\\omega^2}{2\\,c_0}$. That is the thermo-viscous law approximates a power attenuation law with exponent $2$.\n\n- [[**Nachman, Smith and Waag [@NacSmiWaa90]:**]{}]{} Consider a homogeneous and isotropic fluid with density $\\rho_0$ in which $N$ relaxation processes take place. ",
"Then the attenuation coefficient of the model in [@NacSmiWaa90] reads as follows: $$\\label{alpha*Nachman+}\n \\begin{aligned}\n \\alpha_{nsw}^*(\\omega)\n = \\frac{-{{\\mathrm{i}}\\,}\\omega}{c_0}\\,\\left[\n \\frac{c_0}{\\tilde c_0}\\, \\sqrt{ \\frac{1}{N}\\, \\sum_{m=1}^N\n \\frac{1-{{\\mathrm{i}}\\,}\\,\\tilde\\tau_m\\,\\omega}{1-{{\\mathrm{i}}\\,}\\tau_m\\,\\omega} }\n -1 \\right]\\;.",
"\n \\end{aligned}$$ All parameters appearing in (\\[alpha\\*Nachman+\\]) are positive and real. ",
"$\\kappa_m$ and $\\tau_m$ denote the compression modulus and the relaxation time of the $m-$th relaxation process, respectively, and $$\\label{deftildec0tau0}\n \\tilde c_0 := \\frac{c_0}{\\sqrt{1+\\sum_{m=1}^N c_0^2\\,\\rho_0\\,\\kappa_m}}\n \\quad\\mbox{and}\\quad\n \\tilde \\tau_m := \\tau_m\\,(1-N\\,\\tilde c_0^2\\,\\rho_0\\,\\kappa_m)\\,.$$ The last two definitions imply that $$\\label{proptildec0tau0}\n \\frac{\\tilde c_0^2}{c_0^2} = \\frac{1}{N}\\,\\sum_{m=1}^N \\frac{\\tilde \\tau_m}{\\tau_m} \\;.$$ We denote the according attenuation law by [^2] $$\\boxed{\n \\alpha_{nsw}(\\omega) := \\Re (\\alpha_{nsw}^*(\\omega))\\;.}$$\n\n- [[**Greenleaf and Patch [@PatGre06]**]{}]{} consider for $\\gamma \\in {\\{ 1,2 \\}}$ the attenuation coefficient $$\\alpha_{gp}^*(\\omega) = \\alpha_0\\,{\\left|\\omega\\right|}^\\gamma\\,,$$ which, since it is real, equals the attenuation law $$\\label{alpha*Patch}\n \\boxed{\n \\alpha_{gp}(\\omega) = \\Re(\\alpha_{gp}^*(\\omega))\\;.}$$\n\n- [**Chen and Holm [@CheHolm04]:**]{} This model describes the attenuation as a function of the absolute value of the vector-valued wave number ${\\mathbf{k}}\\in{{\\mathbb{R}}}^3$ (instead of the frequency $\\omega\\in{{\\mathbb{R}}}$). ",
"Let ${\\mathcal{F}}_{3D}$ denote the $3D-$Fourier transform $$\\begin{aligned}\n {\\mathcal{F}}_{3D} {\\{ f({\\mathbf{k}}) \\}}({{\\vec{x}}})\n = \\frac{1}{\\sqrt{(2\\,\\pi)^3}}\\,\\int_{{{\\mathbb{R}}}^3} {\\exp \\left( {{\\mathrm{i}}\\,}{{\\vec{x}}}\\cdot{\\mathbf{k}}\\right)}\\, f({\\mathbf{k}}) \\,{\\mbox{d}}{\\mathbf{k}}\\,,\n \\end{aligned}$$ then the Green function of the attenuated equation is defined by $$\\label{GreenChenHolm}\n \\begin{aligned}\n {\\mathcal{G}}({{\\vec{x}}},t) = \\frac{H(t)\\,c_0^2}{(2\\,\\pi)^{3/2}}\\,{\\mathcal{F}}_{3D}\\left\\{ {\\exp \\left( A(\\cdot)\\,t \\right)}\\, \\frac{\\sin(B(\\cdot)\\,t)}{B(\\cdot)} \\right\\}({{\\vec{x}}})\n \\end{aligned}$$ where, for given $\\alpha_1>0$, $$\\label{AkBk}\n A({\\mathbf{k}}) := -\\alpha_1\\,c_0\\,{\\left|{\\mathbf{k}}\\right|}^\\gamma\\,,\n \\qquad\n B({\\mathbf{k}}) := c_0\\,\\sqrt{{\\left|{\\mathbf{k}}\\right|}^2-\\alpha_1^2\\,{\\left|{\\mathbf{k}}\\right|}^{2\\,\\gamma}}\\,.$$\n\n- [**In [@KowSchBon10]**]{} we proposed $$\\label{eq:powlaw2}\n \\begin{aligned}\n \\alpha_{ksb}^*(\\omega)\n = \\frac{\\alpha_0\\,(-{{\\mathrm{i}}\\,}\\omega)}{c_0\\,\\sqrt{1+(-{{\\mathrm{i}}\\,}\\tau_0\\,\\omega)^{\\gamma-1}}}\n \\qquad\\qquad (\\gamma\\in (1,2],\\,\\tau_0>0)\\,,\n \\end{aligned}$$\\[eq:powlaw2a\\] where the square root is again the complex root with positive real part.",
"\n\n Let $\\gamma\\in (1,2]$. Then, for small frequencies we have $$\\boxed{\\alpha_{ksb}(\\omega) \\approx \\frac{\\alpha_0\\,\\sin(\\frac{\\pi}{2}(\\gamma-1))}{2\\,c_0\\,\\tau_0} \\,|\\tau_0\\,\\omega|^\\gamma >0\\,.}$$ Thus our model behaves like a power law for small frequencies.",
"\n\nDistinctive features of unattenuated wave propagation (,i.e. the solution of the standard wave equation) are *causality* and *finite wave front velocity*. ",
"It is reasonable to assume that the attenuated wave satisfies the same distinctive properties as well. ",
"In the following we analyze causality properties of the standard attenuation models.",
"\n\nCausality\n=========\n\nIn the following we present some abstract definitions and basic notations. ",
"In the remainder ${{\\vec{x}}}$ will always denote a vector in three dimensional space. ",
"When we speak about functions, we always mean generalized functions, such as for instance distributions or tempered distributions - we recall the definitions of (tempered) distribution in the course of the paper.",
"\n\n\\[def:defAc\\] A function $f:=f({{\\vec{x}}},t)$ defined on the Euclidean space over time (i.e. in ${{\\mathbb{R}}}^4$) is said to be *causal* if it satisfies $f({{\\vec{x}}},t) = 0$ for $t < 0$.\n\nLet ${\\mathcal{A}}: D \\to D$ be a linear operator, where $\\emptyset \\neq D$ is an appropriate set of functions from ${{\\mathbb{R}}}^4$ to ${{\\mathbb{R}}}$. In this paper we always assume that ${\\mathcal{A}}$ satisfies the following properties:\n\n- ${\\mathcal{A}}$ is *shift invariant* in space and time. ",
"That is, for every function $f$ and every shift $L:=L({{\\vec{x}}},t):=({{\\vec{x}}}-{{\\vec{x}}}_0,t-t_0)$, with ${{\\vec{x}}}_0 \\in {{\\mathbb{R}}}^3$ and $t_0 \\in {{\\mathbb{R}}}$, it holds that $${\\mathcal{A}}(f \\circ L) = ({\\mathcal{A}}f) \\circ L\\;.$$\n\n- ${\\mathcal{A}}$ is *rotation invariant* in space. ",
"That is, for every function $f$ and every rotation matrix $R$, it holds that $${\\mathcal{A}}(R f) = R({\\mathcal{A}}f)\\;.$$\n\n- ${\\mathcal{A}}$ is *causal*. ",
"That is, it maps causal functions to causal functions. ",
"From (\\[eq:a\\_conv\\]) it follows that ${\\mathcal{A}}$ is causal, if and only if the associated Green function is causal.",
"\n\n\\[def:defA\\] The *Green function* of ${\\mathcal{A}}$ is defined by $${\\mathcal{G}}:={\\mathcal{G}}({{\\vec{x}}},t)={\\mathcal{A}}\\delta_{{{\\vec{x}}},t} ({{\\vec{x}}},t)\\;.$$\n\nThe operator ${\\mathcal{A}}$ is uniquely determined by ${\\mathcal{G}}$ and vice versa. ",
"This follows from the fact that $$\\label{eq:a_conv}\n\\begin{aligned}\n{\\mathcal{A}}f ({{\\vec{x}}}_0,t_0) &= {\\mathcal{A}}\\left( \\int_{{\\mathbb{R}}}\\int_{{{\\mathbb{R}}}^3} f({{\\vec{x}}}_0-{{\\vec{x}}},t-t_0)\n\\delta_{{{\\vec{x}}},t} ({{\\vec{x}}},t)\\,d {{\\vec{x}}}dt \\right) \\\\\n&= \\int_{{\\mathbb{R}}}\\int_{{{\\mathbb{R}}}^3} f({{\\vec{x}}}_0-{{\\vec{x}}},t-t_0) {\\mathcal{G}}({{\\vec{x}}},t)\\,d {{\\vec{x}}}dt\\,.",
"\n\\end{aligned}$$\n\nMoreover, we use the following terminology and abbreviations:\n\n- From the rotation invariance of ${\\mathcal{A}}$ it follows that $$\\label{eq:T}\n \\hat{T}({{\\vec{x}}}):=\\sup {\\{ t : {\\mathcal{G}}({{\\vec{x}}},\\tau) = 0 \\text{ for all } \\tau \\leq t \\}}\\,,$$ is rotationally symmetric, which allows us to use the shorthand notation $$\\label{defT}\n T(r) = \\hat{T}({{\\vec{x}}}) \\text{ where } r = {\\left|{{\\vec{x}}}\\right|}\\;.$$ With this notation (\\[eq:T\\]) can be equivalently expressed as $$\\label{eq:defT(r)0a}\n {\\mathcal{G}}\\left({{\\vec{x}}}, t + T({\\left|{{\\vec{x}}}\\right|}) \\right) = 0 \\text{ for every } t<0\\;.$$ In physical terms $T({\\left|{{\\vec{x}}}\\right|})$ denotes the *travel time of a wave front* originating at position $\\mathbf{0}$ at time $t=0$ and traveling to ${{\\vec{x}}}$.\n\n- Because ${\\mathcal{G}}$ is rotationally symmetric we can write $${\\mathcal{G}}({{\\vec{x}}},T({\\left|{{\\vec{x}}}\\right|})) = \\hat{{\\mathcal{G}}}(r,T(r)) \\text{ with } r = {\\left|{{\\vec{x}}}\\right|}\\;.$$ Taking the inverse function of $T$, which we denote by $S=S(t)$, we then find $${\\mathcal{G}}({{\\vec{x}}},T({\\left|{{\\vec{x}}}\\right|})) = \\hat{{\\mathcal{G}}}(S(t),t)\\,,$$\n\n- The *wave front* is the set $${\\cal W}:={\\{ ({{\\vec{x}}},T({\\left|{{\\vec{x}}}\\right|})): {{\\vec{x}}}\\in {{\\mathbb{R}}}^3 \\}}\\;$$\n\n- The *wave front speed* $V$ is the variation of the location of the wave front as a function of time. ",
"That is, $$\\label{eq:v}\n V(t)=\\frac{{\\mbox{d}}S}{{\\mbox{d}}t}(t)= \\left. ",
"\\frac{1}{T'(r)}\\right|_{r=S(t)}\\,.$$ Here $T'$ denotes the derivative with respect to the radial component $r$.\n\n- We say that ${\\mathcal{A}}$ has a *finite speed of propagation* if there exists a constant $\\hat{c}_0$ such that $$\\label{eq:defT(r)}\n 0 < \\left(T'(r)\\right)^{-1} \\leq \\hat{c}_0 < \\infty\\;.$$ In this case it follows from (\\[eq:v\\]) that the wave front velocity satisfies $$\\label{eq:v2}\n V(t) \\leq \\hat{c}_0 < \\infty\\;.$$\n\n- We call an operator ${\\mathcal{A}}$ strongly causal, if it is causal and satisfies the finite propagation speed property.",
"\n\nThe following lemma addresses the case of attenuation coefficients of standard form and gives examples of strongly causal operators ${\\mathcal{A}}$.\n\n\\[lemm:cconst\\] Let $\\beta^*({\\left|{{\\vec{x}}}\\right|},\\omega) = \\alpha^*(\\omega) {\\left|{{\\vec{x}}}\\right|}$ be of the standard form (\\[eq:attenuation\\_law\\]) and ${\\mathcal{A}}$ (\\[eq:a\\_conv\\]) be the operator defined by the Green function ${\\mathcal{G}}$, which is defined in (\\[eq:FGG0\\]). ",
"Then ${\\mathcal{A}}$ is strongly causal if and only if for every ${{\\vec{x}}}\\in {{\\mathbb{R}}}^3$ the function $$t \\to \\frac{1}{\\sqrt{2\\pi}} {\\mathcal{F}^{-1} \\left\\{ {\\exp \\left( - \\alpha^*(\\omega){\\left|{{\\vec{x}}}\\right|} \\right)} \\right\\}}\\,,$$ defined in (\\[eq:kernel\\]), is causal.",
"\n\nWe assume that ${\\mathcal{A}}$ is strongly causal. ",
"It follows from [@KowSchBon10 Theorem 3.1] that there exists a constant $c$, which is smaller than or equal to the wave speed $c_0$ from (\\[eq:ex:ivp3d\\]), which satisfies $T({\\left|{{\\vec{x}}}\\right|}) = \\frac{{\\left|{{\\vec{x}}}\\right|}}{c}$ for all ${{\\vec{x}}}\\in {{\\mathbb{R}}}^3$. Using the definitions of the travel time $T({\\left|x\\right|})$ and (\\[eq:kernel\\]), it follows from (\\[eq:Kandgreen\\]) that $t\\to K({{\\vec{x}}},t)$ is causal.",
"\n\nNow, for every ${{\\vec{x}}}\\in {{\\mathbb{R}}}^3$ let $K$ be causal. ",
"Then from (\\[eq:Kandgreen\\]) it follows that $t \\to {\\mathcal{G}}\\left({{\\vec{x}}},t+{\\left|{{\\vec{x}}}\\right|}/c_0)\\right)$ is causal. ",
"Since $T({\\left|{{\\vec{x}}}\\right|})$ denotes the largest positive time period for which $t \\to {\\mathcal{G}}\\left({{\\vec{x}}},t+T({\\left|{{\\vec{x}}}\\right|})\\right)$ is causal, we have for $r>0$: $$\\label{eq:lem1Tc0}\n 0 < \\frac{r}{c_0} \\leq T(r) = \\int_0^{r} \\frac{1}{V(s)}\\;.$$ Here $V$ is parameterized with respect to the distance $s$ at time $t$ of the wave front from its origin. ",
"As shown in the proof of [@KowSchBon10 Theorem 3.1], the fact that $\\beta^*$ is of standard form together with (\\[eq:lem1Tc0\\]) implies that there exist a constant $c$ such that $T(r) = r/c$ for all $r >0$. But then from (\\[eq:lem1Tc0\\]) it follows $0<r/c_0\\leq r/c<\\infty$ and consequently $0 < c \\leq c_0 < \\infty$.\n\nFinally we explain the above notation for the standard wave equation:\n\nIn the case of the standard wave equation the wave front is the support of the Green function ${\\mathcal{G}}_0$, the wave front velocity is $c_0$, and $T({\\left|{{\\vec{x}}}\\right|})=\\frac{{\\left|{{\\vec{x}}}\\right|}}{c_0}$ denotes the travel time of the wave front.",
"\n\nStrong Causality of Attenuation Laws {#sec:n}\n====================================\n\nIn this section we analyze causality properties of attenuation laws. ",
"We split the section into two parts, where the first concerns numerical studies to determine the kernel function $K$, defined in [(\\[eq:kernel\\])]{}, and the second part contains analytical investigations.",
"\n\nIn Figures \\[fig:powlaw\\], \\[fig:szabo\\] and \\[fig:thviscous\\] we represent the attenuation kernels according to power, Szabo’s, and the thermo-viscous law.",
"\n\n![",
"Simulation of $K({{\\vec{x}}},t)$ for the frequency power law with $(\\gamma,\\alpha_0)\\in\n{\\{ (0.5,0.1581),\\,(1.5,0.0316),\\,(2.7,0.0071),\\,(3.3,0.0027) \\}}$, $c_0=1$ and ${\\left|{{\\vec{x}}}\\right|}=\\frac{1}{4}$. In the first example $\\gamma < 1$ and thus the function is causal. ",
"For all other cases it is non causal.[]{data-label=\"fig:powlaw\"}](PowLaw01.eps \"fig:\"){height=\"4.0cm\"} ![",
"Simulation of $K({{\\vec{x}}},t)$ for the frequency power law with $(\\gamma,\\alpha_0)\\in\n{\\{ (0.5,0.1581),\\,(1.5,0.0316),\\,(2.7,0.0071),\\,(3.3,0.0027) \\}}$, $c_0=1$ and ${\\left|{{\\vec{x}}}\\right|}=\\frac{1}{4}$. In the first example $\\gamma < 1$ and thus the function is causal. ",
"For all other cases it is non causal.[]{data-label=\"fig:powlaw\"}](PowLaw02.eps \"fig:\"){height=\"4.0cm\"}\\\n![",
"Simulation of $K({{\\vec{x}}},t)$ for the frequency power law with $(\\gamma,\\alpha_0)\\in\n{\\{ (0.5,0.1581),\\,(1.5,0.0316),\\,(2.7,0.0071),\\,(3.3,0.0027) \\}}$, $c_0=1$ and ${\\left|{{\\vec{x}}}\\right|}=\\frac{1}{4}$. In the first example $\\gamma < 1$ and thus the function is causal. ",
"For all other cases it is non causal.[]{data-label=\"fig:powlaw\"}](PowLaw03.eps \"fig:\"){height=\"4.0cm\"} ![",
"Simulation of $K({{\\vec{x}}},t)$ for the frequency power law with $(\\gamma,\\alpha_0)\\in\n{\\{ (0.5,0.1581),\\,(1.5,0.0316),\\,(2.7,0.0071),\\,(3.3,0.0027) \\}}$, $c_0=1$ and ${\\left|{{\\vec{x}}}\\right|}=\\frac{1}{4}$. In the first example $\\gamma < 1$ and thus the function is causal. ",
"For all other cases it is non causal.[]{data-label=\"fig:powlaw\"}](PowLaw04.eps \"fig:\"){height=\"4.0cm\"}\n\n![",
"Simulation of $K({{\\vec{x}}},t)$ for Szabo’s frequency law with $(\\gamma,\\alpha_0)\\in\n{\\{ (0.5,0.1581),\\,(1.5,0.0316),\\,(2.7,0.0071),\\,(3.3,0.0027) \\}}$, $c_0=1$ and ${\\left|{{\\vec{x}}}\\right|}=\\frac{1}{4}$. []{data-label=\"fig:szabo\"}](Szabo01.eps \"fig:\"){height=\"4.0cm\"} ![",
"Simulation of $K({{\\vec{x}}},t)$ for Szabo’s frequency law with $(\\gamma,\\alpha_0)\\in\n{\\{ (0.5,0.1581),\\,(1.5,0.0316),\\,(2.7,0.0071),\\,(3.3,0.0027) \\}}$, $c_0=1$ and ${\\left|{{\\vec{x}}}\\right|}=\\frac{1}{4}$. []{data-label=\"fig:szabo\"}](Szabo02.eps \"fig:\"){height=\"4.0cm\"}\\\n![",
"Simulation of $K({{\\vec{x}}},t)$ for Szabo’s frequency law with $(\\gamma,\\alpha_0)\\in\n{\\{ (0.5,0.1581),\\,(1.5,0.0316),\\,(2.7,0.0071),\\,(3.3,0.0027) \\}}$, $c_0=1$ and ${\\left|{{\\vec{x}}}\\right|}=\\frac{1}{4}$. []{data-label=\"fig:szabo\"}](Szabo03.eps \"fig:\"){height=\"4.0cm\"} ![",
"Simulation of $K({{\\vec{x}}},t)$ for Szabo’s frequency law with $(\\gamma,\\alpha_0)\\in\n{\\{ (0.5,0.1581),\\,(1.5,0.0316),\\,(2.7,0.0071),\\,(3.3,0.0027) \\}}$, $c_0=1$ and ${\\left|{{\\vec{x}}}\\right|}=\\frac{1}{4}$. []{data-label=\"fig:szabo\"}](Szabo04.eps \"fig:\"){height=\"4.0cm\"}\n\n![[*",
"Left:*]{} $K({{\\vec{x}}},t)$ defined by the complex thermo-viscous attenuation law with $\\tau_0=10^{-5}$, $c_0=1$ and fixed ${\\left|{{\\vec{x}}}\\right|}=\\frac{1}{4}$. [*Right:*]{} The proposed law (\\[eq:powlaw2\\]) for $\\gamma=2$ with $\\alpha_1=1$, $\\tau_0=10^{-5}$, $c_0=1$ and fixed ${\\left|{{\\vec{x}}}\\right|}=\\frac{1}{4}$ is causal. []{",
"data-label=\"fig:thviscous\"}](ThViscous01.eps \"fig:\"){height=\"4.0cm\"} ![[*",
"Left:*]{} $K({{\\vec{x}}},t)$ defined by the complex thermo-viscous attenuation law with $\\tau_0=10^{-5}$, $c_0=1$ and fixed ${\\left|{{\\vec{x}}}\\right|}=\\frac{1}{4}$. [*Right:*]{} The proposed law (\\[eq:powlaw2\\]) for $\\gamma=2$ with $\\alpha_1=1$, $\\tau_0=10^{-5}$, $c_0=1$ and fixed ${\\left|{{\\vec{x}}}\\right|}=\\frac{1}{4}$ is causal. []{",
"data-label=\"fig:thviscous\"}](CThViscous01.eps \"fig:\"){height=\"4.0cm\"}\n\nThe figures already indicate that power laws with index greater than $1$ violate causality. ",
"In the following we support these computational studies by analytical considerations. ",
"Thereby we make use of distribution theory, which we recall first. ",
"Generally speaking *Distributions* are generalized functions:\n\n\\[def:distribution\\] We use the abbreviations:\n\n- ${\\mathcal{D}}:=C_0^\\infty({{\\mathbb{R}}},{{\\mathbb{C}}})$ is the space of infinitely often differentiable functions from ${{\\mathbb{R}}}$ to ${{\\mathbb{C}}}$ which have compact support.",
"\n\n- ${\\mathcal{S}}$ is the space of infinitely often differentiable functions from ${{\\mathbb{R}}}$ to ${{\\mathbb{C}}}$ which are *rapidly decreasing*. ",
"A function $f: {{\\mathbb{R}}}\\to {{\\mathbb{C}}}$ is rapidly decreasing if for all $i,j \\in {{\\mathbb{N}}}_0$ $${\\left|x\\right|}^i {\\left|f^{(j)}(x)\\right|} \\to 0 \\text{ for } {\\left|x\\right|} \\to \\infty\\;.$$\n\n- ${\\mathcal{S}}$ is a locally convex space (see [@Yos95] for a definition) with the topology induced by the family of semi-norms $$p_{P,j}(f) = \\sup_{x \\in {{\\mathbb{R}}}} {\\left|P(x) f^{(j)}(x)\\right|} ,$$ where $P$ is a polynomial and $j \\in {{\\mathbb{N}}}_0$. The topology on a locally convex set is defined as follows: $U \\subseteq {\\mathcal{S}}$ is open, if for every $f \\in U$ there exists ${\\varepsilon}> 0$ and a finite non-empty set $J'$ of polynomials and a finite set of indices $K'$ such that $$\\bigcap_{P \\in J', k \\in K'} {\\{ g \\in {\\mathcal{S}}: p_{P,k}(g-f) < {\\varepsilon}\\}} \\subseteq U\\;.$$\n\n- The space of *tempered distributions*, ${\\mathcal{S}}'$, is the space of linear continuous functionals and ${\\mathcal{S}}$.\n\n- A functional $L: {\\mathcal{S}}\\to {{\\mathbb{C}}}$ is continuous if there exists a constant $C > 0$ and a seminorm $p_{P,j}$ such that $${\\left|L u\\right|} \\leq C p_{P,j}(u)\\,, \\text{ for every } u \\in {\\mathcal{S}}$$ (see [@Yos95 Sect. ",
"I.6, Thm. ",
"1]\n\nIn the following we give some examples of tempered distributions and review some of their properties. ",
"The examples are taken from [@Yos95 Sec6.2, Ex. ",
"3] and [@DauLio02_2 Remark 6].",
"\n\n[**(Examples of Tempered Distributions)**]{} \\[ex:distr\\_ex\\]\n\n- Let $1 \\leq p \\leq \\infty$ and $f \\in L^p({{\\mathbb{R}}},{{\\mathbb{C}}})$, then the linear operator $T\\phi = \\int_{{{\\mathbb{R}}}} f(x)\\phi(x)\\,dx$ is a tempered distribution. ",
"In the following we identify $f$ and $T$, and this clarifies the terminology $f \\in {\\mathcal{S}}'$ later on.",
"\n\n- ${\\mathcal{S}}\\subseteq {\\mathcal{S}}'$ - thereby already the above relation between functions and tempered distributions is used.",
"\n\n- Distributions with compact support are tempered distributions. ",
"For instance the $\\delta$-Distribution is a tempered distribution.",
"\n\n- Polynomials are tempered distributions.",
"\n\n- The functions $f$ of $L_{loc}^1({{\\mathbb{R}}})$ which are uniformly bounded by a polynomial for ${\\left|x\\right|}$ sufficiently large, are tempered distributions. [",
"^3]\n\n\\[le:distr\\_ex\\]\n\n- The pointwise limit $f:{{\\mathbb{R}}}\\to {{\\mathbb{C}}}$ of a sequence of functions ${\\{ f_n:{{\\mathbb{R}}}\\to {{\\mathbb{C}}}\\}} \\subseteq {\\mathcal{S}}'$, is again a tempered distribution.",
"\n\n- Let $f \\in {\\mathcal{S}}'$, then ${\\mathcal{F} \\left\\{ f \\right\\}} \\in {\\mathcal{S}}'$ and ${\\mathcal{F}^{-1} \\left\\{ f \\right\\}} \\in {\\mathcal{S}}'$.\n\nIn the following we review Theorem 4 on p294 ff from [@DauLio02_5] which characterized when a generalized function $f\\in{\\mathcal{S}}'({{\\mathbb{R}}})$ is causal, that is, when ${\\mbox{supp}}(f) \\subseteq [0,\\infty)$. Below we use the following notation $${{\\mathbb{C}}}_{\\varepsilon}:={\\{ z\\in{{\\mathbb{C}}}: \\Im(z)\\geq {\\varepsilon}\\}}\\;.$$\n\n(Theorem 4 on p294 ff in [@DauLio02_5]) \\[th:lion\\] Let $f \\in {\\mathcal{S}}'({{\\mathbb{R}}})$. Then $f$ is causal [^4] if and only if\n\n1. ",
" \\[it1\\_Lions\\] There exists a function $F:{{\\mathbb{C}}}_0:={\\{ \\xi + {{\\mathrm{i}}\\,}\\eta : \\eta \\geq 0 \\}}\\to{{\\mathbb{C}}}$, which is holomorphic in the interior $\\mathring{{{\\mathbb{C}}}}_0:={\\{ \\xi + {{\\mathrm{i}}\\,}\\eta : \\eta > 0 \\}}$. [^5]\n\n2. ",
" \\[it2\\_Lions\\] For all fixed $\\eta>0$ and $\\xi\\in{{\\mathbb{R}}}$, $F(\\xi + {{\\mathrm{i}}\\,}\\eta)$ is a tempered distribution with respect to the variable $\\xi$ and for $\\eta\\to 0$ $F(\\xi + {{\\mathrm{i}}\\,}\\eta)$ is convergent (with respect to the weak topology on ${\\mathcal{S}}'$) to ${\\mathcal{F} \\left\\{ f \\right\\}}:{{\\mathbb{R}}}\\to{{\\mathbb{C}}}$. [^6]\n\n3. ",
" \\[it3\\_Lions\\] For every ${\\varepsilon}> 0$, there exists a polynomial $P$ such that $${\\left|F(z)\\right|} \\leq P({\\left|z\\right|}) {\\qquad \\text{for} \\qquad} z \\in {{\\mathbb{C}}}_{\\varepsilon}\\,.$$\n\n\\[re:holomophic\\] The definition of the Fourier transform in this chapter has a different sign as in [@DauLio02_5] and consequently also ${{\\mathbb{C}}}_0$ denotes the upper half plane and not the lower half plane as in [@DauLio02_5].",
"\n\nFor analyzing attenuation laws, we use the following corollary, which is derived from Theorem \\[th:lion\\].",
"\n\n\\[co:lion\\] Let $\\alpha^*: {{\\mathbb{R}}}\\to {{\\mathbb{C}}}$ be continuous and let there exist a holomorphic extension to ${{\\mathbb{C}}}_0$, which for the sake of simplicity of notation is again denoted be $\\alpha^*$. In addition, let $\\alpha^* (\\xi+{{\\mathrm{i}}\\,}\\eta) \\to \\alpha^* (\\xi)$ for $\\eta \\to 0$ pointwise. ",
"We denote by $\\alpha: {{\\mathbb{C}}}_0 \\to {{\\mathbb{C}}}$ the real part of $\\alpha^*$. [^7] Moreover, we assume that there exists a constant $C$ such that $$\\label{estimate:rapidly}\n\\alpha(\\omega) \\geq C \\text{ for all }\\omega \\in {{\\mathbb{R}}}\\;.$$\n\n1. ",
" If in addition $$\\label{estimate:polynomial}\n \\alpha(z) \\geq C \\text{ for all } z \\in {{\\mathbb{C}}}_0\\;.$$ Then, for every ${{\\vec{x}}}\\in {{\\mathbb{R}}}^3$, the function $$\\label{31}\n t \\to K({{\\vec{x}}},t) := \\frac{1}{\\sqrt{2\\pi}}{\\mathcal{F}^{-1} \\left\\{ {\\exp \\left( -\\alpha^*(\\cdot){\\left|{{\\vec{x}}}\\right|} \\right)} \\right\\}}(t)$$ is causal.",
"\n\n2. ",
" On the other hand, if there exists $C_1>0$, $\\mu>0$ and $C_2 \\in {{\\mathbb{R}}}$ and a sequence ${\\{ z_n \\}}$ in $\\mathring{{{\\mathbb{C}}}}_0$ such that $$\\label{estimate:polynomial_negativ}\n \\alpha(z_n) \\leq - C_1 {\\left|z_n\\right|}^\\mu - C_2\\,,$$ then $K$ violates causality.",
"\n\nLet ${{\\vec{x}}}\\in {{\\mathbb{R}}}^3$ fixed. ",
"We apply Theorem \\[th:lion\\] to $f(\\cdot)=K({{\\vec{x}}},\\cdot)$. Therefore, we have $$\\sqrt{2\\pi} {\\mathcal{F} \\left\\{ f \\right\\}}(\\omega) = {\\exp \\left( -\\alpha^*(\\omega) {\\left|{{\\vec{x}}}\\right|} \\right)}\\;.$$ Under the assumption (\\[estimate:rapidly\\]), taking into account that $\\alpha_{pl}^*$ is continuous, ${\\mathcal{F} \\left\\{ f \\right\\}}$ is in $L_{loc}^1({{\\mathbb{R}}})$ and bounded by a constant polynomial, thus in ${\\mathcal{S}}'$ (cf. ",
"Example \\[le:distr\\_ex\\]), and consequently, according to Lemma \\[le:distr\\_ex\\], $f \\in {\\mathcal{S}}'$. Therefore, the general assumption of Theorem \\[th:lion\\] is satisfied.",
"\n\nThe function $z \\in {{\\mathbb{C}}}_0 \\mapsto F(z):={\\exp \\left( -\\alpha^*(z) {\\left|{{\\vec{x}}}\\right|} \\right)}$ is an extension of $2\\pi {\\mathcal{F} \\left\\{ f \\right\\}}(\\omega)$, which is holomorphic in $\\mathring{{{\\mathbb{C}}}}_0$. Thus Item \\[it1\\_Lions\\] of Theorem \\[th:lion\\] holds.",
"\n\n- For proving the first assertion, it follows from (\\[estimate:polynomial\\]) that for $z = \\xi + {{\\mathrm{i}}\\,}\\eta$ with $\\eta \\geq 0$, $$\\label{eq:new}\n {\\left|F(z)\\right|} \\leq {\\exp \\left( -C{\\left|{{\\vec{x}}}\\right|} \\right)}\\,,$$ which, in particular, shows that for all $\\eta \\geq 0$, the functions $\\xi \\to F(\\xi+{{\\mathrm{i}}\\,}\\eta)$ is a tempered distribution (cf. ",
"Lemma \\[le:distr\\_ex\\]). ",
"Hence Item \\[it1\\_Lions\\] of Theorem \\[th:lion\\] holds.",
"\n\n Moreover, since by assumption $\\alpha^*(\\xi+{{\\mathrm{i}}\\,}\\eta) \\to \\alpha^*(\\xi)$ for $\\eta \\to 0$ pointwise, $F(\\xi+{{\\mathrm{i}}\\,}\\eta)$ converges to $F(\\xi)=2\\pi {\\mathcal{F} \\left\\{ f \\right\\}}(\\xi)$ pointwise. ",
"Because the limit is a tempered distribution and the convergence is with respect to the weak topology ${\\mathcal{S}}'$ (cf. ",
"Lemma \\[le:distr\\_ex\\]). ",
"Hence Item \\[it2\\_Lions\\] of Theorem \\[th:lion\\] holds.",
"\n\n Moreover, from (\\[eq:new\\]) it follows that ${\\left|F(z)\\right|}$ is bounded by a constant polynomial. ",
"Hence Item (\\[it3\\_Lions\\]) of Theorem \\[th:lion\\] holds and therefore Theorem \\[th:lion\\] guarantees that $t \\mapsto K({{\\vec{x}}},t)$ is causal.",
"\n\n- For the second case, Item \\[it3\\_Lions\\] of Theorem \\[th:lion\\] is violated. ",
"Consequently, $K({{\\vec{x}}},\\cdot)$ is not causal.",
"\n\nPower Laws {#power-laws .unnumbered}\n----------\n\n\\[coro:powlaw1\\] Let $0 < \\gamma \\in {{\\mathbb{R}}}$, be not an odd number, and $\\omega \\in {{\\mathbb{R}}}\\mapsto \\alpha_{pl}^*(\\omega)={\\tilde{\\alpha}}_0 (-{{\\mathrm{i}}\\,}\\omega)^\\gamma$ be the power law attenuation coefficient from (\\[eq:powlaw1\\]), with ${\\tilde{\\alpha}}_0=\\alpha_0/\\cos \\left(\\frac{\\pi}{2}\\gamma\\right)$ as in (\\[ta0\\]). ",
"Then, the function $K$, defined in (\\[31\\]), is causal if and only if $\\gamma\\in (0,1)$.\n\nLet ${{\\vec{x}}}\\in {{\\mathbb{R}}}^3$ be fixed. ",
"The function $z \\in {{\\mathbb{C}}}\\mapsto \\alpha_{pl}^*(z)={\\tilde{\\alpha}}_0 (-{{\\mathrm{i}}\\,}z)^\\gamma$ is the holomorphic extension of $\\omega \\in {{\\mathbb{R}}}\\mapsto \\alpha_{pl}^*(\\omega)$. We prove or disprove causality by using Corollary \\[co:lion\\].",
"\n\nFor $z={\\left|z\\right|} {\\exp \\left( {{\\mathrm{i}}\\,}\\phi \\right)}$ it follows from (\\[eq:powlaw1\\]) that $$\\Re((-{{\\mathrm{i}}\\,}z)^\\gamma)\n = \\Re({\\left|z\\right|}^\\gamma\\,{\\exp \\left( {{\\mathrm{i}}\\,}\\gamma\\,(\\phi-\\pi/2) \\right)})\n = {\\left|z\\right|}^\\gamma\\,\\cos(\\gamma(\\phi-\\pi/2))\\,.$$ This implies that $$\\alpha_{pl}(z) = {\\tilde{\\alpha}}_0 \\Re((-{{\\mathrm{i}}\\,}z)^\\gamma) = {\\tilde{\\alpha}}_0 \\cos(\\gamma(\\phi-\\pi/2)){\\left|z\\right|}^\\gamma\\;.$$ In particular, if $z = \\omega \\in {{\\mathbb{R}}}$, then $\\phi$ is either $0$ or $\\pi$. Taking into account the definition of ${\\tilde{\\alpha}}_0$ and that the $\\cos$-function is symmetric around the origin, it follows that $$\\label{relalphapowlaw1a}\n\\alpha_{pl}(\\omega) = \\alpha_0 {\\left|w\\right|}^\\gamma \\geq 0\\;.$$ Thus (\\[estimate:rapidly\\]) holds.",
"\n\n- Let $\\gamma \\in (0,1)$: Every $z={\\left|z\\right|}\\,{\\exp \\left( {{\\mathrm{i}}\\,}\\phi \\right)}\\in {{\\mathbb{C}}}_0$ satisfies $\\phi\\in [0,\\pi]$. Consequently $\\gamma(\\phi-\\pi/2) \\in [-\\pi/2,\\pi/2]$ and thus $\\cos(\\gamma(\\phi-\\pi/2))$ is uniformly non-negative. ",
"Even more for $\\gamma \\in (0,1)$ the coefficient ${\\tilde{\\alpha}}_0$, defined in (\\[ta0\\]), is positive. ",
"In summary, we have that there exists a constant $C_1 \\geq 0$ such that $$\\label{relalphapowlaw1b}\n \\alpha_{pl}(z) \\geq C_1 {\\left|z\\right|}^\\gamma \\geq 0 {\\qquad \\text{ for } \\qquad} z\\in {{\\mathbb{C}}}_0\\;.$$ Thus (\\[estimate:polynomial\\]) holds and application of Corollary \\[co:lion\\] shows that $K$ is causal.",
"\n\n- Let $\\gamma\\in (1,3)\\cup (5,7)\\cup\\cdots$. Then ${\\tilde{\\alpha}}_0<0$. The sequence $$\\label{eq:34a}\n {\\{ z_n := n {\\exp \\left( {{\\mathrm{i}}\\,}\\pi/2 \\right)} = {{\\mathrm{i}}\\,}n \\}}_{n \\in {{\\mathbb{N}}}}$$ consists of elements of $\\mathring{{{\\mathbb{C}}}}_0$ and satisfies assumption (\\[estimate:polynomial\\_negativ\\]), that is, $$\\label{eq:azn1}\n \\alpha_{pl}(z_n) = \\underbrace{{\\tilde{\\alpha}}_0}_{<0} {\\left|z_n\\right|}^\\gamma\\;.$$ Application of Corollary \\[co:lion\\] shows that $K$ is not causal.",
"\n\n- Let $\\gamma\\in (3,5)\\cup (7,9)\\cup\\cdots$. We fix some $0<\\delta<\\pi/2$, and define $$\\phi := \\left(1+\\frac{1}{\\gamma} \\right)\\frac{\\pi}{2} + \\frac{\\delta}{\\gamma}\\;.$$ The sequence $$\\label{eq:34b}\n {\\{ z_n:=n\\,{\\exp \\left( {{\\mathrm{i}}\\,}\\phi \\right)} \\}}$$ consists of elements of $\\mathring{{{\\mathbb{C}}}}_0$. Under the above assumptions, it follows that ${\\tilde{\\alpha}}_0 > 0$ and therefore $$\\label{eq:azn2}\n \\alpha_{pl} (z_n) = \\underbrace{{\\tilde{\\alpha}}_0}_{>0} \\underbrace{\\cos(\\pi/2+\\delta)}_{<0} {\\left|z_n\\right|}^\\gamma\\;.$$ Thus form Corollary \\[co:lion\\] the assertion follows.",
"\n\nIn the following we analyze the following family of variants of power laws: $$\\label{szabo_model}\n\\alpha_{pl+}^*(\\omega) = {\\tilde{\\alpha}}_0 (-{{\\mathrm{i}}\\,}\\omega)^\\gamma + \\alpha_1(-{{\\mathrm{i}}\\,}\\omega)\\,,$$ which have been considered in [@Sza95; @WatHugBraMil00].",
"\n\n\\[le:pow\\_law\\_extended\\] Let $0 < \\gamma \\notin {{\\mathbb{N}}}$ and $\\alpha_{pl+}^*$ as defined in (\\[szabo\\_model\\]). ",
"Moreover, let $K$ be as in (\\[31\\]). ",
"Then, if $\\gamma >1$, $K$ is not causal. ",
"For $\\gamma \\in (0,1)$ $K$ is causal if and only if $\\alpha_1\\in [0,\\infty)$.\n\nThe holomorphic extension of $\\omega \\in {{\\mathbb{R}}}\\to \\alpha_{pl+}^*(\\omega)$ is the function $$\\alpha_{pl+}^*(z) = {\\tilde{\\alpha}}_0 (-{{\\mathrm{i}}\\,}z)^\\gamma + \\alpha_1(-{{\\mathrm{i}}\\,}z)$$ and consequently $$\\alpha_{pl+}(z)\n= {\\tilde{\\alpha}}_0 {\\left|z\\right|}^\\gamma \\left(1+\\frac{\\alpha_1}{{\\tilde{\\alpha}}_0} {\\left|z\\right|}^{1-\\gamma} \\right)\n\\cos \\left(\\gamma \\left(\\phi-\\frac{\\pi}{2}\\right)\\right)\\;.$$\n\n- For $\\gamma > 1$ we have that $1+\\frac{\\alpha_1}{{\\tilde{\\alpha}}_0} {\\left|z\\right|}^{1-\\gamma} \\to 1$ for ${\\left|z\\right|} \\to \\infty$. Let ${\\{ z_n \\}}$ as defined in (\\[eq:34a\\]) or (\\[eq:34b\\]). ",
"Then, since for both sequences ${\\left|z_n\\right|} \\to \\infty$, it follows from (\\[eq:azn1\\]), (\\[eq:azn2\\]) that the according sequences ${\\{ z_n \\}}$ satisfy (\\[estimate:polynomial\\_negativ\\]) for $n$ sufficiently large, respectively. ",
"Thus $K$ is not causal.",
"\n\n- For $\\gamma \\in (0,1)$ and $\\alpha_1 \\geq 0$ the assertion follows already from the fact that $\\alpha_{pl+}(\\omega) \\geq \\alpha_{pl}(\\omega)$ and that the later already satisfies (\\[estimate:polynomial\\]). ",
"Thus $K$ is causal.",
"\n\n- Let $\\gamma \\in (0,1)$ and $\\alpha_1 < 0$. Then for some $0 < \\delta < -\\alpha_1$ fixed, we can find a constant $C_2$ such that for all $z \\in {{\\mathbb{C}}}$ $${\\tilde{\\alpha}}_0 {\\left|z\\right|}^\\gamma + \\alpha_1 {\\left|z\\right|} \\leq \\underbrace{(\\alpha_1+\\delta)}_{< 0} {\\left|z\\right|} - C_2\\;.$$ Consequently, for ${\\{ z_n={{\\mathrm{i}}\\,}n \\}}$, we have $$\\alpha_{pl+}(z_n) \\leq (\\alpha_1+\\delta) {\\left|n\\right|} - C_2\\;.$$ which shows (\\[estimate:polynomial\\_negativ\\]). ",
"Thus $K$ is not causal.",
"\n\nPowerlaw with $\\gamma = 1$ {#powerlaw-with-gamma-1 .unnumbered}\n--------------------------\n\n\\[thcaus02\\] Let $\\alpha_{pl}^*$ be as in defined in (\\[alphagamma1\\]). ",
"Then the function $K$, defined in (\\[31\\]) is not causal.",
"\n\nFirst, we prove that $$\\begin{aligned}\nz \\in {{\\mathbb{C}}}_0 \\mapsto \\hat{\\alpha}_{pl}^*(z) := \\alpha_0 z\n + {{\\mathrm{i}}\\,}\\frac{2 \\alpha_0}{\\pi} z \\log \\left(\\frac{z}{\\omega_0}\\right)\n\\end{aligned}$$ is the holomorphic extension of $\\omega \\to \\alpha_{pl}^*(\\omega)$. This assertion follows from the facts $$\\begin{aligned}\n\\hat{\\alpha}_{pl}^*(\\omega)& =\\alpha_{pl}^*(\\omega) \\text{ for } & \\omega>0\\,,\\\\\n\\lim_{\\eta \\to 0+} {{\\mathrm{i}}\\,}\\frac{2}{\\pi} \\log \\left(\\frac{\\omega + {{\\mathrm{i}}\\,}\\eta}{\\omega_0}\\right)\n & = {{\\mathrm{i}}\\,}\\frac{2}{\\pi}\\,\\log\\left|\\frac{\\omega}{\\omega_0}\\right|\n - 2 \\text{ for } & \\omega<0\\;.",
"\n\\end{aligned}$$ Since $$\\alpha_{pl}(\\omega) = \\Re( \\alpha_{pl}^*(\\omega)) = \\alpha_0\\,{\\left|\\omega\\right|} \\geq 0\\,,$$ Corollary \\[co:lion\\] is applicable. ",
"For the elements of the sequence ${\\{ z_n:={{\\mathrm{i}}\\,}n \\}}_{n \\in {{\\mathbb{N}}}}$ in ${{\\mathbb{C}}}_0$ $$\\hat{\\alpha}_{pl}^*(z_n)\n = - \\frac{2 \\alpha_0}{\\pi} n \\log \\left(\\frac{n}{\\omega_0}\\right)$$ is real and therefore equals $\\alpha_{pl}(z_n)$ and thus (\\[estimate:polynomial\\_negativ\\]) holds. ",
"Thus Corollary \\[co:lion\\] gives the assertion.",
"\n\nSzabo’s Model: {#szabos-model .unnumbered}\n--------------\n\n\\[theo:szabo\\] For $\\alpha_0>0$ let $\\alpha_{sz}^*$ be the coefficient of Szabo’s model (\\[alpha\\*szabo\\]). ",
"Then, for $\\gamma\\in (0,1)$, the function $K$ (\\[eq:kernel\\]) is a causal function and for $\\gamma>1$ with $\\gamma\\not\\in{{\\mathbb{N}}}$, $K$ violates causality.",
"\n\nWithout loss of generality we assume that $c_0=1$. The holomorphic extension of $\\alpha_{sz}^*:{{\\mathbb{R}}}\\to {{\\mathbb{C}}}$ from (\\[alpha\\*szabo\\]) is $$z\\in {{\\mathbb{C}}}_0 \\to \\alpha_{sz}^*(z) = (-{{\\mathrm{i}}\\,}z)\n \\left[ \\sqrt{1 + 2 {\\tilde{\\alpha}}_0 (-{{\\mathrm{i}}\\,}z)^{\\gamma-1}} - 1\\right]\\,.$$\n\nFirst, we make some general manipulations which can be used in several ways: Let $z = \\xi+{{\\mathrm{i}}\\,}\\eta \\in {{\\mathbb{C}}}_0$. We use the polar representation $$z = {\\left|z\\right|} {\\exp \\left( {{\\mathrm{i}}\\,}\\phi \\right)}\\,, \\quad \\phi := \\phi(z) \\in \\left[0,\\pi\\right]\\;.$$ Then $$\\alpha_{sz}^*(z) = {\\left|z\\right|} {\\exp \\left( {{\\mathrm{i}}\\,}(\\phi-\\pi/2) \\right)} \\Psi(z)\\,,$$ where $$\\label{eq:delta}\n\\begin{aligned}\n\\Psi(z) &:= \\sqrt{\\hat{\\Psi}(z)} - 1\\,,\\\\\n\\hat{\\Psi}(z) &:= 1 + 2 {\\tilde{\\alpha}}_0 {\\left|z\\right|}^{\\gamma-1} {\\exp \\left( {{\\mathrm{i}}\\,}\\delta \\right)}\\,,\\\\\n\\delta &:= \\delta(z):=(\\phi-\\pi/2)(\\gamma-1)\\;.",
"\n\\end{aligned}$$ With this notation we have $$\\label{eq:hatpsi}\n\\begin{aligned}\n\\Re (\\hat{\\Psi}(z)) &= 1 + 2 {\\tilde{\\alpha}}_0 \\cos(\\delta) {\\left|z\\right|}^{\\gamma-1}\\,,\\\\\n\\Im (\\hat{\\Psi}(z)) &= 2 {\\tilde{\\alpha}}_0 \\sin(\\delta) {\\left|z\\right|}^{\\gamma-1}\\,,\\\\\n{\\left|\\hat{\\Psi}(z)\\right|} &= {\\left|z\\right|}^{\\gamma-1} \\sqrt{(1 + 2 {\\tilde{\\alpha}}_0 \\cos(\\delta))^2+4 {\\tilde{\\alpha}}_0^2 \\sin(\\delta)^2}\\;.",
"\n\\end{aligned}$$ Representing $\\hat{\\Psi}$ in polar coordinates, $$\\hat{\\Psi}(z) = {\\left|\\hat{\\Psi}(z)\\right|} {\\exp \\left( {{\\mathrm{i}}\\,}\\theta(z) \\right)}\\,,$$ we get $$\\label{theta}\n\\begin{aligned}\n \\sqrt{\\hat{\\Psi}(z)} &= \\sqrt{{\\left|\\hat{\\Psi}(z)\\right|}} {\\exp \\left( {{\\mathrm{i}}\\,}\\theta(z)/2 \\right)}\n\\qquad\\quad \\text{ with }\\\\\n \\theta (z) &= \\arctan (\\Im (\\hat{\\Psi}(z))/ \\Re (\\hat{\\Psi}(z)))) \\qquad \\theta\\in (-\\pi,\\pi) \\;.",
"\n\\end{aligned}$$ Note, that $\\sqrt{\\hat{\\Psi}(z)}$ is the complex root with non-negative real part, which meets the general assumption of the paper. ",
"Moreover, we have $$\\label{szz}\n\\alpha_{sz}(z) = \\Re (\\alpha_{sz}^*(z))\n = \\eta \\Re (\\Psi(z)) + \\xi \\Im (\\Psi (z))\\;.$$\n\nFirst, we prove that $\\Re (\\Psi(z)) \\geq 0$: We use the elementary inequality $$\\cos (\\theta(z)) \\leq \\cos^2 (\\theta(z)/2)\\,,$$ and $\\cos (\\theta(z)/2)\\geq 0$ which imply that $$\\label{eq:real}\n\\begin{aligned}\n \\Re \\left(\\sqrt{\\hat{\\Psi}(z)}\\right)\n &= \\sqrt{{\\left|\\hat{\\Psi}(z)\\right|}} \\cos (\\theta(z)/2) \\\\\n &= \\sqrt{{\\left|\\hat{\\Psi}(z)\\right|}\\, \\cos^2 (\\theta(z)/2) }\\\\\n &\\geq \\sqrt{\\Re (\\hat{\\Psi}(z))}\\;.",
"\n\\end{aligned}$$\n\n- Now, let $\\gamma\\in (0,1)$. Since $\\Re (\\hat{\\Psi}(z)) \\geq 1$ for $\\gamma\\in (0,1)$, it follows that for all $z \\in {{\\mathbb{C}}}_0$ $$\\eta \\Re (\\Psi(z))\n = \\eta \\Re \\left(\\sqrt{\\hat{\\Psi}(z)}\\right) - \\eta\n \\geq \\eta \\sqrt{\\Re (\\hat{\\Psi}(z))} - \\eta\n \\geq 0\\,.$$ Thus $\\eta \\Re (\\Psi(z)) \\geq 0$.\n\n Now we show that $z \\to \\xi \\Im (\\Psi (z))$ is uniformly bounded from below by $0$ in ${{\\mathbb{C}}}_0$. Thus according to (\\[szz\\]) $\\alpha_{sz}$ is uniformly bounded from below, and thus from Corollary \\[co:lion\\], it follows that $t \\to K({{\\vec{x}}},t)$ is causal.",
"\n\n Using the definition of $\\theta$, (\\[theta\\]), and the facts that $\\delta \\in [0,(1-\\gamma)\\pi/2]$ for $\\phi \\in [0,\\pi/2]$ and $\\delta \\in [(\\gamma-1)\\pi/2,0)$ for $\\phi \\in (\\pi/2,\\pi]$ it follows from the monotonicity of $\\tan$ on $(-\\pi,\\pi)$ that $$\\theta(z) = \\arctan\\left(\\frac{2 {\\tilde{\\alpha}}_0 \\sin(\\delta) {\\left|z\\right|}^{\\gamma-1}}{1 + 2 {\\tilde{\\alpha}}_0 \\cos(\\delta) {\\left|z\\right|}^{\\gamma-1}}\\right) \\in\n \\left\\{ \\begin{array}{rcl}\n ~[0,\\delta] & \\text{ for all } & \\phi \\in [0,\\pi/2]\\,,\\\\\n ~[\\delta,0) & \\text{ for all } & \\phi \\in (\\pi/2,\\pi]\n \\end{array}\n \\right.$$ Now, noting that ${\\mbox{sgn}}(\\xi) = {\\mbox{sgn}}\\left(\\sin (\\theta(z)/2)\\right)$ it follows that $$\\label{eq:imaginary}\n \\xi \\Im (\\Psi(z)) = \\xi \\sqrt{{\\left|\\hat{\\Psi}(z)\\right|}} \\sin (\\theta(z)/2) \\geq 0\\,.$$ Thus the assertion follows from Corollary \\[co:lion\\].",
"\n\n- Assume that $\\gamma > 1$. Let $z=\\xi+{{\\mathrm{i}}\\,}\\eta$ with $\\eta = 0$. Since the square root in [(\\[eq:delta\\])]{} is such that $\\Re(\\Psi(z))>0$ for $z\\in{{\\mathbb{C}}}_0$, property (\\[propsqrt\\]) in the Appendix implies $\\xi \\Im (\\Psi(z))>0$ for $z=\\xi$ and hence $\\alpha_{sz}(z=\\xi) \\geq 0$. Thus (\\[estimate:rapidly\\]) holds and we can apply Corollary \\[co:lion\\].",
"\n\n - Let $\\gamma\\in (1,3)\\cup (5,7)\\cup\\cdots$, which implies that $\\cos\\left(\\gamma\\,\\pi/2\\right)<0$, and consequently, ${\\tilde{\\alpha}}_0 < 0$. For sufficiently large $n$ the elements of the sequence ${\\{ z_n:={{\\mathrm{i}}\\,}n \\}}$ satisfy $$\\alpha_{sz}(z_n) = n\\,\\Re\\left(\\sqrt{1 - 2{\\left|{\\tilde{\\alpha}}_0\\right|}\\, n^{\\gamma-1}}-1\\right) \\leq -n\\,,$$ which shows that (\\[estimate:polynomial\\_negativ\\]) holds with $\\mu=1$, $C_1=1/2$ and $C_2=0$, and hence the assertion follows from Corollary \\[co:lion\\].",
"\n\n - Let $\\gamma\\in (3,5)\\cup (7,9)\\cup\\cdots$, which implies that $\\cos\\left(\\gamma\\,\\pi/2\\right)>0$, and consequently, ${\\tilde{\\alpha}}_0>0$. Now, let $z_n:=n {\\exp \\left( {{\\mathrm{i}}\\,}\\phi \\right)}$ with $$\\phi:= \\frac{\\pi}{\\gamma-1} + \\frac{\\pi}{2}\\,.$$ Since $\\gamma>3$, we have $\\phi\\in (\\pi/2,\\pi)$, and therefor $\\Re(z_n)=n\\,\\cos(\\phi)<0$ and $\\Im(z_n)=n\\,\\sin(\\phi)>0$. Moreover, $$\\hat \\Psi(z_n) = 1-2\\,{\\left|{\\tilde{\\alpha}}_0\\right|}\\,n^{\\gamma-1}\\,$$ and thus for sufficiently large $n$ we have $\\Re(\\Psi(z_n))=-1$ and\\\n $\\Im\\left(\\sqrt{\\hat\\Psi(z_n)}\\right)\\geq C_1\\,n^{(\\gamma-1)/2}$ for some constant $C_1>0$. Hence it follows that $$\\begin{aligned}\n \\alpha_{sz}(z_n)\n &= -\\Im(z_n) + \\Re(z_n)\\, \\Im(\\sqrt{\\hat\\Psi(z_n)})\\\\\n &= - n{\\left|\\sin(\\phi)\\right|} - n{\\left|\\cos(\\phi)\\right|}\\, C_1\\,n^{(\\gamma-1)/2}\n \\end{aligned}$$ and therefore (\\[estimate:polynomial\\_negativ\\]) holds. ",
"Thus from Corollary \\[co:lion\\] the assertion follows.",
"\n\nThermo-Viscous Attenuation Law {#thermo-viscous-attenuation-law .unnumbered}\n------------------------------\n\nLet $c_0,\\tau_0>0$ and let $\\alpha_{tv}^*$ as defined in (\\[alpha\\*th\\]). ",
"Then the kernel function $K$ violates causality.",
"\n\nSince the function $z\\in {{\\mathbb{C}}}_0 \\to 1-{{\\mathrm{i}}\\,}\\tau_0\\,z$ does not vanish, the function $$z \\in {{\\mathbb{C}}}_0 \\to \\alpha_{tv}^*(z)\n = \\frac{-{{\\mathrm{i}}\\,}z}{c_0\\,\\sqrt{1-{{\\mathrm{i}}\\,}\\tau_0 z}} + \\frac{{{\\mathrm{i}}\\,}z}{c_0}$$ is the holomorphic extension of $\\omega \\in {{\\mathbb{R}}}\\to \\alpha_{tv}^*(\\omega)$. That (\\[estimate:rapidly\\]) holds follows from the identity (\\[alpha\\*threal\\]). ",
"For the sequence ${\\{ z_n \\}}_{n\\in{{\\mathbb{N}}}} := {\\{ {{\\mathrm{i}}\\,}n \\}}_{n\\in{{\\mathbb{N}}}}$ we get for $n$ sufficiently large $$\\alpha_{tv}^*(z_n) = \\frac{n}{c_0}\\,\\left[ \\frac{1}{\\sqrt{1+\\tau_0\\,n}} -1 \\right] \\leq - \\frac{1}{2\\,c_0} n$$ Thus (\\[estimate:polynomial\\_negativ\\]) holds.",
"\n\nThe second part of Corollary \\[co:lion\\] implies that $K$ is not causal.",
"\n\nModel of Nachman, Smith and Waag {#model-of-nachman-smith-and-waag .unnumbered}\n--------------------------------\n\nLet $\\alpha_{nsw}^*$ as in (\\[alpha\\*Nachman+\\]). ",
"If $$\\label{asstau}\n \\tilde \\tau_m < \\tau_m \\qquad \\mbox{ for all} \\qquad m\\in\\{1,\\,\\ldots\\,,N\\}\\,,$$ then the kernel function $K$ is causal.",
"\n\nSince for all $z \\in {{\\mathbb{C}}}_0$ and all $1 \\leq m \\leq N$, $1-{{\\mathrm{i}}\\,}\\tau_m\\,z$ does not vanish, $$\\begin{aligned}\nz \\in {{\\mathbb{C}}}_0 \\to \\alpha_{nsw}^*(z)\n = \\frac{-{{\\mathrm{i}}\\,}z}{c_0}\\,\\left[\n \\frac{c_0}{\\tilde c_0}\\, \\sqrt{ \\frac{1}{N}\\, \\sum_{m=1}^N\n \\frac{1-{{\\mathrm{i}}\\,}\\,\\tilde\\tau_m\\,z}{1-{{\\mathrm{i}}\\,}\\tau_m\\,z} }\n -1 \\right]\n\\end{aligned}$$ is the holomorphic extension of $\\omega \\in {{\\mathbb{R}}}\\to \\alpha_{nsw}^*(\\omega)$.\n\nWe use a similar notation as in Proposition \\[theo:szabo\\]. ",
"$$z = \\xi + {{\\mathrm{i}}\\,}\\eta = {\\left|z\\right|} {\\exp \\left( {{\\mathrm{i}}\\,}\\phi \\right)} \\in {{\\mathbb{C}}}_0\\,,$$ with some $\\phi \\in [0,\\pi]$. $$\\alpha_{nsw}^*(z) = \\frac{{\\left|z\\right|}}{c_0} {\\exp \\left( {{\\mathrm{i}}\\,}(\\phi-\\pi/2) \\right)} \\,(\\Psi(z)-1)\\,,$$ where $$\\Psi(z) = \\sqrt{\\sum_{m=1}^N \\hat{\\Psi}_m(z)}\\, \\quad\\mbox{ with }\\quad\n \\hat{\\Psi}_m(z) := \\frac{1}{N}\\, \\frac{c_0^2}{\\tilde c_0^2}\\,\n \\frac{1-{{\\mathrm{i}}\\,}\\,\\tilde\\tau_m\\,z}{1-{{\\mathrm{i}}\\,}\\tau_m\\,z} \\,,$$\n\nIn the following we show that for all $z\\in {{\\mathbb{C}}}_0$ $$\\label{z2A1z1A2}\nc_0 \\alpha_{nsw}(z) = \\eta (\\Re (\\Psi)(z)-1) + \\xi \\Im (\\Psi)(z) >0\\,,$$ which means that (\\[estimate:polynomial\\]) holds. ",
"Then, according to Corollary \\[co:lion\\] the function $t \\to K({{\\vec{x}}},t)$ is causal.",
"\n\nAs in the proof of Proposition \\[theo:szabo\\] we prove $\\eta (\\Re (\\Psi)(z)-1)>0$ and $\\xi \\Im (\\Psi)(z) >0$.\n\n- Taking into account (\\[proptildec0tau0\\]) we define $$\\label{s}\n s := \\frac{1}{N}\\, \\frac{c_0^2}{\\tilde c_0^2}\n = \\left(\\sum_{m=1}^N \\frac{\\tilde\\tau_m}{\\tau_m}\\right)^{-1}\\,.$$ Using this notation, we get $$\\begin{aligned}\n \\hat{\\Psi}_m(z) &=\n s\\,\\frac{(1+\\tilde\\tau_m\\,\\tau_m\\,{\\left|z\\right|}^2) +{{\\mathrm{i}}\\,}\\,(\\tau_m\\,\\bar{z} -\\tilde\\tau_m\\,z)}\n {|1-{{\\mathrm{i}}\\,}\\,\\tau_m\\,z|^2} \\\\\n &= s\\,\\frac{\\tilde\\tau_m}{\\tau_m}\\,\n \\frac{(\\tau_m/\\tilde \\tau_m + \\tau_m^2\\,{\\left|z\\right|}^2 + (\\tau_m^2/\\tilde \\tau_m+\\tau_m)\\,\\eta)\n +{{\\mathrm{i}}\\,}\\,(\\tau_m^2/\\tilde \\tau_m -\\tau_m)\\,\\xi}\n {1+\\tau_m^2\\,{\\left|z\\right|}^2 + 2\\,\\tau_m\\,\\eta} \\,,\n \\end{aligned}$$ Because $\\tau_m/\\tilde \\tau_m>1$, by assumption (\\[asstau\\]), it follows that for all $z \\in {{\\mathbb{C}}}_0$ $$\\Re(\\hat{\\Psi}_m(z)) > s\\,\\frac{\\tilde\\tau_m}{\\tau_m}\\;.$$ Consequently, by using the definition of $s$, (\\[s\\]), it follows that $$\\Re\\left(\\sum_{m=1}^N \\hat{\\Psi}_m(z)\\right) > s\\, \\sum_{m=1}^N\\frac{\\tilde\\tau_m}{\\tau_m} = 1.$$ Now, using (\\[eq:real\\]) it follows $$\\Re(\\Psi(z))\n = \\Re\\left(\\sqrt{\\sum_{m=1}^N \\Psi_m(z)}\\right)\n \\geq \\sqrt{\\Re\\left(\\sum_{m=1}^N \\Psi_m(z)\\right)} > 1$$ and consequently $\\eta (\\Re (\\Psi(z))-1)>0$.\n\n- We have $$\\begin{aligned}\n \\Im(\\hat{\\Psi}_m(z)) = s\\,\\frac{\\tilde\\tau_m}{\\tau_m}\\,\n \\frac{(\\tau_m^2/\\tilde \\tau_m -\\tau_m)\\,\\xi}\n {1+\\tau_m^2\\,{\\left|z\\right|}^2 + 2\\,\\tau_m\\,\\eta}\n \\end{aligned}$$ together with the assumption (\\[asstau\\]), which state that $\\tau_m/\\tilde \\tau_m>1$, it follows that ${\\mbox{sgn}}(\\Im(\\hat{\\Psi}_m(z)))={\\mbox{sgn}}(\\xi)$. According to our assumption, we take that complex root, such that the real part of the argument is non-negative which together with property (\\[propsqrt\\]) in the Appendix implies $${\\mbox{sgn}}\\left( \\Im\\left(\\sqrt{\\sum_{m=1}^N \\hat{\\Psi}_m(z)} \\right) \\right)\n = {\\mbox{sgn}}\\left( \\Im\\left( \\sum_{m=1}^N \\hat{\\Psi}_m(z) \\right) \\right) \\,.$$ Therefore, $$\\begin{aligned}\n {\\mbox{sgn}}(\\Im(\\Psi(z))) &= {\\mbox{sgn}}\\left(\\Im\\left(\\sqrt{\\sum_{m=1}^N \\hat{\\Psi}_m(z)}\\}\\right)\\right)\\\\\n &= {\\mbox{sgn}}\\left(\\Im\\left(\\sum_{m=1}^N \\hat{\\Psi}_m(z)\\right)\\right)\n = {\\mbox{sgn}}(\\xi)\\;.",
"\n \\end{aligned}$$ This shows the assertion.",
"\n\nOur Model {#our-model .unnumbered}\n---------\n\n\\[thcaus01\\] For $\\alpha_0,\\,\\tau_0>0$ and $\\gamma\\in (1,2]$ let $\\alpha_{ksb}^*$ be defined as in (\\[eq:powlaw2\\]). ",
"Then $K$, as defined in (\\[31\\]), is causal.",
"\n\nThe function $\\hat{z}\\to 1+(-{{\\mathrm{i}}\\,}\\tau_0 \\hat{z})$ does not vanish in ${{\\mathbb{C}}}_0$. Thus the holomorphic extension of $\\omega \\to \\alpha_{ksb}^*(\\omega)$ is given by $$\\hat{z} \\in {{\\mathbb{C}}}_0 \\to \\alpha_{ksb}^*(\\hat{z})\n = \\frac{\\alpha_0(-{{\\mathrm{i}}\\,}\\hat{z})}{c_0\\sqrt{1+(-{{\\mathrm{i}}\\,}\\tau_0 \\hat{z})^{\\gamma-1}}}\\;.$$ In the following let $\\hat{z} \\in {{\\mathbb{C}}}_0$. For proving (\\[estimate:polynomial\\]) we make a variable transformation $$\\alpha_{ksb}^*(\\hat{z})\n = \\frac{-{{\\mathrm{i}}\\,}\\tau_0 \\hat{z}}{\\sqrt{1+(-{{\\mathrm{i}}\\,}\\tau_0 \\hat{z})^{\\gamma-1}}}\n = \\frac{\\alpha_0}{\\tau_0 c_0} \\frac{-{{\\mathrm{i}}\\,}z}{\\sqrt{1+(-{{\\mathrm{i}}\\,}z^{\\gamma-1})}}\\,,$$ and define $$\\Psi(z) = \\frac{1}{\\sqrt{\\hat{\\Psi}(z)}} \\quad\\text{ and }\\quad\n \\hat{\\Psi}(z)= 1+(-{{\\mathrm{i}}\\,}\\,z)^{\\gamma-1}\\;.$$ Then, with this notation, in order to prove causality of $K$, it suffices to prove that for all $z \\in {{\\mathbb{C}}}_0$ $$\\label{positive}\n\\frac{\\tau_0 c_0}{\\alpha_0}\\alpha_{ksb}(\\hat{z}) = \\eta \\Re (\\Psi(z)) + \\xi \\Im (\\Psi(z)) \\geq 0\\;.$$ As in the proof of Proposition \\[theo:szabo\\] we show that both terms $\\eta \\Re (\\Psi(z))$ and $\\xi \\Im (\\Psi(z))$ are non-negative, and then from Corollary \\[co:lion\\] the assertion follows.",
"\n\nIn order to prove (\\[positive\\]) we note that the function $\\hat{\\Psi}$ here is the same as in (\\[eq:delta\\]) in the proof of Proposition \\[theo:szabo\\] when ${\\tilde{\\alpha}}_0$ is set to $1/2$. Thus we can already rely on the series of manipulations for $\\hat{\\Psi}$ developed in the proof of Proposition \\[theo:szabo\\].",
"\n\n- Since $\\eta \\geq 0$ it suffices to show that $\\Re (\\Psi(z)) \\geq 0$. We note that for a complex number $a+{{\\mathrm{i}}\\,}b$ $$\\Re \\left(\\frac{1}{a+{{\\mathrm{i}}\\,}b}\\right)\n = \\Re \\left(\\frac{a- {{\\mathrm{i}}\\,}b}{a^2 + b^2}\\right)\n = \\frac{1}{a^2 + b^2} \\Re(a+{{\\mathrm{i}}\\,}b)\\;.$$ Taking into account the definition of $\\Psi$ it therefore suffices to show that $\\Re \\left(\\sqrt{\\hat{\\Psi}(z)}\\right) \\geq 0$ in ${{\\mathbb{C}}}_0$. Since $\\Re \\left(\\hat{\\Psi}(z)\\right) \\geq 0$ in ${{\\mathbb{C}}}_0$ for $\\gamma\\in (1,2]$, it follows that $\\Re \\left(\\sqrt{\\hat{\\Psi}(z)}\\right) \\geq 0$ in ${{\\mathbb{C}}}_0$.\n\n- Now, using that $$\\Im \\left(\\frac{1}{a+{{\\mathrm{i}}\\,}b}\\right) = \\Im \\left(\\frac{a- {{\\mathrm{i}}\\,}b}{a^2 + b^2}\\right) = -\\frac{1}{a^2 + b^2} \\Im(a+{{\\mathrm{i}}\\,}b)\\,,$$ it suffices to show that $-\\xi \\Im (\\sqrt{\\hat{\\Psi}(z)}) \\geq 0$ for proving that $\\xi \\Im (\\Psi(z)) \\geq 0$. The proof is along the lines as the analogous part in Proposition \\[theo:szabo\\] by taking into account that here $\\gamma \\in (1,2)$ (in Proposition \\[theo:szabo\\] $\\gamma \\in (0,1)$). ",
"In this case we have now that sign of $\\delta$ is exactly opposite as in the proof of Proposition \\[theo:szabo\\], which in turn gives that $\\Im (\\Psi(z))$ has the opposite sign as well, and consequently $-\\xi \\Im (\\sqrt{\\hat{\\Psi}(z)}) \\geq 0$. Thus the assertion follows from Corollary \\[co:lion\\].",
"\n\nIn experiments it has been discovered that several biological tissues satisfy a frequency power law (\\[eq:powlaw1b\\]) with exponent $\\gamma \\in (1,2)$ (cf.",
" [@Web00; @BurRoiBauPal010]). ",
"However, as it has been shown in Theorem \\[coro:powlaw1\\], such models are not causal. ",
"Our proposed model approximates the frequency power law for small frequencies, which is actually the range where it has been experimentally validated. ",
"So, our proposed model, is valid in the actual range of experimentally measured data and extrapolates the measured data in a causal way. ",
"Figure \\[fig:comp\\] shows a comparison of $\\alpha_{pl}$ and $\\alpha_{ksb}$ in an experimental frequency range.",
"\n\n![",
"For $\\gamma=1.5$: Comparison $\\omega \\to \\alpha_{ksb}(\\omega)$ (as defined in (\\[eq:powlaw2\\])) where $\\alpha_0 := 2c_0\\tau_0/{\\left|\\cos (\\frac{\\pi}{2} \\gamma)\\right|}$ (dashed line) and the power law $\\alpha_{pl}(\\omega) = {\\left|\\tau_0\\,\\omega\\right|}^\\gamma$ (as defined in (\\[eq:powlaw1b\\])). ",
"For liquids: $\\tau_0=10^{-6}\\,MHz$ (left picture) and for gases: $\\tau_0=10^{-4}\\,MHz$ (right picture) (cf.",
" [@KinFreCopSan00]). ",
"Experiments for determining the power law coefficient are performed in the range $0-60\\,MHz$ (cf. ",
"e.g. [@Sza95]), which is the basis for the range of the represented data. []{",
"data-label=\"fig:comp\"}](compL.eps \"fig:\"){width=\"45.00000%\"} ![",
"For $\\gamma=1.5$: Comparison $\\omega \\to \\alpha_{ksb}(\\omega)$ (as defined in (\\[eq:powlaw2\\])) where $\\alpha_0 := 2c_0\\tau_0/{\\left|\\cos (\\frac{\\pi}{2} \\gamma)\\right|}$ (dashed line) and the power law $\\alpha_{pl}(\\omega) = {\\left|\\tau_0\\,\\omega\\right|}^\\gamma$ (as defined in (\\[eq:powlaw1b\\])). ",
"For liquids: $\\tau_0=10^{-6}\\,MHz$ (left picture) and for gases: $\\tau_0=10^{-4}\\,MHz$ (right picture) (cf.",
" [@KinFreCopSan00]). ",
"Experiments for determining the power law coefficient are performed in the range $0-60\\,MHz$ (cf. ",
"e.g. [@Sza95]), which is the basis for the range of the represented data. []{",
"data-label=\"fig:comp\"}](compR.eps \"fig:\"){width=\"45.00000%\"}\n\nModel of Greenleaf and Patch {#model-of-greenleaf-and-patch .unnumbered}\n----------------------------\n\n\\[theo:gp\\] For $\\alpha_0>0$ let $\\alpha_{gp}^*$ be defined as in (\\[alpha\\*Patch\\]) with the specified values $\\gamma\\in{\\{ 1,2 \\}}$. Then $K$, as defined in (\\[31\\]), is not causal.",
"\n\nFor the two specified models we have $\\alpha_{gp1}(\\omega)= a_0 {\\left|\\omega\\right|} >0$ and $\\alpha_{gp2}(\\omega) = a_0 \\omega^2$. The respective holomorphic extensions are given by (cf. ",
"Proof of Theorem \\[thcaus02\\] and Theorem \\[coro:powlaw1\\]) $$\\begin{aligned}\nz \\in {{\\mathbb{C}}}_0 \\mapsto \\hat{\\alpha}_{gp1}^*(z) := a_0 z\n + {{\\mathrm{i}}\\,}\\frac{2 a_0}{\\pi} z \\log \\left(\\frac{z}{\\omega_0}\\right) \\qquad (\\omega_0\\neq0, \\mbox{fixed})\n\\end{aligned}$$ and $$\\begin{aligned}\n z \\in {{\\mathbb{C}}}_0 \\mapsto \\hat{\\alpha}_{gp2}^*(z) := a_0\\,(-{{\\mathrm{i}}\\,}\\,z)^2 \\,.",
"\n\\end{aligned}$$ The assertion for $\\gamma=1$ follows as in the proof of Theorem \\[thcaus02\\] and the second assertion follows from Theorem \\[coro:powlaw1\\] for $\\gamma=2$.\n\nModel of Chen and Holm {#model-of-chen-and-holm .unnumbered}\n----------------------\n\n\\[th:chenholm\\] Let $0<\\alpha_1<1$, $\\gamma\\in (0,2)$ and ${\\mathcal{G}}$ as in (\\[GreenChenHolm\\]). ",
"Then there does not exist a constant $c>0$ such that $$\\label{CausRef}\n {\\mbox{supp}}( {\\mathcal{G}}\\left(\\cdot,t \\right) ) \\subseteq B_{c\\,t}({{\\mathbf{0}}})\n \\qquad \\mbox{ for } \\qquad t>0\\,,$$ i.e. for each $c>0$ the function $t\\mapsto {\\mathcal{G}}({{\\vec{x}}},t+{\\left|{{\\vec{x}}}\\right|}/c)$ is not causal.",
"\n\nLet $t>0$ be fixed. ",
"Assume that ${{\\vec{x}}}\\mapsto {\\mathcal{G}}({{\\vec{x}}},t)$ has support in $ B_{c\\,t}({{\\mathbf{0}}})$ for some $c>0$. Then according to the Paley-Wiener-Schwartz Theorem (Cf.",
" [@GasWit99; @Hoe03]) the map ${\\mathbf{k}}\\mapsto {\\mathcal{F}^{-1} \\left\\{ {\\mathcal{G}}\\right\\}}({\\mathbf{k}},t)$ is infinitely differentiable. ",
"We show that this is not possible. ",
"According to (\\[GreenChenHolm\\]) and (\\[AkBk\\]), we have $$\\begin{aligned}\n {\\mathcal{F}}_{3D}^{-1}\\left\\{ {\\mathcal{G}}\\right\\} ({\\mathbf{k}},t)\n = \\frac{H(t)\\,c_0^2}{(2\\,\\pi)^{3/2}}\\,{\\exp \\left( A({\\mathbf{k}})\\,t \\right)}\\, \\frac{\\sin(B({\\mathbf{k}})\\,t)}{B({\\mathbf{k}})}\n\\end{aligned}$$ with $$A({\\mathbf{k}}) := -\\alpha_1\\,c_0\\,{\\left|{\\mathbf{k}}\\right|}^\\gamma\\,,\n\\qquad\n B({\\mathbf{k}}) := c_0\\,\\sqrt{{\\left|{\\mathbf{k}}\\right|}^2-\\alpha_1^2\\,{\\left|{\\mathbf{k}}\\right|}^{2\\,\\gamma}}\\,.$$ Since $\\gamma \\in (0,2)$, the function ${\\mathbf{k}}\\mapsto {\\exp \\left( A({\\mathbf{k}})\\,t \\right)}$ is not infinitely often differentiable at ${\\mathbf{k}}={{\\mathbf{0}}}$ and since the holomorphic function $\\frac{\\sin(B({\\mathbf{k}})\\,t)}{B({\\mathbf{k}})}$ does not vanish at ${\\mathbf{k}}={{\\mathbf{0}}}$, it follows that ${\\mathbf{k}}\\mapsto {\\mathcal{F}}_{3D}^{-1}\\{{\\mathcal{G}}\\}({\\mathbf{k}},t)$ is not infinitely often differentiable at ${\\mathbf{k}}={{\\mathbf{0}}}$. Consequently, ${{\\vec{x}}}\\mapsto {\\mathcal{G}}({{\\vec{x}}},t)$ cannot have compact support, which concludes the proof.",
"\n\nIntegro-Differential Equations Describing Attenuation {#sec:integro}\n=====================================================\n\nIn the following we derive the integro-differential equations for the attenuated pressure ${p_{\\text{att}}}$ for various attenuation laws. ",
"Thereby, we first derive equations which the according attenuated Green functions ${\\mathcal{G}}$ (cf. (",
"\\[eq:FGG0\\])) are satisfying, and then, by convolution, we derive the equations for ${p_{\\text{att}}}$. The integro-differential equations are general in the sense, that they apply to arbitrary source terms $f$, and in particular to the source term $f$ (\\[eq:source\\]) of the forward problem of photoacoustic imaging with attenuated waves.",
"\n\nFor this purpose, we rewrite $\\nabla^2 {\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}$ by using its definition (\\[eq:FGG0\\]), i.e. ${\\mathcal{G}}= K *_t {\\mathcal{G}}_0 $, and the product differentiation rule, which gives $$\\label{eq:laplG}\n\\begin{aligned}\n&\\frac{1}{\\sqrt{2\\,\\pi}}\\,\\nabla^2 {\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}\\\\\n= & \\nabla^2 {\\mathcal{F} \\left\\{ K \\right\\}} \\cdot {\\mathcal{F} \\left\\{ {\\mathcal{G}}_0 \\right\\}} +\n2 \\nabla {\\mathcal{F} \\left\\{ K \\right\\}} \\cdot \\nabla {\\mathcal{F} \\left\\{ {\\mathcal{G}}_0 \\right\\}}\n+ {\\mathcal{F} \\left\\{ K \\right\\}} \\cdot \\nabla^2 {\\mathcal{F} \\left\\{ {\\mathcal{G}}_0 \\right\\}}.",
"\n\\end{aligned}$$ To evaluate the expression on the right hand side, we calculate $\\nabla {\\mathcal{F} \\left\\{ K \\right\\}}$ and $\\nabla^2 {\\mathcal{F} \\left\\{ K \\right\\}}$. From (\\[eq:kernel\\]), it follows that $$\\label{eq:defg2b}\n\\begin{aligned}\n \\nabla {\\mathcal{F} \\left\\{ K \\right\\}} = -\\beta^*{}' \\cdot {\\mathcal{F} \\left\\{ K \\right\\}} \\cdot {\\mbox{sgn}}\\,,\n\\end{aligned}$$ where $\\beta^*{}'$ denotes the derivative of $\\beta^*(r,\\omega)$ (cf. (",
"\\[eq:kernel\\])) with respect to $r$. This together with the formula (\\[eq:der\\_sgn\\]) in the Appendix implies that $$\\label{eq:defg2c}\n\\begin{aligned}\n& \\nabla^2 {\\mathcal{F} \\left\\{ K \\right\\}}\\\\\n =&- \\nabla \\cdot \\left( \\beta^*{}' \\cdot {\\mathcal{F} \\left\\{ K \\right\\}} \\cdot {\\mbox{sgn}}\\right)\\\\\n =& - (\\nabla \\cdot {\\mbox{sgn}}) \\cdot \\beta^*{}' \\cdot {\\mathcal{F} \\left\\{ K \\right\\}}\n - ({\\mbox{sgn}}\\cdot \\nabla \\beta^*{}') \\cdot {\\mathcal{F} \\left\\{ K \\right\\}}\n - ({\\mbox{sgn}}\\cdot \\nabla {\\mathcal{F} \\left\\{ K \\right\\}}) \\cdot \\beta^*{}' \\\\\n =& \\left[ -\\frac{2}{{\\left|{{\\vec{x}}}\\right|}} \\cdot\\beta^*{}'\n - \\beta^*{}''\n + \\left(\\beta^*{}'\\right)^2\\right] \\cdot {\\mathcal{F} \\left\\{ K \\right\\}}.",
"\n\\end{aligned}$$ Inserting [(\\[eq:defg2b\\])]{} and [(\\[eq:defg2c\\])]{} into [(\\[eq:laplG\\])]{} and using again the identity ${\\mathcal{G}}= K *_t {\\mathcal{G}}_0$, shows that $$\\label{eq:laplG2}\n \\begin{aligned}\n & \\frac{1}{\\sqrt{2\\,\\pi}}\\,\\nabla^2 {\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}} \\\\\n &\\qquad = \\frac{1}{\\sqrt{2\\,\\pi}}\\,\\left[ -\\frac{2}{{\\left|{{\\vec{x}}}\\right|}} \\cdot \\beta^*{}' - \\beta^*{}''\n + \\left(\\beta^*{}'\\right)^2\\right]\\cdot{\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}\\\\\n &\\qquad\\quad -2 \\beta^*{}' \\cdot {\\mathcal{F} \\left\\{ K \\right\\}} \\cdot ({\\mbox{sgn}}\\cdot \\nabla {\\mathcal{F} \\left\\{ {\\mathcal{G}}_0 \\right\\}})\n + {\\mathcal{F} \\left\\{ K \\right\\}} \\cdot \\nabla^2 {\\mathcal{F} \\left\\{ {\\mathcal{G}}_0 \\right\\}}.",
"\n \\end{aligned}$$ From this identity, together with the two following properties of ${\\mathcal{G}}_0$, $$\\label{eq:g1a}\n\\nabla{\\mathcal{F} \\left\\{ {\\mathcal{G}}_0 \\right\\}}\n= \\left[\\frac{{{\\mathrm{i}}\\,}\\omega}{c_0} -\\frac{1}{{\\left|{{\\vec{x}}}\\right|}}\\right] \\cdot {\\mathcal{F} \\left\\{ {\\mathcal{G}}_0 \\right\\}} \\cdot {\\mbox{sgn}}\\,,$$ and $$\\label{eq:g1b}\n \\nabla^2 {\\mathcal{F} \\left\\{ {\\mathcal{G}}_0 \\right\\}} + \\frac{\\omega^2}{c_0^2}{\\mathcal{F} \\left\\{ {\\mathcal{G}}_0 \\right\\}}\n = -\\frac{1}{\\sqrt{2\\pi}}\\delta_{{\\vec{x}}}\\,,$$ it follows that $$\\label{eq:laplG3}\n \\begin{aligned}\n \\nabla^2 {\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}\n =& \\left[ -\\frac{2}{{\\left|{{\\vec{x}}}\\right|}} \\cdot \\beta^*{}'\n - \\beta^*{}''\n + \\left(\\beta^*{}'\\right)^2\\right] \\cdot {\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}\\\\\n & - 2 \\left[\\frac{{{\\mathrm{i}}\\,}\\omega}{c_0} -\\frac{1}{{\\left|{{\\vec{x}}}\\right|}}\\right] \\cdot\n \\beta^*{}' \\cdot {\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}\\\\\n &-\\frac{\\omega^2}{c_0^2} \\cdot {\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}\\\\\n &-{\\mathcal{F} \\left\\{ K \\right\\}} \\cdot \\delta_{{\\vec{x}}}\\;.",
"\n \\end{aligned}$$ Inserting the identity ${\\mathcal{F} \\left\\{ K \\right\\}}({{\\vec{x}}},\\omega) \\cdot \\delta_{{\\vec{x}}}= {\\mathcal{F} \\left\\{ K \\right\\}}({\\bf 0},\\omega) \\cdot \\delta_{{\\vec{x}}}$ in [(\\[eq:laplG3\\])]{} gives the *Helmholtz equation* $$\\label{eq:helmholtz}\n\\begin{aligned}\n~& \\nabla^2 {\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}\n -\\left[\\beta^*{}' + \\frac{(-{{\\mathrm{i}}\\,}\\, \\omega)}{c_0}\\right]^2 \\cdot{\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}\\\\\n =& -\\beta^*{}'' \\cdot {\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}\n - {\\mathcal{F} \\left\\{ K \\right\\}}({\\bf 0},\\cdot)\\cdot \\delta_{{\\vec{x}}}\\\\\n =& -\\beta^*{}'' \\cdot {\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}\n -\\frac{1}{\\sqrt{2\\pi}} {\\exp \\left( -\\beta^*({\\bf 0},\\omega) \\right)} \\cdot \\delta_{{\\vec{x}}}\\,.",
"\n\\end{aligned}$$ To reformulate [(\\[eq:helmholtz\\])]{} in space–time coordinates, we introduce two convolution operators: $$\\label{eq:defD*D*'}\n D_*f := K_* *_t f {\\qquad \\text{and} \\qquad}\n D_*'f :=K_*' *_t f ,$$ where the kernels $K_*$ and $K_*'$ are given by $$\\label{eq:defKK'}\n K_*:=K_*({{\\vec{x}}},t) := K_*({\\left|{{\\vec{x}}}\\right|},t) {\\quad \\text{and} \\quad}\n K_*(r,t) := \\frac{1}{\\sqrt{2\\pi}} {\\mathcal{F}^{-1} \\left\\{ \\beta^*{}' \\right\\}}(r,t)$$ and $$\\label{eq:defKK'2}\n K_*':=K_*'({{\\vec{x}}},t) := K_*'({\\left|{{\\vec{x}}}\\right|},t)\n{\\quad \\text{and} \\quad}\n K_*'(r,t) = \\frac{1}{\\sqrt{2\\pi}} {\\mathcal{F}^{-1} \\left\\{ \\beta^*{}'' \\right\\}}(r,t).$$ Using these operators and applying the inverse Fourier transform to [(\\[eq:helmholtz\\])]{} gives $$\\label{eq:waveeq+}\n \\nabla^2 {\\mathcal{G}}-\\left[D_*\n + \\frac{1}{c_0} \\frac{\\partial}{\\partial t}\\right]^2 {\\mathcal{G}}= -D_*' {\\mathcal{G}}- K({\\bf 0},\\cdot) \\delta_{{\\vec{x}}}\\;.$$ In the case that $\\beta^*({\\left|{{\\vec{x}}}\\right|},\\omega) = \\alpha^*(\\omega){\\left|{{\\vec{x}}}\\right|}$ is of standard form (\\[eq:attenuation\\_coefficient\\]), it follows that $$\\label{eq:defKK'+}\n\\begin{aligned}\nK_*(t) = \\frac{1}{\\sqrt{2\\pi}} {\\mathcal{F}^{-1} \\left\\{ \\alpha^* \\right\\}}(t) \\qquad\\text{ and } \\qquad\nK_*'\\equiv 0\\;.",
"\n\\end{aligned}$$\n\nFor a general source term $f$, we denote the attenuated wave by ${p_{\\text{att}}}$. That is $${p_{\\text{att}}}:= {p_{\\text{att}}}({{\\vec{x}}},t) = {\\mathcal{G}}*_{{{\\vec{x}}},t} f =: {\\mathcal{A}}f\\,,$$ where ${\\mathcal{A}}$ is the convolution operator according to the Green function ${\\mathcal{G}}$. This then shows that ${p_{\\text{att}}}$ satisfies the integro-differential equation $$\\label{eq:waveeq+2}\n\\boxed{\n\\nabla^2 {p_{\\text{att}}}-\\frac{1}{c_0^2} \\frac{\\partial^2 {p_{\\text{att}}}}{\\partial t^2} = - {\\mathcal{A}}_s f\\;,}$$ where ${\\mathcal{A}}_s$ denotes the space–time convolution operator with kernel $$\\label{eq:defKs}\n\\begin{aligned}\n K_s:=K_s({{\\vec{x}}},t) := -({\\mathcal{B}}{\\mathcal{G}})({{\\vec{x}}},t)\n +(D_*' {\\mathcal{G}})({{\\vec{x}}},t)\n + K({\\bf 0},t) \\cdot \\delta_{{\\vec{x}}}({{\\vec{x}}})\n\\end{aligned}$$ and $$\\label{eq:B}\n {\\mathcal{B}}:= D_*^2 + \\frac{2}{c_0} D_*\\frac{\\partial}{\\partial t}\\;.$$ Equation [(\\[eq:waveeq+2\\])]{} is called *pressure wave equation with attenuation coefficient* $\\beta^*$. We emphasize that $\\beta^*$ determines the operators $D_*$ and $D'_*$ which in turn determine the operator ${\\mathcal{A}}_s$, which in turn determines ${p_{\\text{att}}}$ - this reveals the dependence of ${p_{\\text{att}}}$ from $\\beta_*$.\n\n\\[rema:alpha\\*2\\] Let $\\beta^*(r,\\omega)=\\alpha^*(\\omega)r$ be the standard attenuation model (cf. (",
"\\[eq:attenuation\\_law\\])). ",
"Assuming that the associated kernel $K$ (cf. (",
"\\[31\\])) is causal, it follows that $${\\left|\\nabla K\\right|} = \\frac{1}{\\sqrt{2\\pi}} {\\left|{\\mathcal{F}^{-1} \\left\\{ \\alpha^* \\cdot {\\exp \\left( -\\alpha^* {\\left|{{\\vec{x}}}\\right|} \\right)} \\right\\}}\\right|}\\;.$$ Using some sequence ${\\{ {{\\vec{x}}}_n \\}}$ satisfying ${{\\vec{x}}}_n \\neq {\\bf 0}$ and ${{\\vec{x}}}_n \\to {\\bf 0}$ shows that $$\\lim_{n \\to \\infty} {\\left|\\nabla K\\right|}({{\\vec{x}}}_n,t) = \\frac{1}{\\sqrt{2\\pi}} {\\left|{\\mathcal{F}^{-1} \\left\\{ \\alpha^* \\right\\}}(t)\\right|} \\underbrace{=}_{(\\ref{eq:defKK'+})} {\\left|K_*(t)\\right|}\\;.$$ Due to the causality of $K$ the left hand side is zero for $t<0$, and thus $K_*$ is also causal.",
"\n\nBecause the convolution of causal distributions is well-defined, the operator $D_*$ is well-defined on all causal distributions. ",
"Moreover, since $K_*'=0$, it follows that $D_*'\\equiv 0$. Using that $K_*$ depends only on $t$ it follows that $$(D_* {\\mathcal{G}}) *_{{{\\vec{x}}},t} f = [K_* *_t {\\mathcal{G}}] *_{{{\\vec{x}}},t} f\n = K_* *_t [{\\mathcal{G}}*_{{{\\vec{x}}},t} f] = D_*({\\mathcal{G}}*_{{{\\vec{x}}},t} f).$$ Convolving each term in [(\\[eq:waveeq+\\])]{} with a function $f$, using the previous identity and that $D_*' \\equiv 0$, it follows that $$\\label{eq:waveeq+3}\n \\nabla^2 {p_{\\text{att}}}-\\left[D_* + \\frac{1}{c_0} \\frac{\\partial}{\\partial t}\\right]^2 {p_{\\text{att}}}= - f \\;$$ where $$\\label{eq:D*sp}\n D_* \\cdot = \\frac{1}{\\sqrt{2\\,\\pi}}\\,{\\mathcal{F} \\left\\{ \\alpha^* \\right\\}}(t) *_t \\cdot \\,.$$\n\nIn the following we derive the common forms of the wave equation models corresponding to the various attenuation models listed in Section \\[sec:attenuation\\].",
"\n\nPower Laws {#power-laws-1 .unnumbered}\n----------\n\n- Let $0 < \\gamma \\not\\in {{\\mathbb{N}}}$ and $0 < \\alpha_0$. We note that the *Riemann-Liouville fractional derivative* with respect to time, denote by $D_t^\\gamma$ (see [@KilSriTru06; @Pod99]), is defined in the Fourier domain by $$\\label{eq:defDtga}\n {\\mathcal{F} \\left\\{ D_t^{\\gamma}f \\right\\}} = (-{{\\mathrm{i}}\\,}\\omega)^\\gamma {\\mathcal{F} \\left\\{ f \\right\\}}\\,,$$ and satisfies $$\\label{eq:propDtgamma}\n D_t^{2\\gamma}f = D_t^\\gamma D_t^{\\gamma}f {\\qquad \\text{and} \\qquad}\n \\frac{\\partial}{\\partial t} D_t^{\\gamma}f = D_t^\\gamma \\frac{\\partial}{\\partial t} f = D_t^{\\gamma+1}f.$$ From this together with (\\[eq:powlaw1\\]) and (\\[eq:D\\*sp\\]), we infer $$\\label{deftildealpha0}\n D_* = \\tilde\\alpha_0\\, D_t^\\gamma \\qquad \\mbox{ with }\\qquad\n \\tilde \\alpha_0 := \\frac{\\alpha_0}{\\cos(\\pi\\,\\gamma/2)}$$ and thus wave equation (\\[eq:waveeq+3\\]) reads as follows $$\\label{eq:standwaveeq}\n \\boxed{\n \\begin{aligned}\n \\nabla^2 {p_{\\text{att}}}&-\\left[\\tilde\\alpha_0 \\,D_t^\\gamma\n + \\frac{1}{c_0} \\frac{\\partial}{\\partial t}\\right]^2 {p_{\\text{att}}}= -f \\;.",
"\n \\end{aligned}}$$\n\n- Let $\\gamma=1$, then for the frequency power law (\\[alphagamma1\\]) it follows from the Fourier transform table I in [@Lig64] $$\\begin{aligned}\n D_* &= -\\frac{4\\,\\alpha_0}{2\\,\\pi} \\left[\\,\n \\frac{H(t)}{t^2} - (\\log|\\omega_0|)\\,\\delta'_t\n \\right] *_t \\\\\n &= -\\frac{4\\,\\alpha_0}{2\\,\\pi}\\, \\frac{H(t)}{t^2} *_t \\;\\;\n + \\frac{4\\,\\alpha_0}{\\sqrt{2\\,\\pi}}\\,(\\log|\\omega_0|)\\,\\frac{\\partial }{\\partial t}\\,.",
"\n \\end{aligned}$$\n\nSzabo’s Attenuation Law: {#szabos-attenuation-law .unnumbered}\n------------------------\n\nLet $0 < \\alpha_0$ and $0 < \\gamma\\not\\in{{\\mathbb{N}}}$. From (\\[alpha\\*szabo\\]) and (\\[eq:defDtga\\]), we get $$\\left[D_* + \\frac{1}{c_0} \\frac{\\partial}{\\partial t}\\right]^2\n = \\frac{1}{c_0^2}\\frac{\\partial^2 }{\\partial t^2}\n + \\frac{2\\,\\tilde\\alpha_0}{c_0}\\,\\frac{\\partial }{\\partial t}\\,D_t^\\gamma$$ and thus wave equation (\\[eq:waveeq+3\\]) reads as follows $$\\label{szaboseq}\n\\begin{aligned}\n\\boxed{\n \\nabla^2 {p_{\\text{att}}}-\\frac{1}{c_0^2}\\frac{\\partial^2 {p_{\\text{att}}}}{\\partial t^2}\n - \\frac{2\\,\\tilde\\alpha_0}{c_0}\\,\\frac{\\partial }{\\partial t}\\,D_t^\\gamma {p_{\\text{att}}}= -f({{\\vec{x}}},t)\\,.}",
"\n\\end{aligned}$$\n\nThermo-Viscous Attenuation Law: {#thermo-viscous-attenuation-law-1 .unnumbered}\n-------------------------------\n\nFrom (\\[alpha\\*th\\]) we get $$\\left(\\mbox{Id} + \\tau_0\\,\\frac{\\partial}{\\partial t}\\right)\\,\n \\left[D_* + \\frac{1}{c_0} \\frac{\\partial}{\\partial t}\\right]^2\n = \\frac{1}{c_0^2} \\frac{\\partial^2}{\\partial t^2}$$ and thus (\\[eq:waveeq+3\\]) becomes $$\\label{thviscwaveeq}\n\\begin{aligned}\n\\boxed{\n \\left(\\mbox{Id} + \\tau_0\\,\\frac{\\partial}{\\partial t}\\right)\\,\\nabla^2 {p_{\\text{att}}}- \\frac{1}{c_0^2}\\frac{\\partial^2 {p_{\\text{att}}}}{\\partial t^2}\n = -\\left(\\mbox{Id} + \\tau_0\\,\\frac{\\partial}{\\partial t}\\right)\\,f\\,.}",
"\n\\end{aligned}$$ This equation is called the *thermo-viscous wave equation*.",
"\n\nNachman, Smith and Waag [@NacSmiWaa90]: {#nachman-smith-and-waag .unnumbered}\n---------------------------------------\n\nWe carry out the details only for one relaxation process.",
"\n\n- $N=1$: From (\\[alpha\\*Nachman+\\]) we get $$\\left(\\alpha^*(\\omega) + \\frac{(-{{\\mathrm{i}}\\,}\\,\\omega)}{c_0} \\right)^2\n = \\frac{(-{{\\mathrm{i}}\\,}\\,\\omega)^2}{\\tilde c_0^2}\\,\\frac{1-{{\\mathrm{i}}\\,}\\,\\tilde \\tau_1\\,\\omega}{1-{{\\mathrm{i}}\\,}\\,\\tau_1\\,\\omega}$$ which implies $$\\left(\\mbox{Id} + \\tau_1\\,\\frac{\\partial}{\\partial t}\\right)\\,\n \\left[D_* + \\frac{1}{c_0} \\frac{\\partial}{\\partial t}\\right]^2\n = \\left(\\mbox{Id} + \\tilde\\tau_1\\,\\frac{\\partial}{\\partial t}\\right)\\,\n \\frac{1}{\\tilde c_0^2} \\frac{\\partial^2}{\\partial t^2}\\,.$$ Thus (\\[eq:waveeq+3\\]) reads as follows $$\\label{Nachmanwaveeq}\n \\boxed{\n \\begin{aligned}\n \\left(\\mbox{Id} + \\tau_1\\,\\frac{\\partial}{\\partial t}\\right)\\,\\nabla^2 {p_{\\text{att}}}&\n - \\frac{1}{\\tilde c_0^2}\\left(\\mbox{Id} + \\tilde\\tau_1\\,\\frac{\\partial}{\\partial t}\\right)\\,\n \\frac{\\partial^2 {p_{\\text{att}}}}{\\partial t^2}\\\\\n &= -\\left(\\mbox{Id} + \\tau_1\\,\\frac{\\partial}{\\partial t}\\right)\\,f\\,.",
"\n \\end{aligned}}$$ If the term with $\\tilde\\tau_1=0$ is dropped and $\\tilde c_0$ is replaced by $c_0$, then we obtain the thermo-viscous wave equation (\\[thviscwaveeq\\]).",
"\n\n- $N>1$: For the general case we refer to equation (26) in [@NacSmiWaa90].",
"\n\nGreenleaf and Patch [@PatGre06]: {#greenleaf-and-patch .unnumbered}\n--------------------------------\n\n- For $\\gamma=2$ the attenuation coefficient equals to $$\\alpha^*(\\omega) = \\alpha_0\\,\\omega^2 = \\tilde\\alpha_0\\,(-{{\\mathrm{i}}\\,}\\omega)^2\\,,$$ where $\\tilde\\alpha_0$ is defined as in (\\[deftildealpha0\\]) and thus $$D_* = -\\alpha_0\\,\\frac{\\partial^2 }{\\partial t^2} = -\\alpha_0\\,D_t^ 2\\,,$$ which gives wave equation (\\[eq:standwaveeq\\]) with $\\gamma=2$.\n\n- For $\\gamma=1$ we have $$\\alpha^*(\\omega) = \\alpha_0\\,|\\omega| = \\alpha_0\\,(-{{\\mathrm{i}}\\,}\\omega)\\,{{\\mathrm{i}}\\,}{\\mbox{sgn}}(\\omega)$$ and thus $$D_* = \\alpha_0\\,\\mathbf{D}^{-1}\n = -\\alpha_0\\,\\frac{\\partial}{\\partial t} {\\mathcal{H}}\\,,$$ where $\\mathbf{D}^{-1}$ and ${\\mathcal{H}}$ denote the *Riesz fractional differentiation operator* and the Hilbert transform (cf. ",
"Appendix), respectively. ",
"Therefore the wave equation reads as follows $$\\boxed{\n \\begin{aligned}\n \\nabla^2 {p_{\\text{att}}}&-\\left[\\alpha_0\\,\\mathbf{D}^{-1}\n + \\frac{1}{c_0} \\frac{\\partial}{\\partial t}\\right]^2 {p_{\\text{att}}}= -f \\;.",
"\n \\end{aligned}}$$\n\nChen and Holm [@CheHolm04]: {#chen-and-holm .unnumbered}\n---------------------------\n\nLet $\\gamma \\in (0,2)$. The Green function defined by (\\[GreenChenHolm\\]) satisfies the Helmholtz equation $$\\label{Holmeq02}\n\\begin{aligned}\n~ &\\frac{\\partial^2 {\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}}{\\partial t^2}({\\mathbf{k}},t)\n + 2\\,\\alpha_1\\,c_0\\,{\\left|{\\mathbf{k}}\\right|}^\\gamma \\,\\frac{\\partial {\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}}{\\partial t}({\\mathbf{k}},t)\n + c_0^2\\,{\\left|{\\mathbf{k}}\\right|}^2 \\,{\\mathcal{F} \\left\\{ {\\mathcal{G}}\\right\\}}({\\mathbf{k}},t)\\\\\n= & \\frac{c_0^2}{(2\\,\\pi)^{3/2}} \\,\\delta(t)\n\\end{aligned}$$ for $t\\in{{\\mathbb{R}}}$ and ${\\mathbf{k}}\\in{{\\mathbb{R}}}^3$. Since the fractional Laplacian for a rotational symmetric function $f$ and $\\gamma\\in (0,2)$ is defined by (cf. ",
"Definition (2.10.1) in ([@KilSriTru06]) $$\\begin{aligned}\n & \\left(-\\nabla^2\\right)^{\\gamma/2} f({{\\vec{x}}})\\\\\n:= &\\frac{1}{\\sqrt{(2\\,\\pi)^3}}\\,\\int_{{{\\mathbb{R}}}^3} {\\exp \\left( {{\\vec{x}}}\\cdot{\\mathbf{k}}\\right)}\\,\n \\left[ {\\left|{\\mathbf{k}}\\right|}^\\gamma\\,\n \\frac{1}{\\sqrt{(2\\,\\pi)^3}}\\,\\int_{{{\\mathbb{R}}}^3} {\\exp \\left( -{{\\vec{x}}}\\cdot{\\mathbf{k}}\\right)}\\, f({{\\vec{x}}})\\,{\\mbox{d}}{{\\vec{x}}}\\right]\n \\,{\\mbox{d}}{\\mathbf{k}}\\\\\n= & {\\mathcal{F}}_{3D}\\left\\{ {\\left|{\\mathbf{k}}\\right|}^\\gamma\\,{\\mathcal{F}}_{3D}^{-1}\\{ f\\}({\\mathbf{k}}) \\right\\}({{\\vec{x}}})\\,,\n\\end{aligned}$$ we obtain the following wave equation for ${p_{\\text{att}}}:={\\mathcal{G}}\\ast_{{{\\vec{x}}},t}f$ $$\\label{Holmeq01}\n\\boxed{\n\\begin{aligned}\n \\nabla^2 {p_{\\text{att}}}-\\frac{1}{c_0^2}\\frac{\\partial^2 {p_{\\text{att}}}}{\\partial t^2}\n - \\frac{2\\,\\alpha_1}{c_0}\\,\\frac{\\partial }{\\partial t}\\,\\left(-\\nabla^2\\right)^{\\gamma/2} {p_{\\text{att}}}= -f({{\\vec{x}}},t)\\;.",
"\n\\end{aligned}}$$ We note that Chen and Holm used instead of $2\\,\\alpha_1/c_0$ the term $2\\,\\alpha_1/c_0^{1-\\gamma}$.\n\nOur Model [@KowSchBon10]: {#our-model-1 .unnumbered}\n-------------------------\n\nFrom (\\[eq:powlaw2\\]) we get $$\\left(\\mbox{Id} + \\tau_0^{\\gamma-1}\\,D_t^{\\gamma-1}\\right)\\,\n \\left[D_* + \\frac{1}{c_0} \\frac{\\partial}{\\partial t}\\right]^2\n = \\frac{1}{c_0^2} \\frac{\\partial^2}{\\partial t^2} \\,\n \\left(\\alpha_0\\,\\mbox{Id} + L^{1/2}\\right)^2$$ where the time convolution operator $L^{1/2}$ is the convolution operator with kernel $$l(t):=L^{1/2} (\\delta_t)\n = \\frac{1}{\\sqrt{2\\,\\pi}}\\, {\\mathcal{F}}\\left\\{ \\sqrt{1+(-{{\\mathrm{i}}\\,}\\tau_0\\,\\omega)^{\\gamma-1}} \\right\\}\\;.$$ Consequently, $L:=(L^{1/2})^2=\\mbox{Id} + \\tau_0\\,D_t^{\\gamma-1}$ and (\\[eq:waveeq+3\\]) can be rewritten as follows $$\\label{causthviscwaveeq}\n\\boxed{\n\\begin{aligned}\n \\left(\\mbox{Id} + \\tau_0^{\\gamma-1}\\,D_t^{\\gamma-1}\\right)\\,\\nabla^2 {p_{\\text{att}}}&\n - \\frac{1}{c_0^2}\\,\\left(\\alpha_0\\,\\mbox{Id} + L^{1/2}\\right)^2\n \\,\\frac{\\partial^2 {p_{\\text{att}}}}{\\partial t^2}\\\\\n &= -\\left(\\mbox{Id} + \\tau_0^{\\gamma-1}\\,D_t^{\\gamma-1}\\right)\\,f\\,.",
"\n\\end{aligned}}$$\n\nPressure Relation\n=================\n\nIn this section we derive the relation between ${p_{\\text{att}}}$ and $p_0$ when the source term is of the form (\\[eq:source\\]). ",
"This chapter is a special instance of Section \\[sec:integro\\]. ",
"However, utilizing the special structure of the source term different formulas can be derived.",
"\n\nAttenuation is defined as a multiplicative law (in the frequency domain) relating the amplitudes of an attenuated and an unattenuated wave initialized by a delta impulse. ",
"Here we are concerned in deriving the convolution relation between the solution $p_0$ of [(\\[eq:ex:wave3d\\])]{} (or equivalently of [(\\[eq:ex:ivp3d\\])]{} and [(\\[eq:init\\_values\\])]{}) and the attenuated wave function ${p_{\\text{att}}}$, which, according to (\\[eq:FGG0\\]) and (\\[eq:a\\_conv\\]), is given by $$\\label{eq:defpatt}\n {p_{\\text{att}}}= {\\mathcal{G}}*_{{{\\vec{x}}},t} f = (K *_t {\\mathcal{G}}_0) *_{{{\\vec{x}}},t} f\\,,$$ with $f$ from (\\[eq:source\\]). ",
"Using (\\[eq:g0\\]) and the rotational symmetry of $K$, it follows that $$\\begin{aligned}\n & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad{p_{\\text{att}}}({{\\vec{x}}},t) \\\\\n &= \\int_{{\\mathbb{R}}}\\int_{{{\\mathbb{R}}}^3} (K \\ast_{t}{\\mathcal{G}}_0)({{\\vec{x}}}-{{\\vec{x}}}',t-t'')\n \\rho({{\\vec{x}}}')\\,d{{\\vec{x}}}' \\frac{\\partial \\delta_t}{\\partial t}(t'')\\,dt''\\\\\n &= \\int_{{\\mathbb{R}}}\\int_{{{\\mathbb{R}}}^3} \\int_{{\\mathbb{R}}}K({{\\vec{x}}}-{{\\vec{x}}}',t-t'-t''){\\mathcal{G}}_0({{\\vec{x}}}-{{\\vec{x}}}',t')\\,dt'\n \\rho({{\\vec{x}}}')\\,d{{\\vec{x}}}' \\frac{\\partial \\delta_t}{\\partial t}(t'')\\,dt''\\\\\n &= \\int_{{\\mathbb{R}}}\\int_{{{\\mathbb{R}}}^3} \\int_{{\\mathbb{R}}}\\frac{\\partial }{\\partial t}K({{\\vec{x}}}-{{\\vec{x}}}',t-t'-t''){\\mathcal{G}}_0({{\\vec{x}}}-{{\\vec{x}}}',t')\n \\rho({{\\vec{x}}}') \\delta_t (t'')\\,dt'\\,d{{\\vec{x}}}' \\,dt''\\\\\n &= \\int_{{{\\mathbb{R}}}^3} \\int_{{\\mathbb{R}}}\\frac{\\partial }{\\partial t} K({\\left|{{\\vec{x}}}-{{\\vec{x}}}'\\right|},t-t')\\,\n \\frac{\\delta_t(t'-{\\left|{{\\vec{x}}}-{{\\vec{x}}}'\\right|}/c_0)}{4\\pi {\\left|{{\\vec{x}}}-{{\\vec{x}}}'\\right|}} \\,\\rho({{\\vec{x}}}')\\,{\\mbox{d}}t'\\,{\\mbox{d}}{{\\vec{x}}}'\\\\\n &= \\int_{{{\\mathbb{R}}}^3}\\frac{\\partial }{\\partial t} K({\\left|{{\\vec{x}}}-{{\\vec{x}}}'\\right|},t-{\\left|{{\\vec{x}}}-{{\\vec{x}}}'\\right|}/c_0)\\,\\frac{\\rho({{\\vec{x}}}')}{4\\pi {\\left|{{\\vec{x}}}-{{\\vec{x}}}'\\right|}} \\,{\\mbox{d}}{{\\vec{x}}}'\\,.\\\\\n\\end{aligned}$$\n\nUsing the representation ${{\\vec{x}}}'-{{\\vec{x}}}= r' \\vec{s}$ with $r' \\geq 0$ and $\\vec{s} \\in S^2$, it follows that $$\\label{eq:pattrep01}\n{p_{\\text{att}}}({{\\vec{x}}},t) = \\frac{1}{4 \\pi}\\, \\int_0^\\infty \\frac{\\partial }{\\partial t} K(r',t-r'/c_0) \\,r'\\,\n \\int_{S^2} \\rho({{\\vec{x}}}+r'\\vec{s})\\,d\\vec{s}\\,dr'\\;.$$ Moreover, $$\\begin{aligned}\np_0({{\\vec{x}}},t)\n&= \\frac{\\partial }{\\partial t} \\int_{{{\\mathbb{R}}}^3} \\frac{\\delta_t(t-{\\left|{{\\vec{x}}}-{{\\vec{x}}}'\\right|}/c_0)}{4\\pi {\\left|{{\\vec{x}}}-{{\\vec{x}}}'\\right|}} \\rho({{\\vec{x}}}')\\,{\\mbox{d}}{{\\vec{x}}}'\\\\\n&= \\frac{\\partial }{\\partial t} \\int_0^\\infty r'^2 \\int_{S^2} \\frac{\\delta_t(t-r'/c_0)}{4\\pi r'} \\rho({{\\vec{x}}}+ r' \\vec{s})\\,d\\vec{s}\\,dr'\\\\\n&= \\frac{\\partial }{\\partial t} \\int_0^\\infty r' \\frac{\\delta_t(t-r'/c_0)}{4\\pi} \\int_{S^2} \\rho({{\\vec{x}}}+ r' \\vec{s})\\,d\\vec{s}\\,dr'\\\\\n&= \\frac{\\partial }{\\partial t} \\int_0^\\infty \\frac{c_0^2 r''}{4\\pi} \\delta_t(t-r'') \\int_{S^2} \\rho({{\\vec{x}}}+ c_0 r'' \\vec{s})\\,d\\vec{s}\\,d r''\\\\\n&= \\frac{\\partial }{\\partial t} \\left(\\frac{c_0^2 t}{4\\pi} \\int_{S^2} \\rho({{\\vec{x}}}+ (c_0t) \\vec{s})\\,d\\vec{s} \\right)\\;.",
"\n\\end{aligned}$$ This gives $$\\label{eq:summ}\n\\begin{aligned}\n r'\\, \\int_{S^2} \\rho({{\\vec{x}}}+r'\\vec{s})\\,d \\vec{s}\n = \\frac{4\\,\\pi}{c_0} \\int_0^{r'/c_0} p_0({{\\vec{x}}},t')\\,{\\mbox{d}}t'\\;.",
"\n\\end{aligned}$$ Now, denoting $$F(t,r') := \\int_0^{r'} \\frac{\\partial}{\\partial t} K(r'',t-r''/c_0) \\,dr''\\,,\\qquad\nG(r') := \\int_0^{r'/c_0} p_0({{\\vec{x}}},r'')\\,dr''\\,$$ and $$F(t,\\infty) = \\lim_{r' \\to \\infty} F(t,r')$$ it follows from (\\[eq:pattrep01\\]) and (\\[eq:summ\\]) and the fact that $p_0({{\\vec{x}}},0) = 0$ that $$\\label{eq:pattrep02a}\n\\begin{aligned}\n{p_{\\text{att}}}({{\\vec{x}}},t) =& \\frac{1}{c_0} \\int_0^\\infty F'(t,r') G(r')\\,dr'\\\\\n=& - \\frac{1}{c_0} \\int_0^\\infty F(t,r') \\underbrace{G'(r')}_{=p_0({{\\vec{x}}},r'/c_0)/c_0}\\,dr' + \\frac{1}{c_0} \\left. ",
"F(t,r') G(r') \\right|_{r'=0}^\\infty\\\\\n=& \\frac{1}{c_0^2} \\left( F(t,\\infty) \\int_0^\\infty p_0({{\\vec{x}}},r'/c_0)\\,dr'- \\int_0^\\infty F(t,r') p_0({{\\vec{x}}},r'/c_0)\\,dr'\\right)\\\\\n=& \\frac{1}{c_0} \\left( F(t,\\infty) \\int_0^\\infty p_0({{\\vec{x}}},t')\\,dt' - \\int_0^\\infty F(t,c_0 t') p_0({{\\vec{x}}},t')\\,d t'\\right)\\\\\n=:& \\int_0^\\infty {\\mathcal{M}}(t,t') p_0({{\\vec{x}}},t')\\,dt'\\,,\n\\end{aligned}$$ where $$\\label{eq:mos}\n\\begin{aligned}\n {\\mathcal{M}}(t,t') :&= \\frac{1}{c_0} \\,(F(t,\\infty)-F(t,c_0 t')) \\,.",
"\n\\end{aligned}$$ In the following we derive an equivalent representation of ${\\mathcal{M}}$ in terms of the attenuation coefficient,under the assumption that the attenuation coefficient $\\alpha^*$ is such that $K$ is causal. ",
"From [(\\[eq:kernel\\])]{}, [(\\[eq:attenuation\\_law\\])]{} and Item \\[item:Fdelta\\] in the Appendix, it follows that $$\\begin{aligned}\n K(r',t-r'/c_0)\n = \\frac{1}{\\sqrt{2\\,\\pi}}\\, {\\mathcal{F}^{-1} \\left\\{ {\\exp \\left( -\\alpha^*(\\omega)\\,r'+{{\\mathrm{i}}\\,}\\,\\frac{\\omega}{c_0}\\,r' \\right)} \\right\\}}(t) \\,.",
"\n\\end{aligned}$$ which implies $$\\label{eq:pattrep02b}\n\\begin{aligned}\n~ & F(t,c_0\\,t')\\\\\n =& \\int_0^{c_0\\,t'} \\frac{\\partial }{\\partial t} K(r',t-r'/c_0)\\,{\\mbox{d}}r' \\\\\n =& \\frac{1}{\\sqrt{2\\,\\pi}}\\, {\\mathcal{F}^{-1} \\left\\{ \\frac{-{{\\mathrm{i}}\\,}\\,\\omega\\,\\left[ {\\exp \\left( -\\alpha^*(\\omega)\\,c_0\\,t'+{{\\mathrm{i}}\\,}\\,\\omega\\,t' \\right)} -1 \\right]}\n {-\\alpha^*(\\omega)+{{\\mathrm{i}}\\,}\\,\\omega/c_0}\\, \\right\\}}(t) \\\\\n =& \\frac{1}{\\sqrt{2\\,\\pi}}\\, {\\mathcal{F}^{-1} \\left\\{ \\frac{-{{\\mathrm{i}}\\,}\\,\\omega\\,}{-\\alpha^*(\\omega)+{{\\mathrm{i}}\\,}\\,\\omega/c_0}\\, \\right\\}}(t) \\ast_t\n \\left[ K(c_0\\,t',t-t') - \\delta_t(t)\\right]\\,.",
"\n\\end{aligned}$$ Since $K$ is causal, it satisfies $K(c_0\\,t',t-t') = 0$ for $t < t'$, and therefore $$F(t,\\infty) = \\lim_{t'\\to \\infty} F(t,c_0\\,t')\n = \\frac{1}{\\sqrt{2\\,\\pi}}\\, {\\mathcal{F}^{-1} \\left\\{ \\frac{-{{\\mathrm{i}}\\,}\\,\\omega\\,}{-\\alpha^*(\\omega)+{{\\mathrm{i}}\\,}\\,\\omega/c_0}\\, \\right\\}}(t)\\,.$$ Hence [(\\[eq:pattrep02b\\])]{} can be written as follows: $$\\begin{aligned}\n F(t,c_0\\,t')\n &= F(t,\\infty) \\ast_t\n K(c_0\\,t',t-t') - F(t,\\infty)\\,.",
"\n\\end{aligned}$$ and therefore [(\\[eq:mos\\])]{} simplifies to $$\\label{eq:defM}\n\\begin{aligned}\n {\\mathcal{M}}(t,t')\n &= - \\frac{1}{c_0}\\,F(t,\\infty) \\ast_t K(c_0\\,t',t-t') \\\\\n & =\\frac{1}{\\sqrt{2\\,\\pi}}\\, {\\mathcal{F}^{-1} \\left\\{ \\frac{{{\\mathrm{i}}\\,}\\,\\omega \\,{\\exp \\left( -\\alpha^*(\\omega)\\,c_0\\,t'+{{\\mathrm{i}}\\,}\\,\\omega\\,t' \\right)}}\n {-\\alpha^*(\\omega)\\,c_0+{{\\mathrm{i}}\\,}\\,\\omega}\\, \\right\\}}(t) \\,.",
"\n\\end{aligned}$$ Note that ${\\mathcal{M}}(t,0)= - \\frac{1}{c_0}\\,F(t,\\infty)$. The following lemma shows that if $K$ is causal $$\\label{eq:propM}\n {\\mathcal{M}}(t,t') = 0 \\qquad \\mbox{ if } \\qquad 0< t < t'\\,$$ and therefore the upper limit of integration in the last term [(\\[eq:pattrep02a\\])]{} can be replaced by $t$. This means that the set of attenuated pressure values $${\\{ {p_{\\text{att}}}({{\\vec{x}}},s) : 0\\leq s\\leq t \\}}$$ depend only on the unattenuated pressure values $${\\{ p_0({{\\vec{x}}},s) : 0\\leq s\\leq t \\}}\\;.$$\n\n\\[lemm02\\] Let $K$ from [(\\[eq:kernel\\])]{} be causal with $\\beta^*(r,\\omega):=\\alpha^*(\\omega)\\,r$. Moreover, assume that $\\alpha^*(\\omega) \\neq {{\\mathrm{i}}\\,}\\,\\omega/c_0$ for every $\\omega\\in{{\\mathbb{R}}}$, and let ${\\mathcal{M}}$ be as defined in [(\\[eq:defM\\])]{}. ",
"Then,\n\n- the function $t \\to {\\mathcal{M}}(t,0)$ is causal.",
"\n\n- For every $t'>0$ $$\\label{propM1}\n {\\mathcal{M}}(t,t')=0 \\text{ for all } t < |t'|\\,.$$\n\n<!-- --",
">\n\n- In order to prove causality of $t \\to {\\mathcal{M}}(t,0)$ we verify the three assumptions of Theorem \\[th:lion\\] for the tempered distribution $${\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}}(\\omega,0) = \\frac{1}{\\sqrt{2\\,\\pi}}\\,\\frac{\\omega }{k(\\omega)\\,c_0}\n \\qquad\\mbox{with}\\qquad k(\\omega):= {{\\mathrm{i}}\\,}\\,\\alpha^*(\\omega) + \\frac{\\omega}{c_0}\\,.$$ Since $K$ is causal, as has been shown in Remark \\[rema:alpha\\*2\\], also the function $$t\\mapsto K_*(t)=\\frac{1}{\\sqrt{2\\pi}} {\\mathcal{F}^{-1} \\left\\{ \\alpha^* \\right\\}}(t)$$ is causal. ",
"Now, using Theorem \\[th:lion\\], it follows that\n\n 1. ",
" $\\alpha_*$ is holomorphic in $\\mathring{{{\\mathbb{C}}}}_0$,\n\n 2. ",
" $\\alpha^*(\\xi+{{\\mathrm{i}}\\,}\\,\\eta)\\to \\alpha^*(\\xi)$ for $\\eta\\to 0$ in ${\\mathcal{S}}'$ and\n\n 3. ",
" for each $ \\epsilon>0$ there exists a polynomial $P$ such that ${\\left|\\alpha^*(z)\\right|}\\leq P({\\left|z\\right|})$ for $z\\in{{\\mathbb{C}}}_\\epsilon$.\n\n Since $\\alpha_*(\\omega)\\neq {{\\mathrm{i}}\\,}\\,\\omega/c_0$ for all $\\omega\\in {{\\mathbb{R}}}$ together with in [@BelWoh66 Theorem 2.7] it follows that $z \\to \\alpha_*(z)$ is unique holomorphic extension to ${{\\mathbb{C}}}_0$ and therefore $z \\to \\alpha_*(z)$ cannot be identical to $z \\to {{\\mathrm{i}}\\,}\\,z/c_0$, the holomorphic extension of ${{\\mathrm{i}}\\,}\\,\\omega/c_0$. $\\alpha_*(z)\\neq{{\\mathrm{i}}\\,}\\,z/c_0$ for $z\\in {{\\mathbb{C}}}_0$ implies that $k$ has no zeros and hence $z/k(z)$ is holomorphic on $\\mathring{{{\\mathbb{C}}}}_0$. This shows that Item \\[it1\\_Lions\\] in Theorem \\[th:lion\\] is satisfied for $t \\to {\\mathcal{M}}(t,0)$.\n\n Since $1/(k(\\xi+{{\\mathrm{i}}\\,}\\,\\eta)\\,k(\\xi))$ is bounded and $k(\\xi+{{\\mathrm{i}}\\,}\\,\\eta)\\to k(\\xi)$ for $\\eta\\to 0$ in $S'$, it follows that for all $\\psi\\in {\\mathcal{S}}$ $$\\lim_{\\eta\\to 0} \\int_{{\\mathbb{R}}}\\left[ \\frac{k(\\xi)-k(\\xi+{{\\mathrm{i}}\\,}\\,\\eta)}{k(\\xi+{{\\mathrm{i}}\\,}\\,\\eta)\\,k(\\xi)}\\right]\\,\\psi(\\xi)\\,{\\mbox{d}}\\xi \\to 0$$ i.e. $1/k(\\xi+{{\\mathrm{i}}\\,}\\,\\eta)\\to 1/k(\\xi)$ for $\\eta\\to 0$ in $S'$. Hence ${\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}}(\\xi+{{\\mathrm{i}}\\,}\\,\\eta,0)\\to {\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}}(\\xi,0)$ for $\\eta\\to 0$ in ${\\mathcal{S}}'$, which shows Item \\[it2\\_Lions\\] in Theorem \\[th:lion\\].",
"\n\n Since $k(z)$ does not vanish on ${{\\mathbb{C}}}_0$ and ${\\left|k(z)\\right|}$ is bounded by a polynomial in ${\\left|z\\right|}$ for $z\\in\\mathring{{{\\mathbb{C}}}}_0$, it follows that ${\\left|z/k(z)\\right|}$ is bounded by a polynomial in ${\\left|z\\right|}$. Hence Item \\[it3\\_Lions\\] in Theorem \\[th:lion\\] is satisfied and consequently $t \\to {\\mathcal{M}}(t,0)$ is causal.",
"\n\n- Property (\\[propM1\\]) is satisfied if $$\\label{MMM}\n \\begin{aligned}\n {\\mathcal{M}}(t + {\\left|t'\\right|},t')=0 \\qquad \\mbox{ for } \\quad t < 0\\,.",
"\n \\end{aligned}$$ From [(\\[eq:defM\\])]{} and $$K(c_0\\,t',t)={\\mathcal{F}}^{-1}\\left\\{ \\frac{e^{-\\alpha^*(\\omega)\\,c_0\\,t'}}{\\sqrt{2\\,\\pi}}\\, \\right\\}(t)\\,,$$ it follows that $${\\mathcal{M}}(t + {\\left|t'\\right|},t')\n = {\\mathcal{M}}(t,0) \\ast_t K(c_0\\,t',t)\\,.$$ Since $t\\mapsto{\\mathcal{M}}(t,0)$ and $t\\mapsto K(c_0\\,t',t)$ are causal, their convolution is also causal (cf. ",
"Item \\[item:conv\\] in the Appendix). ",
"This proves property (\\[MMM\\]) and concludes the proof.",
"\n\nAssume that the attenuation coefficient is given by $$\\alpha^*(\\omega) = {{\\mathrm{i}}\\,}\\,\\omega/c_0 \\qquad \\mbox{ for }\\qquad \\omega\\in{{\\mathbb{R}}}\\, .$$ Then $$K(r,t) = \\frac{1}{\\sqrt{2\\,\\pi}}\\,{\\mathcal{F}^{-1} \\left\\{ {\\exp \\left( -{{\\mathrm{i}}\\,}\\,\\omega\\,r/c_0 \\right)} \\right\\}}(t) = \\delta(t+r/c_0)$$ which implies together with [(\\[eq:FGG0\\])]{} and [(\\[eq:g0\\])]{} that $${\\mathcal{G}}(r,t) = \\frac{\\delta(t)}{4\\,\\pi\\,r} \\,.$$ But this function does not correspond to the intuition of an attenuated wave, which is manifested by the convolution equation (\\[eq:FGG0\\]), which should give a smooth decay of frequency components over travel distance. ",
"With this Green function ${\\mathcal{G}}$ the input impulse collapses immediately and consequently, in this case, the assumption $\\alpha^*(\\omega)\\neq{{\\mathrm{i}}\\,}\\,\\omega/c_0$ in Theorem \\[lemm02\\] reflects physical reality.",
"\n\nSolution of the Integral Equation\n=================================\n\nThe inverse problem of photoacoustics with attenuated waves reduces to solving the integral equation (\\[eq:pattrep02a\\]) for $p_0$, and to the standard photoacoustical inverse problem, which consists in calculating the initial pressure $\\rho$ in the wave equation (\\[eq:ex:ivp3d\\]) from measurements of $p_0({{\\vec{x}}},t)$ over time on a manifold surrounding the object of interest. ",
"The standard photoacoustical imaging problem is not discussed here further, but we focus on the the integral equation (\\[eq:pattrep02a\\]).",
"\n\nIn the following we investigate the ill–conditionness of the integral equation (\\[eq:pattrep02a\\]), where the kernel ${\\mathcal{M}}$ is given from the attenuation law [(\\[eq:powlaw2\\])]{} with $\\gamma \\in (1,2]$. In this case the model is causal and the parameter range $\\gamma \\in (1,2]$ is relevant for biological imaging.",
"\n\nIn order to estimate the ill–conditionness of the integral equation (\\[eq:pattrep02a\\]) it is rewritten in Fourier domain: $$\\label{eq:Mfrequency}\n {\\mathcal{F} \\left\\{ {p_{\\text{att}}}\\right\\}}({{\\vec{x}}}_0,\\omega)\n = \\int_{{\\mathbb{R}}}{\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}} (\\omega,t')\\,p_0({{\\vec{x}}}_0,t')\\,{\\mbox{d}}t'\\;.$$ After discretization the ill-conditionness of this equation is reflected by the decay rate of the singular values of the matrix ${\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}} (\\omega,t')$ at certain discrete frequencies and time instances.",
"\n\nWe consider simple test examples of attenuation coefficients $\\rho$ (as in (\\[eq:u(x)\\])), which are characteristic functions of balls with center at the origin and radiii $R$. For these examples we investigate the dependence of the ill–conditionedness of (\\[eq:Mfrequency\\]) on the radius $R$ and the location ${{\\vec{x}}}_0$. For applications in photoacoustic imaging ${{\\vec{x}}}_0$ would be the location of a detector outside of the object of interest, to be imaged. ",
"Then, by solving the integral equation (\\[eq:Mfrequency\\]) $p_0$ can be calculated, and in turn, the absorption energy $\\rho$ can be reconstructed with standard backprojection formulas. ",
"Since $${\\mbox{supp}}(p_0({{\\vec{x}}}_0,\\cdot))\n = \\left[({\\left|{{\\vec{x}}}_0\\right|}-R)/c_0,({\\left|{{\\vec{x}}}_0\\right|}+R)/c_0\\right]\\,,$$ the integral equation [(\\[eq:Mfrequency\\])]{} can be rewritten as $$\\label{eq:Mfrequency*}\n {\\mathcal{F} \\left\\{ {p_{\\text{att}}}\\right\\}}({{\\vec{x}}}_0,\\omega)\n = \\int_{(R_0-R)/c_0}^\\infty {\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}} (\\omega,t')\\,p_0({{\\vec{x}}}_0,t')\\,{\\mbox{d}}t'\\,,$$ where $R_0 = {\\left|{{\\vec{x}}}\\right|}_0$.\n\nIn the following we analyze the integral equation (\\[eq:Mfrequency\\*\\]) in terms of the two parameters $R$ and $R_0={\\left|{{\\vec{x}}}_0\\right|}$. This gives a clue on the effect of attenuation in terms of the size of the object and the distance of the location ${{\\vec{x}}}_0$ to the simple object. ",
"In order to show the effect of attenuation on the single frequencies, we make a singular value decomposition of the kernel of the integral equation (\\[eq:Mfrequency\\*\\]).",
"\n\n\\[exam:distance2\\] For small frequencies the attenuation law of castor oil, which behaves very similar to biological soft tissue, is approximately a power law with exponent $\\gamma=1.66$ and $\\hat{\\alpha}_0 \\approx 4\\cdot 10^{-2}\\,\\frac{1}{cm\\,(MHz)^\\gamma}$, i.e. $$\\alpha_{pl} (\\omega) \\approx 4\\cdot 10^{-2}\\cdot \\omega^{1.66} \\cdot cm^{-1} \\qquad \\mbox{($\\omega$ in $MHz$)}.$$ The sound speed of castor oil is $1490\\cdot\\frac{m}{s}$ at $25$ degree Celsius. ",
"In units of $cm$ and $MHz$ we have $$c_0\\approx 0.15\\cdot cm\\cdot MHz\\,.$$ Since [(\\[eq:powlaw2\\])]{} approximates the power law (cf. ",
"Figure \\[fig:comp\\]) it follows that $$\\hat{\\alpha}_0\\,{\\left|\\omega\\right|}^\\gamma \\approx\n \\frac{\\alpha_0\\,\\sin(\\frac{\\pi}{2}\\,(\\gamma-1))}{2\\,c_0\\,\\tau_0}\\,{\\left|\\tau_0\\,\\omega\\right|}^\\gamma$$ and consequently the coefficients of [(\\[eq:powlaw2\\])]{} satisfy $$\\alpha_0 \\approx \\frac{2\\,c_0\\,\\hat{\\alpha}_0}\n {\\tau_0^{(\\gamma-1)}\\,\\sin(\\frac{\\pi}{2}\\,(\\gamma-1))}\n \\approx 6\\,.$$ We note that the relaxation time is $\\tau\\approx 10^{-4}\\,\\frac{1}{MHz}$ for liquids (cf.",
" [@KinFreCopSan00]).",
"\n\nFor the calculation of the singular value decomposition of the discretized kernel of the integral equation (\\[eq:Mfrequency\\*\\]) we used a frequency range\\\n$\\omega \\in [-80,80] MHz$ and step size $\\Delta \\omega=\\frac{2\\,\\pi}{N-1}\\, MHz$ with $N=2^9$. The time interval has been set to $[0,\\frac{2\\,\\pi}{\\Delta \\omega}] MHz^{-1}$ and a step size $\\Delta t=\\frac{2\\,\\pi}{80}\\,MHz^{-1}$ was used.",
"\n\nThe upper left picture in Fig.",
" \\[fig:distance2\\] visualizes the discretized kernel of the integral equation (\\[eq:Mfrequency\\*\\]) for $R_0=R$, i.e., when ${{\\vec{x}}}_0$ is directly on the surface of the object of interest. ",
"The upper right picture shows the singular values of the discretized kernel in a logarithmic scale. ",
"Two properties of the singular values become apparent:\n\n1. ",
" For large indices the decay rate is exponential, which can be seen from the linear decay in the logarithmic scale.",
"\n\n2. ",
" Secondly, there is a range of indices, where the singular values do not decay that rapidly. ",
"As a consequence, for solving the integral equation this means that the Fourier coefficients of $p_0$ according to the first block of singular values can be determined in a stable manner.",
"\n\nFor increasing distance $L={\\left|{{\\vec{x}}}_0\\right|}-R$ of ${{\\vec{x}}}_0$ to the object the singular values of the discretization of the integral equation (\\[eq:Mfrequency\\*\\]) show a drastically more exponentially decay rate for increasing $L$ (see bottom right picture Fig.",
" \\[fig:distance2\\]). ",
"This means that if the object is further away from ${{\\vec{x}}}_0$ attenuation is more drastically, and solution of the integral equation is more unstable. ",
"We analyze the dependence of the number of largest singular values from $L$. For this purpose we denote by $n_{cut}$ the index of the singular value that is about $0.1\\%$ of the maximal singular value. ",
"For the numerical solution of (\\[eq:Mfrequency\\*\\]) it means that if we make a truncated singular valued decomposition with only $n_{cut}$ singular values, the error amplification can be bounded by a factor $1000$. The dependence of $n_{cut}$ on $L$ is shown in the lower left picture of Fig.",
" \\[fig:distance2\\]. ",
"The picture reveals that for increasing distance (from about $2cm$) only about four Fourier modes of $p_0$ are significant when a maximal error amplification of a factor $1000$ is required. ",
"This reveals that in general the solution of the integral equation (\\[eq:Mfrequency\\*\\]) is significantly ill–posed and worse if the data recording is far away from the object.",
"\n\n![",
"Case: $\\gamma=1.66$ (castor oil). ",
"The upper left and right pictures visualize the kernel ${\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}}(\\omega,t')$ and its singular values for $L:=R_0-R=0cm$. The lower right and left pictures visualize ${\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}}(\\omega,t')$ for the detector distances $L=0\\cdot cm$, $L=2\\cdot cm$ and $L=4\\cdot cm$ and the respective indices $n_{cut}$ for which the singular values are about $0.1$ per cent of the maximal singular value.[]{data-label=\"fig:distance2\"}](ExSingVal_2_00.eps){width=\"\\textwidth\"}\n\n\\[exam:distance1\\] An analogous numerical example as in Example \\[exam:distance2\\] for the case $\\gamma=1.1$ is presented in Fig.",
" \\[fig:distance1\\]. ",
"From the lower left picture of Fig.",
" \\[fig:distance1\\], we see that if the distance is about $2\\cdot cm$ from the boundary of the object, then $17$ singular values are available for the numerical estimation.",
"\n\n![",
"Case: $\\gamma=1.1$. The upper left and right pictures visualize the kernel ${\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}}(\\omega,t')$ and its singular values. ",
"The lower right and left pictures visualize ${\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}}(\\omega,t')$ for the detector distances $L=0\\cdot cm$, $L=2\\cdot cm$ and $L=4\\cdot cm$ and the respective indices $n_{cut}$ for which the singular values are about $0.1$ per cent of the maximal singular value.[]{data-label=\"fig:distance1\"}](ExSingVal_1_00.eps){width=\"\\textwidth\"}\n\n\\[exam:distance3\\] An analogous numerical example as in Example \\[exam:distance2\\] for the case $\\gamma=2$ is presented in Fig.",
" \\[fig:distance3\\]. ",
"From the lower left picture of Fig.",
" \\[fig:distance3\\], we see that if the distance is about $2\\cdot cm$ from the boundary of the object, then only $4$ singular values are available for the numerical estimation.",
"\n\n![",
"Case: $\\gamma=1.1$. The upper left and right pictures visualize the kernel ${\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}}(\\omega,t')$ and its singular values. ",
"The lower right and left pictures visualize ${\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}}(\\omega,t')$ for the detector distances $L=0\\cdot cm$, $L=2\\cdot cm$ and $L=4\\cdot cm$ and the respective indices $n_{cut}$ for which the singular values are about $0.1$ per cent of the maximal singular value.[]{data-label=\"fig:distance3\"}](ExSingVal_3_00.eps){width=\"\\textwidth\"}\n\n\\[exam:distance5\\] An analogous numerical example as in Example \\[exam:distance2\\] for the frequency power law $$\\alpha_*(\\omega) = \\alpha_0^{pl}\\cdot (-{{\\mathrm{i}}\\,}\\,\\omega)^{0.66}\n \\qquad\\quad (\\mbox{$\\alpha_0^{pl}$ as in Example~\\ref{exam:distance2}})$$ is presented in Fig.",
" \\[fig:distance5\\]. ",
"From the lower left picture of this figure, we see that if the distance is about $2\\cdot cm$ from the boundary of the object, then $77$ singular values are available for the numerical estimation. ",
"If the distance is about $4cm$, then $46$ singular values are available.",
"\n\n![",
"Case: $\\gamma=0.66$. The upper left and right pictures visualize the kernel ${\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}}(\\omega,t')$ and its singular values. ",
"The lower right and left pictures visualize ${\\mathcal{F} \\left\\{ {\\mathcal{M}}\\right\\}}(\\omega,t')$ for the detector distances $L=0\\cdot cm$, $L=2\\cdot cm$ and $L=4\\cdot cm$ and the respective indices $n_{cut}$ for which the singular values are about $0.1$ per cent of the maximal singular value.[]{data-label=\"fig:distance5\"}](ExSingVal_5_00.eps){width=\"\\textwidth\"}\n\nComparing all numerical examples shows that the larger $\\gamma$ (stronger attenuation), the more rapidly decrease the singular values.",
"\n\nAppendix: Nomenclature and Elementary Facts {#sec:app}\n===========================================\n\n[Sets:]{} $B_R$ denotes the open ball with center at $\\mathbf{0}$ and radius $R$. $S^n \\subseteq {{\\mathbb{R}}}^n$ denotes the $n$-dimensional unit sphere.",
"\n\n[Real and Complex Numbers:]{} ${{\\mathbb{C}}}$ denotes the space of complex numbers, ${{\\mathbb{R}}}$ the space of reals. ",
"For a complex number $c=a+{{\\mathrm{i}}\\,}b$ $a=\\Re{(c)}$, $b=\\Im{(c)}$ denote the real and imaginary parts, respectively.",
"\n\nFor a complex number $c$ we denote by ${\\left|c\\right|}$ the absolute value and by $\\phi \\in (-\\pi,\\pi]$ the argument. ",
"That is $$c = {\\left|c\\right|} {\\exp \\left( i \\phi \\right)}\\;.$$ As a consequence, when $w=r\\,{\\exp \\left( {{\\mathrm{i}}\\,}\\phi \\right)}$ then $$\\label{defpower}\n w^\\gamma = {\\exp \\left( \\gamma\\,\\left(\\text{log}(r)+{{\\mathrm{i}}\\,}\\phi\\right) \\right)}\\;.$$ Consequently $w^\\gamma$ has absolute value $r^\\gamma$ and the argument is $\\gamma \\phi$ modulo $2\\pi$. In this paper all power functions are defined on ${{\\mathbb{C}}}\\backslash {{\\mathbb{R}}}_-$. We note that $$\\label{propsqrt}\n w\\in{{\\mathbb{C}}}\\backslash {{\\mathbb{R}}}_- \\quad\\mbox{and}\\quad \\Re(\\sqrt{w})>0 \\qquad \\Rightarrow \\qquad\n \\Im(w)\\,\\Im(\\sqrt{w})\\geq 0 \\,.$$\n\n[Differential Operators:]{} $\\nabla$ denotes the gradient. ",
"$\\nabla \\cdot$ denotes divergence, and $\\nabla^2$ denotes the Laplacian.",
"\n\n[Product:]{} When we write $\\cdot$ between two functions, then it means a pointwise product, it can be a scaler product or if the functions are vector valued an inner product. ",
"The product between a function and a number is not explicitly stated.",
"\n\n[Composition:]{} The composition of operators ${\\mathcal{A}}$ and ${\\mathcal{B}}$ is written as ${\\mathcal{A}}{\\mathcal{B}}$.\n\n[Special functions:]{}\n\n- The *signum* function is defined by $${\\mbox{sgn}}:= {\\mbox{sgn}}({{\\vec{x}}}) := \\frac{{{\\vec{x}}}}{{\\left|{{\\vec{x}}}\\right|}}\\;.$$ In ${{\\mathbb{R}}}^3$ it satisfies $$\\label{eq:der_sgn} \\nabla \\cdot {\\mbox{sgn}}= \\frac{2}{{\\left|{{\\vec{x}}}\\right|}}\\;.$$\n\n- The *Heaviside* function $${H}:= {H}(t) := \\left\\{ \\begin{array}{rcl}\n 0 & \\text{ for } & t < 0\\\\\n 1 & \\text{ for } & t > 0\\\\\n \\end{array} \\right.$$ satisfies $${H}:= \\frac{1}{2} (1+{\\mbox{sgn}})\\;.$$\n\n- The $\\delta$-distribution is the derivative of the Heaviside function at $0$ and is denoted by $\\delta_t := \\delta_t(t)$. In our terminology $\\delta_t$ denotes a *one*-dimensional distribution. ",
"Sometimes, if the context is clear, we will omit the subscript at the $\\delta$-distributions.",
"\n\n- The three dimensional $\\delta$-distribution $\\delta_{{\\vec{x}}}$ is the tensor product of the three one-dimensional distributions $\\delta_{x_i}$, $i=1,2,3$. Moreover, $$\\label{eq:hatdelta} \\delta_{{{\\vec{x}}},t} := \\delta_{{{\\vec{x}}},t}({{\\vec{x}}},t) = \\delta_{{\\vec{x}}}\\cdot \\delta_t,$$ is a four dimensional distribution in space and time. ",
"If we do not add a subscript $\\delta$ denotes a one-dimensional $\\delta$-distribution.",
"\n\n- $\\chi_\\Omega$ denotes the characteristic set of $\\Omega$, i.e., it attains the value $1$ in $\\Omega$ and is zero else.",
"\n\n[Properties related to functions:]{} ${\\mbox{supp}}(g)$ denote the *support* of the function $g$, that is the closure of the set of points, where $g$ does not vanish.",
"\n\n[Derivative with respect to radial components:]{} We use the notation $$r:=r({{\\vec{x}}}) = {\\left|{{\\vec{x}}}\\right|},$$ and denote the derivative of a function $f$, which is only dependent on the radial component ${\\left|{{\\vec{x}}}\\right|}$, with respect to $r$ (i.e., with respect to ${\\left|{{\\vec{x}}}\\right|}$) by $\\cdot'$.\n\nLet $\\beta = \\beta(r)$, then it is also identified with the function $\\beta = \\beta ({\\left|{{\\vec{x}}}\\right|})$ and therefore $$\\nabla \\beta = \\frac{{{\\vec{x}}}}{{\\left|{{\\vec{x}}}\\right|}} \\beta'\\;.$$\n\n[Convolutions:]{} Three different types of convolutions are considered: $*_t$ and $*_\\omega$ denote *convolutions* with respect to time and frequency, respectively. ",
"Let $f$, $\\hat{f}$, $g$ and $\\hat{g}$ be functions defined on the real line with complex values. ",
"Then $$\\begin{aligned}\n &f *_t g := \\int_{{\\mathbb{R}}}f(t-t')g(t') d t' , \\quad\\quad\n &\\hat{f} *_\\omega \\hat{g}\n := \\int_{{\\mathbb{R}}}\\hat{f}(\\omega-\\omega')\\hat{g}(\\omega') d \\omega'.",
"\n\\end{aligned}$$ $*_{{{\\vec{x}}},t}$ denotes space–time convolution and is defined as follows: Let $f,g$ be functions defined on the Euclidean space ${{\\mathbb{R}}}^3$ with complex values, then $$f *_{{{\\vec{x}}},t} g := \\int_{{{\\mathbb{R}}}^3} \\int_{{\\mathbb{R}}}f({{\\vec{x}}}- {{\\vec{x}}}',t-t')g({{\\vec{x}}}',t') d{{\\vec{x}}}' d t'\\;.$$\n\n[Fourier transform:]{} For more background we refer to [@Lig64; @Tit48; @Pap62; @Yos95; @Hoe03]. ",
"All along this paper ${\\mathcal{F} \\left\\{ \\cdot \\right\\}}$ denotes the *Fourier transformation* with respect to $t$, and the *inverse Fourier transform* ${\\mathcal{F}^{-1} \\left\\{ \\cdot \\right\\}}$ is with respect to $\\omega$. In this paper we use the following definitions of the transforms: $$\\begin{aligned}\n{\\mathcal{F} \\left\\{ f \\right\\}}(\\omega) &= \\frac{1}{\\sqrt{2\\pi}} \\int_{{\\mathbb{R}}}{\\exp \\left( {{\\mathrm{i}}\\,}\\omega t \\right)} f(t) d t\\,,\\\\\n{\\mathcal{F}^{-1} \\left\\{ \\hat{f} \\right\\}}(t) &= \\frac{1}{\\sqrt{2\\pi}} \\int_{{\\mathbb{R}}}{\\exp \\left( -{{\\mathrm{i}}\\,}\\omega t \\right)} \\hat{f}(\\omega) d\\omega\\;.",
"\n\\end{aligned}$$ The Fourier transform and its inverse have the following properties:\n\n1. ",
" \\[item:derF\\] $${\\mathcal{F} \\left\\{ \\frac{\\partial }{\\partial t}f \\right\\}} (\\omega) = (-{{\\mathrm{i}}\\,}\\omega){\\mathcal{F} \\left\\{ f \\right\\}}(\\omega)\\;.$$\n\n2. ",
" $$\\begin{aligned}\n {\\mathcal{F} \\left\\{ f \\cdot g \\right\\}} & =\\frac{1}{\\sqrt{2\\pi}} {\\mathcal{F} \\left\\{ f \\right\\}} *_\\omega {\\mathcal{F} \\left\\{ g \\right\\}} \\text{ and }\\\\\n {\\mathcal{F} \\left\\{ f \\right\\}} \\cdot {\\mathcal{F} \\left\\{ g \\right\\}} &= \\frac{1}{\\sqrt{2\\pi}} {\\mathcal{F} \\left\\{ f *_t g \\right\\}},\\\\\n {\\mathcal{F}^{-1} \\left\\{ \\hat{f} \\cdot \\hat{g} \\right\\}}\n &= \\frac{1}{\\sqrt{2\\pi}}{\\mathcal{F}^{-1} \\left\\{ \\hat{f} \\right\\}} *_t {\\mathcal{F}^{-1} \\left\\{ \\hat{g} \\right\\}} \\text{ and }\\\\\n {\\mathcal{F}^{-1} \\left\\{ \\hat{f} \\right\\}} \\cdot {\\mathcal{F}^{-1} \\left\\{ \\hat{g} \\right\\}}\n &= \\frac{1}{\\sqrt{2\\pi}} {\\mathcal{F}^{-1} \\left\\{ \\hat{f} *_\\omega \\hat{g} \\right\\}}\\;.",
"\n \\end{aligned}$$\n\n3. ",
" For $a \\in {{\\mathbb{R}}}$ $${\\mathcal{F} \\left\\{ f(t-a) \\right\\}}(\\omega) = {\\exp \\left( i a \\omega \\right)} \\cdot{\\mathcal{F} \\left\\{ f(t) \\right\\}}(\\omega)$$\n\n4. ",
" \\[item:Fdelta\\] The $\\delta$-distribution at $a \\in {{\\mathbb{R}}}$ satisfies $$\\delta_t(t-a)=\\frac{1}{\\sqrt{2\\pi}}{\\mathcal{F}^{-1} \\left\\{ \\exp({{\\mathrm{i}}\\,}a \\omega) \\right\\}}(t)\\;.$$\n\n5. ",
" \\[item:even\\] Let $f$ be real and even, odd respectively, then ${\\mathcal{F} \\left\\{ f \\right\\}}$ is real and even, imaginary and odd, respectively.",
"\n\n6. ",
" \\[item:temp\\] The Fourier transformation of a tempered distribution is a tempered distribution.",
"\n\n7. ",
" \\[item:conv\\] Let $\\tau_1,\\,\\tau_2\\in {{\\mathbb{R}}}$. If $f_1$ and $f_2$ are two distributions with support in $[\\tau_1,\\infty)$ and $[\\tau_2,\\infty)$, respectively, then $f_1*f_2$ is well-defined and (cf. ",
" [@Hoe03]) $$\\label{propIM}\n \\mbox{supp} (f_1 * f_2)\n \\subseteq \\mbox{supp} (f_1) + \\mbox{supp} (f_2) \\subseteq [\\tau_1+\\tau_2,\\infty)\\,.$$\n\n[The Hilbert transform for $L^2-$functions is defined by]{} $${\\mathcal{H} \\left\\{ f \\right\\}} (t) = \\frac{1}{\\pi} \\Xint-_{{\\mathbb{R}}}\\frac{f(s)}{t-s}ds\\;,$$ where $\\Xint-_{{\\mathbb{R}}}f(s) ds$ denotes the Cauchy principal value of $\\int_{{\\mathbb{R}}}f(s) ds$.\n\nA more general definition of the Hilbert transform can be found in [@BelWoh66]. ",
"The Hilbert transform satisfies\n\n- ${\\mathcal{H} \\left\\{ {\\mathcal{F} \\left\\{ f \\right\\}} \\right\\}}(\\omega) = -i{\\mathcal{F} \\left\\{ {\\mbox{sgn}}f \\right\\}}(\\omega)$,\n\n- ${\\mathcal{H} \\left\\{ {\\mathcal{H} \\left\\{ f \\right\\}} \\right\\}} = - f$.\n\nFrom the first of these properties the Kramers-Kronig relation can be formally derived as follows. ",
"Since $f(t)$ is a causal function if and only if $ f = {H}\\cdot f$ and ${H}=(1+{\\mbox{sgn}})/2$, it follows that ${\\mathcal{F} \\left\\{ f \\right\\}} = [{\\mathcal{F} \\left\\{ f \\right\\}} +{{\\mathrm{i}}\\,}{\\mathcal{H} \\left\\{ {\\mathcal{F} \\left\\{ f \\right\\}} \\right\\}}]/2 $, which is equivalent to ${\\mathcal{F} \\left\\{ f \\right\\}} = i{\\mathcal{H} \\left\\{ {\\mathcal{F} \\left\\{ f \\right\\}} \\right\\}}$, i.e. $$\\Re({\\mathcal{F} \\left\\{ f \\right\\}}) = -\\Im({\\mathcal{H} \\left\\{ {\\mathcal{F} \\left\\{ f \\right\\}} \\right\\}}) {\\qquad \\text{and} \\qquad}\n \\Im({\\mathcal{F} \\left\\{ f \\right\\}}) = \\Re({\\mathcal{H} \\left\\{ {\\mathcal{F} \\left\\{ f \\right\\}} \\right\\}}) .$$\n\n[The inverse Laplace transform]{} of $f$ is defined by $${\\mathcal{L}^{-1} \\left\\{ f \\right\\}}(t) = \\left\\{ \\begin{array}{ccl}\n0 &\\text{ for }& t < 0,\\\\\n\\frac{1}{2 \\pi i} \\int_{\\gamma - i\\infty}^{\\gamma + {{\\mathrm{i}}\\,}\\infty} {\\exp \\left( st \\right)} f(s)ds, &\\text{ for }& t > 0,\n\\end{array}\\right.$$ where $\\gamma$ is appropriately chosen.",
"\n\nThe inverse Laplace transform satisfies (see e.g. [@Heu91]) $$\\label{eq:pr1} {\\mathcal{L}^{-1} \\left\\{ h(s-a) \\right\\}}(t) = {\\exp \\left( a t \\right)} {\\mathcal{L}^{-1} \\left\\{ h(s) \\right\\}}(t) \\text{ for all } a, t \\in {{\\mathbb{R}}}$$ and $$\\label{eq:pr2}\n{\\mathcal{L}^{-1} \\left\\{ s^{-r} \\right\\}}(t) = \\frac{H(t) t^{r-1}}{\\Gamma(r)}\\qquad (r>0)\\;.$$\n\nAcknowledgement {#acknowledgement .unnumbered}\n===============\n\nThis work has been supported by the Austrian Science Fund (FWF) within the national research network Photoacoustic Imaging in Biology and Medicine, project S10505-N20.",
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"\n\nXu, M. and Xu, Y. and Wang, L. V.: Time-Domain Reconstruction Algorithms and Numerical Simulation for Thermoacoustic Tomography in Various Geometries. [*",
"IEEE Trans. ",
"Biomed. ",
"Eng.*]{}, ",
"Vol. ",
"50, N0. ",
"9, Sept. 2003.",
"\n\nXu, Y. and Wang, L. V. and Ambartsoumian, G. and Kuchment, P.: Reconstructions in limited-view thermoacoustic tomography. [*",
"Med. ",
"Phys.*]{}, ",
"31 (4), April 2004.",
"\n\nXu, M. and Wang, L. V.: Universal back-projection algorithm for photoacoustic computed tomography. [*",
"Phys. ",
"Rev. E 71*]{}, 2005. ",
"Article ID 016706.",
"\n\nXu, M. and Wang, L. V.: Photoacoustic imaging in biomedicine. [*",
"Rev. Sci. ",
"Instruments*]{}, 77(4):1–22, 2006. ",
"Article ID 041101.",
"\n\nYosida, K.: Functional analysis. ",
"Springer-Verlag, Berlin, Heidelberg, New York 1995, 5th edition\n\nZhang, E.Z. and Laufer, J. and Beard, P.: Three-dimensional photoacoustic imaging of vascular anatomy in small animals using an optical detection system. ",
"In [*[@OraWan07]*]{}, 2007.",
"\n\nZhang, H. and Maslov, K. and Stoika, G. and Wang. ",
"V.L.: Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging. , ",
"24:848 – 851, 2006.",
"\n\n[^1]: In this paper the root of a complex number is always the one with non-negative real part.",
"\n\n[^2]: In [@NacSmiWaa90] they use the notion $c_\\infty$ for $c_0$ and $c$ for $\\tilde c_0$.\n\n[^3]: A function $f$ is an element of $L_{loc}^1({{\\mathbb{R}}})$ if it is in $L^1$ on every compact set.",
"\n\n[^4]: In Theorem 4 on p294 ff in [@DauLio02_5] the assumption that $f$ is strongly causal is expressed by $f \\in \\mathring{{\\mathcal{D}}}_+$, which is the set of distributions with support in $[0,+\\infty)$.\n\n[^5]: A function $F:{{\\mathbb{C}}}_0 \\to{{\\mathbb{C}}}$ is holomorphic in $\\mathring{{{\\mathbb{C}}}}_0$ if it is complex differentiable in $\\mathring{{{\\mathbb{C}}}}_0$. Sometimes the functions are also refered to as analytic or regular functions or conformal maps.",
"\n\n[^6]: A function $F:{{\\mathbb{C}}}_0\\to{{\\mathbb{C}}}$ which satisfies Items \\[it1\\_Lions\\],\\[it2\\_Lions\\] of Theorem \\[th:lion\\] is called *holomorphic extension* of ${\\mathcal{F} \\left\\{ f \\right\\}}$.\n\n[^7]: $\\alpha: {{\\mathbb{C}}}_0 \\to {{\\mathbb{C}}}$ extends the function $\\omega \\in {{\\mathbb{R}}}\\to \\alpha(\\omega)$ but is not an holomorphic extension.",
"\n"
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"The Ohio-Kentucky-Indiana Regional Council of Governments wants to hear from people and groups who bicycle in Cincinnati as it begins updating its Bike Route Guide map. ",
"OKI is asking for comments, ideas and questions about the current map.",
"\n\n\"It was last updated in 2010,\" said OKI Senior Planner Don Burrell. \"",
"It's fairly popular with cyclists and gets distributed through our office and a number of bike shops.\"",
"\n\nThe update will give the Cincinnati bicycle community the opportunity to comment on changes along routes, detours and safety issues. ",
"Burrell said the input will be used to help finalize the new map. ",
"He said it's being released in May for National Bike Month.",
"\n\nThe campaign runs until February 17. ",
"Here's the link for comments."
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"\n\nJudy Foote, MP for Random-Burin-St. George’s, said in a letter to Hounsell that she agrees the Government of Newfoundland and Labrador’s decision to freeze further fracking applications is a positive step.",
"\n\nIt was in early November 2013 that Natural Resources Minister Derrick Dalley declared a moratorium on fracking in the province. ",
"He announced government will not approve fracking onshore or onshore-to-offshore pending further review. ",
"Dalley said government would hold public consultations before it develops any policy for fracking.",
"\n\n“I think an independent, external review would be better suited to provide the information necessary to make informed decisions about the use of fracking because of the ongoing controversy over the practice,” Foote said in the letter.",
"\n\nShe said she cannot support the use of the process if there exists the possibility it can cause significant harm to the health of those who live in the area where it would occur, and to the environment.",
"\n\nFoote said exploration of shale gas reservoirs through hydraulic fracturing requires a significant amount of water with chemical additives and sand that could lead to contamination of water sources and habitat fragmentations.",
"\n\nShe also read of other environmental concerns resulting from air emissions, the disposal of wastewater and infrastructure requirements.",
"\n\nHounsell, the chairman of the Port au Port/Bay St. George Fracking Awareness Group, appreciated the MP’s support of the group’s position.",
"\n\n“We know she has a concern for the environment in western Newfoundland,” he said.",
"\n\nHounsell said Foote joins a rather large group that has come out in support of their position, including Liberal Leader Dwight Ball, NDP Leader Lorraine Michael and Lana Payne, Atlantic director of Unifor, the largest energy union in Canada.",
"\n\nThe Newfoundland and Labrador Federation of Labour has also approved of a moratorium on hydraulic fracturing at its convention back on Dec. 3, 2013.",
"\n\nHounsell said he also heard from Charlene Fitzgerald, president of the Newfoundland and Labrador College of Family Physicians, who has pledged support.",
"\n\n“Our group is building a large base of support for western Newfoundland in favour of having an independent, external review carried out on hydraulic fracturing,” he said."
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"\n\nEt af de to højtragende byggerier er rykket et skridt nærmere realisering, efter et bredt flertal i Teknik- og Miljøudvalget og Økonomiudvalget i Københavns Kommune har vedtaget, at der skal arbejdes videre med udviklingen af højhus-projekterne.",
"\n\nTeknik- og Miljøforvaltningen og Økonomiforvaltningnen havde ellers anbefalet en afvisning af de to projekter, da de bryder afgørende med de planer, der i øjeblikket findes for udbygningen af Nordhavn."
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"Snoring is the earliest symptom of OSA, but its potential in clinical diagnosis is not fully recognized yet. ",
"The first task in the automatic analysis of snore-related sounds (SRS) is to segment the SRS data as accurately as possible into three main classes: snoring (voiced non-silence), breathing (unvoiced non-silence) and silence. ",
"SRS data are generally contaminated with background noise. ",
"In this paper, we present classification performance of a new segmentation algorithm based on pattern recognition. ",
"We considered four features derived from SRS to classify samples of SRS into three classes. ",
"The features--number of zero crossings, energy of the signal, normalized autocorrelation coefficient at 1 ms delay and the first predictor coefficient of linear predictive coding (LPC) analysis--in combination were able to achieve a classification accuracy of 90.74% in classifying a set of test data. ",
"We also investigated the performance of the algorithm when three commonly used noise reduction (NR) techniques in speech processing--amplitude spectral subtraction (ASS), power spectral subtraction (PSS) and short time spectral amplitude (STSA) estimation--are used for noise reduction. ",
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"Note: Citations are based on reference standards. ",
"However, formatting rules can vary widely between applications and fields of interest or study. ",
"The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.",
"\n\nHidden in plain sight : how to create extraordinary products for tomorrow's customers\n\nWho are your next customers - not just the ones you are serving today but the ones you'll need three, five, ten years from now? ",
"How do you figure out what goods and services will attract them in the future? ",
"How do you figure this out ahead of your competitors? ",
"This title shows us how to see the world differently.",
"\n\nDetails\n\nCrossing state (of mind) lines --\nThe social lives of everyday objects --\nRiding the waves of the past, present, and future --\nYou are what you carry --\nCalibrating your cultural compass --\nA matter of trust --\nFinding the essence --\nThe great tradeoff --\nConclusion.",
"\n\nResponsibility:\n\nJan Chipchase and Simon Steinhardt.",
"\n\nAbstract:\n\nWho are your next customers - not just the ones you are serving today but the ones you'll need three, five, ten years from now? ",
"How do you figure out what goods and services will attract them in the future? ",
"How do you figure this out ahead of your competitors? ",
"This title shows us how to see the world differently.",
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"1. ",
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"Leave for 2 minutes to gelatin soaking. ",
"Put the saucepan on a small fire and cook for 2-3 minutes, stirring constantly, until gelatin is completely dissolved.",
"\n\n2. ",
"Remove from the heat and pour the remaining 150 ml of champagne or sparkling wine. ",
"Mix well. ",
"Transfuse liquid into molds for baking (pre-each can be laid out on the cherry cocktail or frozen) and put in the freezer for at least 1 hour.",
"\n\n3. ",
"To get jelly, dip the form in warm water and gently stroll along the wall with a knife. ",
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"\n\nA:\n\nYou can try this\n\n.spotlight {\r\n background-color: #4c5c96;\r\n position : relative;\r\n}\r\n\r\n.spotlight::before {\r\n position : absolute;\r\n content: '';\r\n background: url(\"data:image/svg+xml;utf8,<svg xmlns='http://www.w3.org/2000/svg' width='30' height='30'><circle cx='15' cy='15' r='10' /></svg>\") no-repeat;\r\n}\n\n"
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"Saw the HKTrampers website this morning and remembered that they were going to my favorite part of Hong Kong (well, favorite after Sharp Peak and Plover Cove). ",
"So, make that 3rd favorite part of Hong Kong – Wong Lung Hang waterfalls. ",
"So, carried my camera for a change and decided to take it easy (at least for the first half).",
"\n\nGetting to Wong Lung Hang is easy. ",
"It’s walkable from Tung Chung station and takes about 15 minutes from there to get to the starting point (you can see the Garmin page for the map – link below). ",
"From there, there’s a lot of hopping over boulders and you eventually get to a beautiful waterfall.",
"\n\nReady for action\n\nThe group\n\nWaterfalls\n\nThere was too much demand for real estate by the main waterfalls so decided to go up a bit and found a private pool to occupy. ",
"After chilling for a while in the pools, it was time to undo all the “take it easy” stuff and so decided to go full throttle to Mui Wo. ",
"Enthusiasm was great but as usual sense of direction was not. ",
"Could not find the trail up to Lin Fa Shaan but after going in some random direction and heading back, the trail suddenly appeared in front of my eyes! ",
"Wasn’t too easy going up to 715m from 250m but wasn’t that incredibly tough either.",
"\n\nFrom there a straight run down to Mui Wo in about 30 minutes or so concluded the hike. (",
"Dejavu from yesterday). ",
"Took a total of 1 hour 30 minutes from the waterfalls to get to Mui Wo ferry pier.",
"\n\nNow, for HK Capitalism 101.",
"\n\nICC, The Arch, The Cullinan and co.\n\nThey say success leaves clues and the clues these beautiful buildings have left is that one owner has built all the 3 of them. ",
"Sun Hung Kai Properties. ",
"ICC is already attracted all the big financial giants (Morgan Stanley, Credit Suisse, Deutsche Bank and co.) This area will probably become like Central in the next couple of years.",
"\n\nMasterpiece and First Ferry\n\nNew World Development has built the Masterpiece (nice name) and they also own the First Ferry brand. ",
"They keep advertising First Ferry services on the ferry’s TVs YET have a monopoly on that route!",
"\n\nAnd finally, here’s Ocean Terminal in TST.",
"\n\nOcean Terminal\n\nThis place is special because if you bring the word’s top 100 dirtiest, fattest pigs and have them all fart in Ocean Terminal, the interior of the building WILL STILL smell nice! ",
"That’s because this place hosts all the high end perfume/make up brands that HK has to offer.",
"\n\nSo, after a nice long hike, I don’t have to worry about how much I stink as I am walking home through Ocean Terminal…\n\n20-something Trampers showed up today for a day of waterfall exploration. ",
"There were several new faces too, including combat-ready Kevin from the UK, who had evidently paid utmost attention to the Tramper rucksack details listed on the website. ",
"He was wearing camouflage pants, had sturdy waterproof hiking boots on and several pouches containing life-saving equipments like a torch were attached to his belt.",
"\n\nWe left at 10.45 AM to embark on a 20-minute walk that led to the base of the stream. ",
"As we approached the stream, the fragrant smell of fresh stream water hit our noses and the sound of the water hitting against the boulders intensified.",
"\n\nShoes and socks came off and many of us immersed our bare feet in the purifying stream water. ",
"We were all like children in nature’s playground as we hopped from one boulder onto another in order to reach the Wong Lung Hang waterfalls. ",
"The water current was very strong because of the previous day rains and there were several challenging boulder hops on the course to the waterfalls. ",
"While the gentlemen offered their helping hands to those who needed, Francis seized on this opportunity to help a beautiful lady Tramper by giving her a Stream Hiking 101 lesson. ",
"In fact, we could hear the conversation. ",
"It went something like this:\n\nAfter 2 minutes, I could hear her say “But we should have gone THAT way!”",
"\n\nAbout one grueling hour later, we reached the mesmerizing waterfalls. ",
"Many of us simply couldn’t resist a dip in the enticing water. ",
"At this point Francis approached me with an unusual request. ",
"He wanted me to buy some very unique cloth made of jute from my upcoming trip to India. ",
"He came to me bringing a cigarette lighter and a piece of cloth. ",
"That conversation went something like this.",
"\n\nFrancis: You see, this is a genuine cloth made of jute. ",
"Can you observe the natural color? ",
"Feel the cloth. (",
"He then brings another cloth for comparison). ",
"Now, this is artificial. ",
"If I burn it, you’ll notice that the strands turn black and the feeling will be of burnt plastic. ",
"That’s why it is artificial.",
"\n\nArianna, who was sitting 2 rocks away, came closer to me and confessed her curiosity. ",
"She felt the specimen cloth too.",
"\n\n“Is this guy in the fashion industry?” ",
"she asked in amazement.",
"\n\nAs I burst into laughter, I asked her to look at his green shorts, his unusual hat and rethink her question. ",
"She then retorted “You know you can get a lot of strange guys in the fashion industry”.",
"\n\nA refreshing swim and a couple of group photos later, Dominic and I were off to high speed yet exhausting finish."
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"Q:\n\nSymbolically Minimizing Expression with Multiple Piecewise Functions\n\nFollowing up my previous question about ArgMin, I now face a situation where the solution to an ArgMin call, where the output is rewritten / partly solved.",
"\nThe code leading to the problem is very long and messy and I hope that someone has encountered this before and can tell me under what circumstances this happens in general without needing the specific example?",
"\nUpdate 1\nAs requested, some sample code. ",
"I do not know of a shorter, more synthesized version of the problem, unfortunately.",
"\nRemove[\"Global`*\"]\n$Assumptions = _Symbol \\[Element] Reals;\nf[x_, u_] := x + u\nv[x_, w_] := (x - w)^2\nu[x_, b1_, b2_, f_, v_] := \n Module[{x1}, \n ArgMin[(x1 = f[x, uMin]; v[x1, b1] + v[x1, b2]), uMin, Reals]]\nuConstrained[x_, b1_, b2_, f_, v_] := \n Module[{x1}, \n ArgMin[{(x1 = f[x, uMin]; v[x1, b1] + v[x1, b2]), -1 <= uMin <= 1}, \n uMin, Reals]]\np[x_, b1_, b2_, f_, u_, v_] := v[f[x, u[x, b1, b2, f, v]], b2]\nc[b11_, b12_, f_, u_, v_, p_, up_] := Module[{u0, x1, u1, x2}, (\n u0 = FullSimplify[u[x0, b11, b21, f, v]];\n x1 = FullSimplify[f[x0, u0]];\n u1 = FullSimplify[u[x1, b12, b22, f, v]];\n x2 = FullSimplify[f[x1, u1]];\n FullSimplify[v[x1, w1]] + \n FullSimplify[p[x0, b11, b21, f, up, v]] + \n FullSimplify[v[x2, w2]] + \n FullSimplify[p[x1, b12, b22, f, up, v]])]\nArgMin[FullySimplify[c[b11, b12, f, uConstrained, v, p, u]], {b11, b12}, Reals]\n\nA:\n\nEdit, taking advantage of improvements to the code in the question and using PiecewiseExpand in the answer.",
"\nThe quantity to be minimized, c[b1, b2, f, uConstrained, v, p, u], contains two distinct Piecewise functions, one with three regions in {x0, b21, b22, b1, b2} space, and the other with five regions. ",
" ({w1, w2} do not appear in the range definitions.) ",
" It can be converted into a single Piecewise function of ten expressions and regions, to each of which ArgMin can be applied in turn. ",
" Then, finding the minimum of the ten minimums would, in principle, give the desired result.",
"\nonepw = PiecewiseExpand[c[b1, b2, f, uConstrained, v, p, u]] // FullSimplify\n\n(* Piecewise[\n {{1/2 (b1^2 + b2^2 + b21^2 + b22^2) - (b1 + b21) w1 + w1^2 - (b2 + b22) w2 + w2^2, \n -2 < b1 - b2 + b21 - b22 <= 2 && -2 < b1 + b21 - 2 x0 <= 2}, \n {1/4 ((b1 - b21)^2 + (b2 - b22)^2 + (b1 + b21 - 2 w1)^2 + (b1 + b21 - 2(1 + w2))^2), \n -2 < b1 + b21 - 2 x0 <= 2 && ((2 + b1 + b21 > b2 + b22 && 4 + b2 + b22 < 2 x0) || \n b1 + b21 > 2 + b2 + b22)}, \n {1/4 ((b1 - b21)^2 + (b2 - b22)^2 + (b1 + b21 - 2 w1)^2 + (2 + b1 + b21 - 2 w2)^2), \n 2 + b1 + b21 <= b2 + b22 && -2 < b1 + b21 - 2 x0 <= 2}, \n {1/4 ((b1 - b21)^2 + (b2 - b22)^2 + (b2 + b22 - 2 w2)^2 + 4 (1 - w1 + x0)^2), \n b1 + b21 > 2 + 2 x0 && 0 < b2 + b22 - 2 x0 <= 4}, \n {1/4 (b1 - b21)^2 + 1/4 (b2 - b22)^2 + (w2 - x0)^2 + (1 - w1 + x0)^2, \n b1 + b21 > 2 + 2 x0 && b2 + b22 <= 2 x0}, \n {1/4 (b1 - b21)^2 + 1/4 (b2 - b22)^2 + (1 - w1 + x0)^2 + (2 - w2 + x0)^2, \n b1 + b21 > 2 + 2 x0 && b2 + b22 > 4 + 2 x0}, \n {1/4 ((b1 - b21)^2 + (b2 - b22)^2 + (b2 + b22 - 2 w2)^2 + 4 (1 + w1 - x0)^2), \n (-4 < b2 + b22 - 2 x0 <= 0 && 2 + b1 + b21 <= 2 x0) || (0 < b2 + b22 - 2 x0 <= 4 && \n b1 + b21 > 2 + 2 x0) || (-2 < b1 + b21 - 2 x0 <= 2 && \n -2 < b1 - b2 + b21 - b22 <= 2)}, \n {1/4 (b1 - b21)^2 + 1/4 (b2 - b22)^2 + (1 + w1 - x0)^2 + (w2 - x0)^2, \n (-2 < b1 + b21 - 2 x0 <= 2 && 2 + b1 + b21 <= b2 + b22) || (b1 + b21 > 2 + 2 x0 && \n b2 + b22 <= 2 x0) || (2 + b1 + b21 <= 2 x0 && b2 + b22 > 2 x0)}, \n {1/4 (b1 - b21)^2 + 1/4 (b2 - b22)^2 + (1 + w1 - x0)^2 + (2 + w2 - x0)^2, \n (-2 < b1 + b21 - 2 x0 <= 2 && ((2 + b1 + b21 > b2 + b22 && 4 + b2 + b22 < 2 x0) || \n b1 + b21 > 2 + b2 + b22)) || (2 + b1 + b21 <= 2 x0 && 4 + b2 + b22 <= 2 x0) || \n (b1 + b21 > 2 + 2 x0 && b2 + b22 <= 2 x0)}},\n 1/4 ((b1 - b21)^2 + (b2 - b22)^2 + (b2 + b22 - 2 (2 + w2))^2 + 4 (1 + w1 - x0)^2)] *)\n\nUnfortunately, minimizing the first of these ten expressions,\nArgMin[onepw[[1, 1]], {b1, b2}]\n\nproduces a Piecewise answer with a LeafCount of 830699 (in about 15 minutes), and Simplify runs for hours (twenty-one before I terminated that computation) without returning a result. ",
" Presumably, minimizing the other nine expressions produces similarly enormous results. ",
" The reason for the enormous results returned by ArgMin is that the regions over which any of the ten expressions is valid are complex shapes in the five-dimensional space of parameters and solutions, and those shapes change with the values of the parameters. ",
" So, there are an enormous number of case to be considered.",
"\nA more practical approach is to define the function,\nsol[x0t_, w1t_, w2t_, b21t_, b22t_] := \n ArgMin[Simplify[c[b1, b2, f, uConstrained, v, p, u], \n x0 == x0t && w1 == w1t && w2 == w2t && b21 == b21t && b22 == b22t], \n {b1, b2}, Reals]\n\nwhich usually can produce the answer for a given set of parameters in several seconds. ",
" For instance,\nsol[5/2, E, Pi, -7, 1.11]\n(* {-7., ",
"3.14159} *)\n\nor\nsol[19, 70, -71, 86, 19]\n(* {86, 19} *)\n\nInterestingly,\nCount[Table[param = RandomInteger[{-300, 300}, 5]; \n param[[4 ;; 5]] == sol @@ param, 100], True]\n\nsuggests that {b1, b2} == {b21, b22} is the answer roughly 90% of the time.",
"\n\n"
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"Dennis Martínez National Stadium\n\nDennis Martínez National Stadium () is located in Managua, Nicaragua. ",
"It has a capacity of 15,000 and it was named after former MLB player Dennis Martínez.",
"\n\nThe original stadium was built in 1948 and is Nicaragua's national stadium. ",
"It is used mainly for baseball but also serves as a venue for concerts, boxing, football, religious events and had a capacity of 30,100 people. ",
"It is the home stadium of Indios del Bóer baseball team and Deportivo Walter Ferretti football team. ",
"Inside the stadium is a hall of fame near the entrance showing medals, cups, photos and memories of Nicaraguan players. ",
"There is also a gym located inside.",
"\n\nRenaming\nThe stadium has been renamed three times. ",
"Built in 1948, it was originally named \"El Estadio Nacional\" (The National Stadium). ",
" After the 1972 earthquake that struck Managua and destroyed 90% of the city, it was rebuilt and renamed after Anastasio Somoza García. ",
"In 1979, after the Sandinistas overthrew the Somoza dynasty, it was renamed in honor of Rigoberto López Pérez, the man who, in 1956, assassinated Anastasio Somoza García. ",
" On November 20, 1998, the 50th anniversary of the founding of the stadium, then-President Arnoldo Alemán issued a decree changing the stadium's name to Estadio Nacional Dennis Martínez. ",
" Dennis Martínez, born in 1955 (several years after this stadium had opened), was Major League Baseball's first Nicaraguan-born player.",
"\n\nNew Dennis Martinez National Stadium\nThe new Dennis Martinez National Stadium has been under construction since 2016 and was completed in October, 2017. ",
"The first sport event in the stadium was the Baseball series of three games between Nicaragua & Chinese Taipei in October 20-22, 2017. ",
"It will play host to the Central American Games, scheduled to take place December 3 to December 17, 2017. ",
" The stadium, located near the Tiscapa lagoon, is approximately 3 kilometers southeast of the original stadium. ",
" It was designed by Dynamica and will seat 15,000, with the capability to expand to 20,000. ",
" In addition, the berm field will accommodate 4,000 spectators. ",
" The field will meet Major League Baseball specifications and could potentially host a Major League game. ",
"Lately and sadly this stadium have been used by dictator Daniel Ortega and the national police as a military center to kill students from universities who protests again the dictator Ortega.",
"\n\nSee also\nNicaragua\nCulture of Nicaragua\n\nReferences\n\nExternal links\nInstituto Nicaragüense de Juventud y Deporte\nWorld Stadiums page\nFussballtempel.net - Photo gallery\n\nCategory:Athletics (track and field) venues in Nicaragua\nCategory:Baseball venues in Nicaragua\nCategory:Football venues in Managua\nNicaragua\nCategory:Sports venues completed in 1948\nCategory:Buildings and structures in Managua\nCategory:Tourist attractions in Managua"
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[
"util-deprecate\n==============\n### The Node.js `util.deprecate()` function with browser support\n\nIn Node.js, this module simply re-exports the `util.deprecate()` function.",
"\n\nIn the web browser (i.e. via browserify), a browser-specific implementation\nof the `util.deprecate()` function is used.",
"\n\n\n## API\n\nA `deprecate()` function is the only thing exposed by this module.",
"\n\n``` javascript\n// setup:\nexports.foo = deprecate(foo, 'foo() is deprecated, use bar() instead');\n\n\n// users see:\nfoo();\n// foo() is deprecated, use bar() instead\nfoo();\nfoo();\n```\n\n\n## License\n\n(The MIT License)\n\nCopyright (c) 2014 Nathan Rajlich <nathan@tootallnate.net>\n\nPermission is hereby granted, free of charge, to any person\nobtaining a copy of this software and associated documentation\nfiles (the \"Software\"), to deal in the Software without\nrestriction, including without limitation the rights to use,\ncopy, modify, merge, publish, distribute, sublicense, and/or sell\ncopies of the Software, and to permit persons to whom the\nSoftware is furnished to do so, subject to the following\nconditions:\n\nThe above copyright notice and this permission notice shall be\nincluded in all copies or substantial portions of the Software.",
"\n\nTHE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND,\nEXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES\nOF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND\nNONINFRINGEMENT. ",
"IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT\nHOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,\nWHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\nFROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR\nOTHER DEALINGS IN THE SOFTWARE.",
"\n"
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"What Do Customers Buy Together With This Item?",
"\n\nThe Ugoos UM3 TV Box has an Android 4.4 + Ubuntu operating system, a RK3288 28nm Cortex-A17 Quad Core CPU, 2GB of RAM as well as 8GB of Internal Memory.",
"\n\nUgoos UM3 TV Box\n\nThere are so many Android TV boxes on the market that it has become difficult to differentiate, so Ugoos has decided to support not only Android 4.4 like all their competitors, but also Ubuntu, and they’ve provided an alpha release of dual boot Android / Ubuntu images for Ugoos UM3 models.",
"\n\nThis TV box has Bluetooth connectivity so you can quickly connect with other devices. ",
"There is dual Wi-Fi band plus this TV box supports DLNA amd Miracast so you can make a strong connection between compatible devices. ",
"Ugoos has made sure this TV box can also support 2.4GHz wireless air mouse or remote controllers.",
"\n\nThere are several ports on the TV box including a micro SD card slot that supports cards of up to 32GB, an AV out, and two USB ports to allow for many diverse ways to connect and view media.",
"\n\nBoasting a RK3288 28nm Cortex-A17 Quad Core processor and a dedicated Quad-Core Mali-T764 GPU, this TV box will stream all video content perfectly and even supports various media files including MPEG-4, H.263, H.264, AVS, VC-1, Sorenson Spark and MVC formats. ",
"Backing up the 1.8GHz CPU is 2GB of RAM to ensure there is no lagging in the performance when streaming videos or engaging in apps. ",
"Transform your television today with the incredible UM3 by Android TV box specialists, Ugoos.",
"\n\nUbuntu + Android 4.4 OS\n\nWhen you turn on this TV box, you will enter the Ubuntu operating system, from there you can decide whether to stay with the Ubuntu system or switch to the Android 4.4 operating system. ",
"Ubuntu is a Debian-based Linux operating system and is one of the most secured operating system and defines the highest level of security as compared to other operating system. ",
"A major advantage is that Ubuntu is an open source platform.",
"\n\nAndroid 4.4 is a super popular operating system that has been given the nickname KitKat and has been designed to be a huge improvement on previous versions. ",
"With a faster performance and being more efficient when performing multiple tasks, Android really benefits this quad core TV box\n\nThis product is in stock now, we guarantee next working day shipping and offer a 12 months warranty on all our products. ",
"Brought to you by the leader in electronic devices and wholesale Android TV boxes, Chinavasion.",
"\n\nWeight/dimension is for the main item of this boxed product, to help you compare product sizes before buying: please do not base your shipping calculations on this price - shipping prices depend on your cart contents, shipping destination, and shipping method: please use the checkout to select options and preview shipping price for your total order."
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[
"#\n# DEPRECATED: implementation for ffi.verify()\n#\nimport sys, os, binascii, shutil, io\nfrom . ",
"import __version_verifier_modules__\nfrom . ",
"import ffiplatform\nfrom .error import VerificationError\n\nif sys.version_info >= (3, 3):\n import importlib.machinery\n def _extension_suffixes():\n return importlib.machinery.",
"EXTENSION_SUFFIXES[:]\nelse:\n import imp\n def _extension_suffixes():\n return [suffix for suffix, _, type in imp.get_suffixes()\n if type == imp.",
"C_EXTENSION]\n\n\nif sys.version_info >= (3,):\n NativeIO = io.",
"StringIO\nelse:\n class NativeIO(io.",
"BytesIO):\n def write(self, s):\n if isinstance(s, unicode):\n s = s.encode('ascii')\n super(NativeIO, self).write(s)\n\n\nclass Verifier(object):\n\n def __init__(self, ffi, preamble, tmpdir=None, modulename=None,\n ext_package=None, tag='', force_generic_engine=False,\n source_extension='.c', flags=None, relative_to=None, **kwds):\n if ffi._parser._uses_new_feature:\n raise VerificationError(\n \"feature not supported with ffi.verify(), but only \"\n \"with ffi.set_source(): %s\" % (ffi._parser._uses_new_feature,))\n self.ffi = ffi\n self.preamble = preamble\n if not modulename:\n flattened_kwds = ffiplatform.flatten(kwds)\n vengine_class = _locate_engine_class(ffi, force_generic_engine)\n self._vengine = vengine_class(self)\n self._vengine.patch_extension_kwds(kwds)\n self.flags = flags\n self.kwds = self.make_relative_to(kwds, relative_to)\n #\n if modulename:\n if tag:\n raise TypeError(\"can't specify both 'modulename' and 'tag'\")\n else:\n key = '\\x00'.join([sys.version[:3], __version_verifier_modules__,\n preamble, flattened_kwds] +\n ffi._cdefsources)\n if sys.version_info >= (3,):\n key = key.encode('utf-8')\n k1 = hex(binascii.crc32(key[0::2]) & 0xffffffff)\n k1 = k1.lstrip('0x').rstrip('L')\n k2 = hex(binascii.crc32(key[1::2]) & 0xffffffff)\n k2 = k2.lstrip('0').rstrip('L')\n modulename = '_cffi_%s_%s%s%s' % (tag, self._vengine._class_key,\n k1, k2)\n suffix = _get_so_suffixes()[0]\n self.tmpdir = tmpdir or _caller_dir_pycache()\n self.sourcefilename = os.path.join(self.tmpdir, modulename + source_extension)\n self.modulefilename = os.path.join(self.tmpdir, modulename + suffix)\n self.ext_package = ext_package\n self._has_source = False\n self._has_module = False\n\n def write_source(self, file=None):\n \"\"\"Write the C source code. ",
" It is produced in 'self.sourcefilename',\n which can be tweaked beforehand.\"\"\"",
"\n with self.ffi._lock:\n if self._has_source and file is None:\n raise VerificationError(\n \"source code already written\")\n self._write_source(file)\n\n def compile_module(self):\n \"\"\"Write the C source code (if not done already) and compile it.",
"\n This produces a dynamic link library in 'self.modulefilename'.\"\"\"",
"\n with self.ffi._lock:\n if self._has_module:\n raise VerificationError(\"module already compiled\")\n if not self._has_source:\n self._write_source()\n self._compile_module()\n\n def load_library(self):\n \"\"\"Get a C module from this Verifier instance.",
"\n Returns an instance of a FFILibrary class that behaves like the\n objects returned by ffi.dlopen(), but that delegates all\n operations to the C module. ",
" If necessary, the C code is written\n and compiled first.",
"\n \"\"\"\n with self.ffi._lock:\n if not self._has_module:\n self._locate_module()\n if not self._has_module:\n if not self._has_source:\n self._write_source()\n self._compile_module()\n return self._load_library()\n\n def get_module_name(self):\n basename = os.path.basename(self.modulefilename)\n # kill both the .so extension and the other .",
"'s, as introduced\n # by Python 3: 'basename.cpython-33m.so'\n basename = basename.split('.', ",
"1)[0]\n # and the _d added in Python 2 debug builds --- but try to be\n # conservative and not kill a legitimate _d\n if basename.endswith('_d') and hasattr(sys, 'gettotalrefcount'):\n basename = basename[:-2]\n return basename\n\n def get_extension(self):\n ffiplatform._hack_at_distutils() # backward compatibility hack\n if not self._has_source:\n with self.ffi._lock:\n if not self._has_source:\n self._write_source()\n sourcename = ffiplatform.maybe_relative_path(self.sourcefilename)\n modname = self.get_module_name()\n return ffiplatform.get_extension(sourcename, modname, **self.kwds)\n\n def generates_python_module(self):\n return self._vengine._gen_python_module\n\n def make_relative_to(self, kwds, relative_to):\n if relative_to and os.path.dirname(relative_to):\n dirname = os.path.dirname(relative_to)\n kwds = kwds.copy()\n for key in ffiplatform.",
"LIST_OF_FILE_NAMES:\n if key in kwds:\n lst = kwds[key]\n if not isinstance(lst, (list, tuple)):\n raise TypeError(\"keyword '%s' should be a list or tuple\"\n % (key,))\n lst = [os.path.join(dirname, fn) for fn in lst]\n kwds[key] = lst\n return kwds\n\n # ----------\n\n def _locate_module(self):\n if not os.path.isfile(self.modulefilename):\n if self.ext_package:\n try:\n pkg = __import__(self.ext_package, None, None, ['__doc__'])\n except ImportError:\n return # cannot import the package itself, give up\n # (e.g. it might be called differently before installation)\n path = pkg.__path__\n else:\n path = None\n filename = self._vengine.find_module(self.get_module_name(), path,\n _get_so_suffixes())\n if filename is None:\n return\n self.modulefilename = filename\n self._vengine.collect_types()\n self._has_module = True\n\n def _write_source_to(self, file):\n self._vengine._f = file\n try:\n self._vengine.write_source_to_f()\n finally:\n del self._vengine._f\n\n def _write_source(self, file=None):\n if file is not None:\n self._write_source_to(file)\n else:\n # Write our source file to an in memory file.",
"\n f = NativeIO()\n self._write_source_to(f)\n source_data = f.getvalue()\n\n # Determine if this matches the current file\n if os.path.exists(self.sourcefilename):\n with open(self.sourcefilename, \"r\") as fp:\n needs_written = not (fp.read() == source_data)\n else:\n needs_written = True\n\n # Actually write the file out if it doesn't match\n if needs_written:\n _ensure_dir(self.sourcefilename)\n with open(self.sourcefilename, \"w\") as fp:\n fp.write(source_data)\n\n # Set this flag\n self._has_source = True\n\n def _compile_module(self):\n # compile this C source\n tmpdir = os.path.dirname(self.sourcefilename)\n outputfilename = ffiplatform.compile(tmpdir, self.get_extension())\n try:\n same = ffiplatform.samefile(outputfilename, self.modulefilename)\n except OSError:\n same = False\n if not same:\n _ensure_dir(self.modulefilename)\n shutil.move(outputfilename, self.modulefilename)\n self._has_module = True\n\n def _load_library(self):\n assert self._has_module\n if self.flags is not None:\n return self._vengine.load_library(self.flags)\n else:\n return self._vengine.load_library()\n\n# ____________________________________________________________\n\n_FORCE_GENERIC_ENGINE = False # for tests\n\ndef _locate_engine_class(ffi, force_generic_engine):\n if _FORCE_GENERIC_ENGINE:\n force_generic_engine = True\n if not force_generic_engine:\n if '__pypy__' in sys.builtin_module_names:\n force_generic_engine = True\n else:\n try:\n import _cffi_backend\n except ImportError:\n _cffi_backend = '?'",
"\n if ffi._backend is not _cffi_backend:\n force_generic_engine = True\n if force_generic_engine:\n from . ",
"import vengine_gen\n return vengine_gen.",
"VGenericEngine\n else:\n from . ",
"import vengine_cpy\n return vengine_cpy.",
"VCPythonEngine\n\n# ____________________________________________________________\n\n_TMPDIR = None\n\ndef _caller_dir_pycache():\n if _TMPDIR:\n return _TMPDIR\n result = os.environ.get('CFFI_TMPDIR')\n if result:\n return result\n filename = sys._getframe(2).f_code.co_filename\n return os.path.abspath(os.path.join(os.path.dirname(filename),\n '__pycache__'))\n\ndef set_tmpdir(dirname):\n \"\"\"Set the temporary directory to use instead of __pycache__.\"\"\"",
"\n global _TMPDIR\n _TMPDIR = dirname\n\ndef cleanup_tmpdir(tmpdir=None, keep_so=False):\n \"\"\"Clean up the temporary directory by removing all files in it\n called `_cffi_*.{c,so}` as well as the `build` subdirectory.\"\"\"",
"\n tmpdir = tmpdir or _caller_dir_pycache()\n try:\n filelist = os.listdir(tmpdir)\n except OSError:\n return\n if keep_so:\n suffix = '.c' # only remove .c files\n else:\n suffix = _get_so_suffixes()[0].lower()\n for fn in filelist:\n if fn.lower().startswith('_cffi_') and (\n fn.lower().endswith(suffix) or fn.lower().endswith('.c')):\n try:\n os.unlink(os.path.join(tmpdir, fn))\n except OSError:\n pass\n clean_dir = [os.path.join(tmpdir, 'build')]\n for dir in clean_dir:\n try:\n for fn in os.listdir(dir):\n fn = os.path.join(dir, fn)\n if os.path.isdir(fn):\n clean_dir.append(fn)\n else:\n os.unlink(fn)\n except OSError:\n pass\n\ndef _get_so_suffixes():\n suffixes = _extension_suffixes()\n if not suffixes:\n # bah, no C_EXTENSION available. ",
" Occurs on pypy without cpyext\n if sys.platform == 'win32':\n suffixes = [\".pyd\"]\n else:\n suffixes = [\".so\"]\n\n return suffixes\n\ndef _ensure_dir(filename):\n dirname = os.path.dirname(filename)\n if dirname and not os.path.isdir(dirname):\n os.makedirs(dirname)\n"
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"1. ",
"Field of the Invention\nThe present invention relates to a connector capable of connecting, disconnecting or switching an electric signal.",
"\n2. ",
"Description of the Background Art\nRecent semiconductor devices are rapidly integrated, to result in increasing importance of an inspection step for semiconductor devices. ",
"Referring to FIG. ",
"7 showing a conventional inspection step, several stages of boards are arranged between a semiconductor device 127 to be inspected and a test head 121 provided with terminals of a tester. ",
"First, a motherboard 122 is arranged immediately above the test head 121. ",
"This motherboard 122 is provided with contacts arranged in response to the arrangement of the terminals on the test head 121, and remains unchanged regardless of the type of the semiconductor device 127 to be inspected. ",
"A female contact terminal (not shown) is provided on an upper portion of the motherboard 122, to be connected with a male contact terminal (not shown) provided on a lower portion of a performance board 124 located above the same by a cable 123.",
"\nThe performance board 124 is formed by a plurality of boards. ",
"Connectors 110 are arranged between the boards forming the performance board 124. ",
"A contactor (test board) 125 is arranged on the performance board 124. ",
"Terminals 126 of the semiconductor device 127 to be inspected are directly connected to the test board 125. ",
"In general, a large number of semiconductor chips 127 are inspected at the same time. ",
"Assuming that each semiconductor chip 127 is formed with tens of terminals, therefore, the connectors 110 arranged between the boards forming the performance board 124 are provided with thousands of terminals, in order to simultaneously inspect hundreds of semiconductor chips 127.",
"\nThe performance board 124 or a socket board is provided between the motherboard 122 and the semiconductor device 127 to be inspected, in order to attain connection with flexibility in response to the type of the semiconductor device 127 to be inspected and the performance to be inspected.",
"\nThe present invention is directed to the connectors 110 employed for connecting the boards forming the aforementioned performance board 124 with each other, for example. ",
"The connectors 110 are provided with thousands of terminals as hereinabove described, and hence particular notice must be taken in order to reliably connect the terminals.",
"\nFIGS. ",
"8 and 9 show the structure of each of the conventional connectors 110 employed for connecting the aforementioned boards forming the performance board 124 with each other. ",
"FIG. ",
"9 is a sectional view taken along the line IXxe2x80x94IX in FIG. ",
"8. ",
"The structure of the connector 110 is now described with reference to connection. ",
"First, a male connector 110a and a female connector 110b are mated with each other so that each male connector terminal 102 fixed to a male connector body 101 is located between each pair of female connector terminals 105 fixed to a female connector body. ",
"Then, connector terminal switching mechanism parts 107 press the female connector terminals 105 against the male connector terminal 102 for bringing the male and female connectors 110a and 110b into contact with each other, thereby attaining electrical contact.",
"\nThen, a push-up mechanism part 106 moves up to push up the male connector body 101. ",
"Thus, the male connector terminal 102 moves upward in the state frictionally in contact with the female connector terminals 105. ",
"The male and female connector terminals 102 and 105 must essentially be frictionally connected with each other in order to stably maintain electrical connection in an excellent state in a connector having a large number of pins, to which the present invention is directed. ",
"Such frictional connection between the connector terminals 102 and 105 is referred to as wiping operation or simply as wiping. ",
"A mechanism for this wiping operation is provided around the aforementioned connection part. ",
"Thus, a signal line connected to the male connector 110a can be electrically connected with a signal line connected to the female connector 110b. ",
"\nIn the aforementioned connector 110, the length of the connection part formed by the male and female connectors 110a and 110b is varied in wiping due to reciprocation of the male connector 110a. ",
"Therefore, the wiping mechanism provided around the aforementioned connection part must be so formed that the male connector 110a is reciprocative to vary the distance between the same and the female connector 110b. ",
"Consequently, the wiping mechanism provided around the connection part and other mechanism parts are complicated. ",
"Considering fluctuation of the width of the connector 110, a driving source such as a motor necessary for the wiping operation and a gear mechanism for transmitting the driving force must be arranged in an extremely complicated structure. ",
"In other words, the gear mechanism is complicated and the number of components for transmitting the driving force is increased. ",
"If the aforementioned reciprocation is unallowable, further, the type of the employed connector 110 must be limited.",
"\nIn the aforementioned conventional connector 110, the signal lines connected to the male and female connector terminals 102 and 105 respectively are fixed in connectional association. ",
"In other words, the male and female connector terminals 102 and 105 must be connected or disconnected with or from each other while keeping fixed connectional association. ",
"Therefore, the aforementioned connectional association cannot be changed by mechanically operating the connector 110, for example.",
"\nA first object of the present invention is to provide a connector enabling wiping operation without changing the distance between a male connector and a female connector.",
"\nA second object of the present invention is to provide a connector capable of changing connectional association between a male connector terminal and a female connector terminal by mechanically operating the connector.",
"\nA connector according to a first aspect of the present invention comprises a male connector having a male connector body and a male connector terminal held by the male connector body, a female connector having a female connector terminal electrically connected with the male connector terminal and a female connector body holding the female connector terminal, a relay connector terminal movably mounted on either the male connector body or the female connector body, and a drive mechanism part bringing the relay connector terminal into contact with both of the male connector terminal and the female connector terminal and performing rubbing.",
"\nAccording to this structure, the distance between the male and female connectors may not be changed in wiping operation. ",
"Therefore, mechanism parts around the connector can be simplified. ",
"Further, height restriction against the used connector can be loosened. ",
"Each of the aforementioned drive mechanism part (wiping mechanism part) and a switching mechanism part described later is formed by a motor arranged around the connector and gears transmitting the driving force thereof. ",
"If the distance between the male and female connectors is varied, therefore, the arrangement of the aforementioned mechanisms is extremely complicated. ",
"Such complication of the mechanism parts around the connector can be relaxed due to the aforementioned structure. ",
"Consequently, components themselves can be simplified and the number of the components can be reduced. ",
"Further, the aforementioned mechanism parts can be improved in reliability and durability.",
"\nA connector according to a second aspect of the present invention comprises a male connector having a male connector body and at least two male connector terminals held by the male connector body and arranged along a prescribed direction and a female connector having at least two female connector terminals arranged along the same direction as the male connector terminals and a female connector body holding at least two female connector terminals. ",
"The connector also comprises a switching mechanism part inhibiting the male connector terminals and the female connector terminals from coming into electrical contact with each other in a first state and individually bringing the female connector terminals and the male connector terminals into electrical contact with each other when shifting from the first state to a second state. ",
"The connector further comprises a connection switching mechanism part rubbing either the male connector or the female connector with respect to the counterpart along the direction of arrangement of the connector terminals when the switching mechanism part is in the first state.",
"\nAccording to this structure, the male or female connector terminals can be displaced along the direction of arrangement thereof with no hindrance. ",
"Thus, the connector, i.e., a hard component, can switch the connectional relation. ",
"The connection switching mechanism part is formed by a mechanism similar to the aforementioned rubbing mechanism part."
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"Characterizing the learning curve of MRI-US fusion prostate biopsies.",
"\nMRI-US fusion prostate biopsies are becoming a common procedure to diagnose prostate cancer. ",
"There is a paucity of information regarding the learning curve for fusion biopsies. ",
"We aim to study the amount of experience needed to be both accurate and time-efficient in this procedure. ",
"We prospectively collected data on all MRI-US fusion biopsies performed from April 2014 to August 2017. ",
"We used two parameters to define the learning curve. ",
"Process Measurement (efficiency) was measured by time from the beginning of anesthesia to end of procedure. ",
"Outcome Measurement (accuracy) was measured by cancer detection rate for PI-RAD 3 lesions. ",
"The end of the learning curve was defined graphically and mathematically. ",
"We performed a separate analysis for transrectal and transperineal biopsies. ",
"We completed 779 fusion biopsies (523 transrectal, 256 transperineal). ",
"Patients median age was 66 years (IQR 61-70) and median PSA 6.95 ng/ml (IQR 4.2-10.6). ",
"Prostate cancer was diagnosed in 385 (49%). ",
"Process Measurement-Procedure time decreased from 45 min in the first transrectal fusion biopsy to 15 min after 109 biopsies and remained stable (p < 0.0001). ",
"Time decreased from 55 min in the first transperineal biopsy to 18 min after 124 biopsies (p < 0.0001). ",
"Outcome Measurement-In transrectal fusion-biopsies detection rate for PI-RADS 3 lesions increased from 35 to 50% after 104 biopsies. ",
"In transperineal fusion-biopsies, detection rate increased from 40 to 55% after 119 cases for PI-RADS 3 lesions. ",
"We measured the learning curve of fusion biopsies graphically and mathematically. ",
"We demonstrated that proficiency occurs after 110 transrectal and 125 transperineal fusion-biopsies."
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"The present invention is directed to the provision of a device and method for displaying biological materials, such as plants, which are to be viewed in a preserved state. ",
"It is important for teachers, botanists, store owners, home owners, and others, to be able to display materials, such as rare plants, in a container without worrying about the preservation of the material, and to display the material in a fashionable and attractive way which will attract notice and interest.",
"\nDisplay devices are known in the art for displaying articles needed to be preserved, in which a solution of alcohol and water is provided to preserve the material. ",
"However, such prior art units are relatively unattractive. ",
"Display devices are also known in other arts where, for example, lighting effects are to be achieved.",
"\nU.S. Pat. ",
"No. ",
"3,535,805 provides for display of various shaped birefringent sheet forms freely suspended in a rotating liquid. ",
"Light is directed onto the forms and polarizing filters are located on opposite sides of the forms and in the path of light. ",
"Means are provided to rotate the liquid, thus causing the suspended forms to move and to provide a moving image on a wall or screen. ",
"However, this arrangement would not be suitable for the display of biological materials. ",
"It is complicated and cumbersome and is for enjoyment only. ",
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"Q:\n\nHighcharts area charts with single data, not rendering\n\nI have an area chart with three series x-axis is 'datetype', when there is only data in each series the labels on x and y axis appears as expected but the plot point is missing. ",
"Can't expect a filled area for one data but even the plot point is missing. ",
"\n'minRange' property is set and so the label is showing the date from the data.",
"\nIs this expected or there is a property that needs to be set to see the plot points.",
"\nfiddle link here http://jsfiddle.net/bM9j9/6/\n$(function () {\n $('#container').highcharts({\n chart: {\n type: 'area'\n},\ntitle: {\n text: 'Area chart'\n},\nxAxis: {\n type: 'datetime',\n minRange: 864e5\n},\nyAxis: {\n min: 0\n //tickInterval: 0.5,\n //minRange: 0.5\n},\ncredits: {\n enabled: false\n},\nplotOptions: {\n area: {\n stacking: 'normal',\n marker: {\n enabled: false,\n symbol: 'circle',\n radius: 2,\n states: {\n hover: {\n enabled: true\n }\n }\n }\n }\n},\nseries: [{\n name: 'Open',\n data: [\n [Date.",
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"\nthis is happening because you have disabled marker and have enabled hover for it.",
"\nTo get the points visible turn them on.",
"\nplotOption: {\n area:{\n marker: {\n enabled: true,\n-----continue with other properties----\n }\n }\n}\n\nupdated your fiddle here: http://jsfiddle.net/bM9j9/7/\nHope this will help you.",
"\n\n"
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"\n\nA methane explosion in a Czech coal mine has killed 13 miners, including 11 Polish nationals and two Czechs, the mine operator OKD said on Friday.",
"\n\nThe state-run OKD company said a methane blast more than 800 metres (875 yards) underground devastated areas of the CSM hard-coal mine, near the town of Karvina, on Thursday.",
"\n\n“Due to yesterday’s mine disaster, 13 miners lost their lives – 11 Poles and two Czechs,” OKD spokesman Ivo Celechovsky said at a televised news conference.",
"\n\nThe incident is the worst mining accident in the Czech Republic since 1990, when 30 miners died in a fire at a mine near Karvina, according to a list compiled by the CTK news agency.",
"\n\nOKD said most of the victims and injured were Polish miners provided by the ALPEX company.",
"\n\nOKD managing director Boleslav Kowalczyk said the search of the scene had had to be abandoned due to conditions in the shaft.",
"\n\n“Unfortunately, we have reached a point where it was impossible to move forward because there was a fire raging and zero visibility,” he said."
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"Q:\n\nStored Procedure with a conditional?",
"\n\nI trying to rewrite a stored procedure and my SQL is not very good. ",
"What i'm hoping to do is write it so that if ModuleID is 555 then select a custom date range (eg. ",
"2012-01-01 2012-12-31). ",
"The Current SP is below.",
"\nset ANSI_NULLS ON\nset QUOTED_IDENTIFIER ON\nGO\n\n/*** EventsGetByRange ***/\n\nALTER PROCEDURE [dbo].[EventsGetByRange]\n(\n @Filter nvarchar(500),\n @BeginDate datetime,\n @EndDate datetime\n)\nAS\nSET DATEFORMAT mdy\nDeclare @sql nvarchar(4000)\nSelect @sql = 'SELECT E.PortalID, E.EventID, E.RecurMasterID, E.ModuleID, E.EventDateBegin, E.EventDateEnd, '\n + 'E.EventTimeBegin, E.Duration, E.EventName, E.EventDesc, '\n + 'E.Importance, E.CreatedDate, '\n + 'CreatedBy = U.DisplayName, '\n + 'CreatorID = E.CreatedBy, '\n + 'E.Every, '\n + 'E.Period, '\n + 'E.RepeatType, '\n + 'E.Notify, '\n + 'E.approved, '\n + 'E.Signups, '\n + 'E.MaxEnrollment, '\n + '(Select count(*) from dbo.",
"EventsSignups WHERE EventID = E.EventID and E.Signups = 1) as Enrolled, '\n + 'E.EnrollRoleID, '\n + 'E.EnrollFee, '\n + 'E.EnrollType, '\n + 'E.PayPalAccount, '\n + 'E.PayPalPassword, '\n + 'E.Cancelled, '\n + 'E.DetailPage, '\n + 'E.DetailNewWin, '\n + 'E.DetailURL, '\n + 'E.ImageURL, '\n + 'E.ImageType, '\n + 'E.ImageWidth, '\n + 'E.ImageHeight, '\n + 'E.ImageDisplay, '\n + 'E.Location, '\n + 'c.LocationName, '\n + 'c.MapURL, '\n + 'E.Category, '\n + 'b.CategoryName, '\n + 'b.Color, '\n + 'b.FontColor, '\n + 'E.Reminder, '\n + 'E.TimezoneOffset, '\n + 'E.SendReminder, '\n + 'E.ReminderTime, '\n + 'E.ReminderTimeMeasurement, '\n + 'E.ReminderFrom, '\n + 'E.SearchSubmitted, '\n + 'E.CustomField1, '\n + 'E.CustomField2, '\n + 'E.EnrollListView, '\n + 'E.DisplayEndDate, '\n + 'E.AllDayEvent, '\n + 'E.OwnerID, '\n + 'OwnerName = O.DisplayName, '\n + 'E.LastUpdatedAt, '\n + 'LastUpdatedBy = L.DisplayName, '\n + 'E.LastUpdatedID, '\n + '(Select ModuleTitle from dbo.",
"Modules WHERE ModuleID = E.ModuleID) as ModuleTitle, '\n + 'RMOwnerID = r.OwnerID, '\n + 'r.RRULE, '\n + 'E.OriginalDateBegin, '\n + 'E.NewEventEmailSent '\n + 'FROM dbo.",
"Events E '\n + 'inner join dbo.",
"EventsRecurMaster AS r on E.RecurMasterID = r.RecurMasterID '\n + 'left outer join dbo.",
"Users U on E.CreatedBy = U.UserID '\n + 'left outer join dbo.",
"Users O on E.OwnerID = O.UserID '\n + 'left outer join dbo.",
"Users L on E.LastUpdatedID = L.UserID '\n + 'left join dbo.",
"EventsCategory b on E.Category = b.Category '\n + 'left join dbo.",
"EventsLocation c on E.Location = c.Location '\n + 'WHERE (E.ModuleID = 555 AND E.EventTimeBegin BETWEEN 2012-01-01 AND 2012-12-31) OR ((E.EventTimeBegin <= DATEADD(DAY,1,''' + convert(varchar, @EndDate) + ''') AND DATEADD(minute,E.Duration,E.EventTimeBegin) >= ''' + convert(varchar, @BeginDate) + ''') OR '\n + ' (E.EventTimeBegin BETWEEN ''' + convert(varchar, @BeginDate) + ''' AND DATEADD(DAY,1,''' + convert(varchar, @EndDate) + ''')))'\n + ' AND E.Approved = 1'\n + ' AND E.Cancelled = 0'\n + ' ' + @Filter + ' '\n + ' ORDER BY E.EventDateBegin, E.EventTimeBegin, E.EventDateEnd'\nEXEC (@sql)\n\nUPDATE: I used the where statemnnt that Diego recommended but that is not having the desired result. ",
" It does not act as and If\\Else scanrio (which makes sense when I think about it).",
"\nI need to first identify if the module ID is 555 and if so only pull the dates from in the hard coded range otherwise execute it as written. ",
" Please let me know if more detail is required.",
"\n\nA:\n\nis the proc failing?",
"\ndid you try adding E.ModuleID = 555 on the where clause?",
"\nDo you really want to hard code the value 555? ",
"how about passing it\non a parameter?",
"\nAnd most important question: why adding the sql statement to a\nvariable and execute it? ",
"why not just run the SQL? ",
"Is it because of\nthe @Filter variable?",
"\nalso, sql server 2005 or 2008?",
"\nwhy nvarchar and not varchar on your variables (\"n\" occupies double\nof space)\n\nEDIT:\nok, you have a OR in there so it may be tricky. ",
"Do you want everything from code 555 despite the date range value, or everything within the date range and code 555?",
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"\nThe purpose of this study was to describe the development and implementation of an on-site, behavior-based safety audit based on a safety program designed specifically to reduce injuries and fatalities in the residential construction industry. ",
"The audit was used to assess safety hazards and safety compliance on residential construction work sites. ",
"Safety behaviors were scored as all-or-none. ",
"A high score was related to high safety compliance. ",
"A total of 195 audits were performed on residential construction companies from varying trades. ",
"Analysis of mean total scores indicated that companies that had received some form of safety training scored significantly higher than companies that had not received any (P<0.01). ",
"Analysis of mean total scores between company trades indicated that masonry/stucco application companies had significantly lower scores than most other trades represented (P<0.01). ",
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"Description\n\nThis book asks an important question often ignored by ancient historians and political scientists alike: Why did Athenian democracy work as well and for as long as it did? ",
"Josiah Ober seeks the answer by analyzing the sociology of Athenian politics and the nature of communication between elite and nonelite citizens. ",
"After a preliminary survey of the development of the Athenian \"constitution,\" he focuses on the role of political and legal rhetoric. ",
"As jurymen and Assemblymen, the citizen masses of Athens retained important powers, and elite Athenian politicians and litigants needed to address these large bodies of ordinary citizens in terms understandable and acceptable to the audience. ",
"This book probes the social strategies behind the rhetorical tactics employed by elite speakers. ",
"A close reading of the speeches exposes both egalitarian and elitist elements in Athenian popular ideology. ",
"Ober demonstrates that the vocabulary of public speech constituted a democratic discourse that allowed the Athenians to resolve contradictions between the ideal of political equality and the reality of social inequality.",
"\nHis radical reevaluation of leadership and political power in classical Athens restores key elements of the social and ideological context of the first western democracy.show more\n\nBack cover copy\n\nFirst, the book is meant to be a contribution to Greek history: an attempt to explain the social roots and internal functioning of the political system of an ancient city-state. ",
"I hope that many of those who consider the history and culture of fifth-and fourth-century Athens intrinsically interesting, as I do, will find this study valuable in formulating or reformulating their own assessments of classical Greece.show more"
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"My Life As A Spiritualist: A Former Psychic's Testimony\n\nSupernatural phenomena interested me since childhood. ",
"A few psychic episodes as a child added to the intrigue I felt. ",
"My mother had also experienced psychic happenings since her childhood, but didn’t pursue it seriously until later. ",
"By the time I began Secondary School, she felt ready to experiment further. ",
"When walking her dogs in the park one day, a Medium approached, telling Mum he saw her potential as a Medium and invited her to a Spiritualist Church in Glasgow.",
"\n\nVery quickly, Mum became totally engrossed by the supernatural. ",
"As a new member of the Spiritualist Church, she attended Sunday services, mid week psychic development groups and Yoga classes. ",
"She shared everything she learned with me and I also became fascinated. ",
"Her heart’s desire was to train in Mediumship and the other Mediums encouraged this, asking her to join an Open Circle to learn to meditate and channel spirits of the deceased. ",
"Mum was keen to develop her abilities of Clairvoyance, Clairaudience and Clairsentience and to learn from other Mediums.",
"\n\nEvery time Mum booked a Private Sitting for a Life Reading with a resident or visiting Medium, she would buy a tape of the session and we’d listen to it later that day. ",
"It amazed me that a half hour tape could contain so much communication from spirits. ",
"Minute details were relayed about our lives. ",
"Precise names, places and dates were often given. ",
"It was obvious the Mediums weren’t charlatans, falling upon names by sheer chance. ",
"They accurately described the physical appearance, personalities and even repeated common phrases of our dead relatives as they conversed with them.",
"\n\nWhen old enough to attend, I also joined the Spiritualist Church. ",
"We devoured mystical books on opening the chakras, Crystal Healing, Reincarnation, Alternative Therapies, etc. ",
"Attending New Age Centres and Psychic Fairs, we absorbed as much information as we could to help in our spiritual enlightenment. ",
"With our passion for environmental and conservation concerns, social justice and international peace, we gladly participated in healing people and animals, through either contact or distant psychic healing. ",
"Eventually, we also attended Transfiguration Sessions. ",
"Mum developed in the area of Automatic Writing and Mediums predicted I would become a Psychic Artist, drawing portraits of dead relatives and Spirit Guides for clients. ",
"I experimented with Kirlian Photography; using infra red film I captured images of ectoplasm as it formed.",
"\n\nOver the next ten years, we’d often hear of Mediums who could no longer control when spirits spoke to or through them. ",
"Many Mediums had nervous breakdowns, or attacked people claiming their Spirit Guides had forced them to, or were admitted to psychiatric wards. ",
"We heard reports of poltergeist activity in Medium’s homes. ",
"At first, we accepted explanations that mischievous or obnoxious spirits could sometimes come through and it was a potential hazard of the job. ",
"But when it happened to us, it became difficult to tolerate and almost impossible to function properly.",
"\n\nOur friends, the other Psychics and Mediums, failed to free our home from spirits. ",
"The spirits spoke to Mum constantly, depriving her of sleep, attacking her physically, chapping loudly from within wardrobes, slamming all the doors, etc.",
"\n\nOn one of the occasions when they forced her into a trance against her will, she was frying food. ",
"When she came out of trance, the kitchen was consumed by fire. ",
"I arrived home after the Fire Brigade had extinguished it, just before it spread to other rooms. ",
"We all realised Mum, our dogs and cats could have been killed. ",
"One afternoon, Mum’s elderly Aunt who had also visited Spiritualist Churches, felt invisible hands grab her and throw her down our stairs. ",
"Her wrist was broken. ",
"Another day, while approaching shops, I watched in horror as Mum was lifted from the ground and catapulted from the pavement, landing on the bonnet of a passing car.",
"\n\nWe decided to withdraw from Spiritualism and told our Spirit Guides to leave. ",
"To our shock, they laughed and insulted us. ",
"They began to physically attack us. ",
"This was perplexing as they had provided guidance and kindness for many years. ",
"It became obvious they’d deceived us, pretending to be benevolent, when in reality they were wicked all along.",
"\n\nThe biggest shock however came when even our dead relatives turned against us. ",
"They also mocked and hit us. ",
"The spirits warned we couldn’t leave the occult, as we had unknowingly given them control from the first day we invited them into our lives.",
"\n\nDuring my second year at university, Mum’s health deteriorated and her Spirit Guides threatened to use her to kill people. ",
"Mum’s doctor couldn’t accept her accounts of poltergeist activity and diagnosed as Schizophrenic, she was detained in a Psychiatric Hospital. ",
"After months of heavy sedation, so they would discharge her, Mum deliberately told the psychiatrists she no longer heard voices. ",
"They did discharge her, but when she returned home, she endured further harassment from spirits and committed suicide.",
"\n\nA Christian I’d met in a Psychology Tutorial, invited me to her Pentecostal Church, explaining other Psychics who had experienced similar tragedies were set free from spiritual attacks when they accepted Jesus Christ as their Saviour. ",
"I renounced Spiritism and became a Christian. ",
"Later, a Pastor and her friend visited my mum’s home and successfully cleansed it the first time they prayed. ",
"At the Name of Jesus, all the spirits left and never returned. ",
"About a year later, I sold Mum’s home without worrying the new tenants would be harassed.",
"\n\nThe Pentecostal Church did not portray dead religious traditions, instead it was lively, loving and fun. ",
"Members are trained by the Holy Spirit to prophecy about people’s future, bodies are healed and other miracles, signs and wonders occur. ",
"Actually, in a couple of years I saw more peopled healed in Pentecostal Churches than I ever saw in over ten years at Spiritualist Churches.",
"\n\nWhen I became a Christian, even before I had read the Biblical warnings of the occult in Deuteronomy chapter 18, I instinctively realised what had happened. ",
"The missing piece of the jigsaw fit into place. ",
"When our dead relatives and Spirit Guides began to attack Mum and I, it indicated they had deceived us, merely impersonating our loved ones. ",
"It made complete sense when Christians explained that it’s impossible for dead souls to return to talk with us, as they remain in heaven or hell for eternity. ",
"Also that, evil spirits are fallen angels, and they have existed for centuries possessing psychic knowledge of our families and historical figures down the generations. ",
"They can easily disguise their evil form to pose as our deceased family or any famous celebrity that ever lived. ",
"This was confirmed as I read in 2 Corinthians 11 to 14 : “And no wonder! ",
"For Satan himself masquerades as an angel of light. ",
"It is not surprising then, if his servants masquerade as servants of righteousness. ",
"Their end will be what their actions deserve.” ",
"The evil spirits working through Spiritualism, serve Satan in deceiving people into believing they are their dead relatives and Spirit Guides.",
"\n\nBefore I left the occult, I was unaware the Bible advises in 1 John 4 verse 1 to 3 :\n\n“Beloved, do not believe every spirit, but test the spirits, whether they are of God … every spirit that confesses that Jesus Christ has come in the flesh is of God, and every spirit that does not confess that Jesus has come in the flesh is not of God…And this is the spirit of the Antichrist…”\n\nAs a Spiritualist, I had believed Jesus was just a cosmic force; part of the universal consciousness, or a great Psychic Healer. ",
"Since my conversion I’ve met and read of many Psychics, Gypsies, Mediums, Witches and even Satanists who also discovered the entities of their Spirit Guides or dead relatives were not who they claimed to be. ",
"These people were also set free from demonic attack after receiving Christ into their lives. ",
"Two successful Glasgow Mediums, Sadie Bryce and Geordie Aitken also became Christians. ",
"Sadie went to the same Spiritualist Church I did. ",
"She spoke of her occult past in BBC programs and Geordie wrote a book detailing his. ",
"As Christians, Sadie and I wrote to each other and I met Geordie several times, discovering we all had been misled by spirits in the same ways.",
"\n\nJesus healed me of fears, phobias and physical conditions. ",
"He has given me joy and peace I simply never knew existed. ",
"His very tangible Presence of pure love, is more real than any powerful encounter I experienced within Spiritualism.",
"\n\nAfter sharing my life story on TV and radio, I continue to hear of more people who are turning away from Spiritualist Meetings, Tarot Card Parties, etc. ",
"Recently, I completed a novella containing some detail of events Mum and I experienced within the occult. ",
"It also contains some wonderful encounters with Jesus and angels, very different from what I experienced in Spiritualism.",
"\n\nIf you would like to order a copy of my book ‘A Spiritual Quest,’ please contact me on searching.08 At@ hotmail.co.uk\n\n© 2009, Laura Maxwell, BA (Hons).",
"\n\nhttp://www.reachouttrust.org/articleView.php?id=352\n\nhttp://www.facebook.com/inbox/?tid=1287388500477#/pages/Scotland-United-Kingdom/A-Spiritual-Quest/55259309502?ref=ts\n\nhttp://www.youtube.com/user/LauraMaxwellExSpirit\n\n\n\n\n\n\n\n\n\nThe Truth about Psychics\n\nPsychics, mediums, and ghosts have become a sensation in our culture today. ",
"As a result, there are many confusing and deceptive beliefs presented. ",
"Ron Rhodes, respected and popular biblical scholar, tackles the truth about ghosts and those who say they communicate with them and answers the questions:\n\nDo ghosts in any shape or form exist? ",
"Why is there a rise in psychic phenomena today? ",
"What do psychics believe about God, Jesus, and salvation? ",
"What is Satan's role with the paranormal? ",
"How can parents protect their family from the psychic trend?",
"\n\nThis reader–friendly presentation of intriguing facts and biblical insights will help Christians know how to respond to this fascination with the ultimate truth.",
"\n\n\n\n"
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"They take turns lying on it, and although they share the dog bed, I've never seen them actually lie down on it together, until last night.",
"\n\nDaniel stuffed the dog bed into its newly laundered cover, and the minute he set it down on the living room floor, both dogs threw themselves onto it, and didn't move for the next 2 quarters of the Celtics/Lakers basketball game. ",
"It seems even dogs prefer fresh clean bed sheets. ",
"Who knew... I mean, earlier this weekend they'd spent a half an hour rolling around in the dirt at the dog park.",
"\n\nToday I want to share with you these severely stylish dog beds you can buy at The Diva Dog. ",
"These beds are gorgeous to look at, but they're also stain, moisture, odor, and bacteria resistant. ",
"And you can even get the matching sofa pillows for yourself!",
"\n\nNo comments:\n\nAbout House & Hound\n\nHouse & Hound was created by an interior designer who loves her dogs, and wants to share her dog inspired home decor ideas with you. ",
"I search for products that are modern, hip, and classy, so you can show off your dog love in style.",
"\n\nAbout Me\n\nProduct Reviews\n\nFeel free to pitch a product and I'll get back to you if I'm interested in reviewing it. ",
"Product samples are welcome, but I do not accept paid product or editorial placements. ",
"Email tips@houseandhoundblog.com ."
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"Saint-Pierre, Réunion\n\nSaint-Pierre () is the third-largest commune in the French overseas department of Réunion. ",
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" SUBROUTINE SLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )\n*\n* -- LAPACK auxiliary routine (version 3.0) --\n* Univ. ",
"of Tennessee, Univ. ",
"of California Berkeley, NAG Ltd.,\n* Courant Institute, Argonne National Lab, and Rice University\n* October 31, 1992\n*\n* .. Scalar Arguments ..\n CHARACTER UPLO\n INTEGER LDA, LDW, N, NB\n* ..\n* .. Array Arguments ..\n REAL A( LDA, * ), E( * ), TAU( * ), W( LDW, * )\n* ..\n*\n* Purpose\n* =======\n*\n* SLATRD reduces NB rows and columns of a real symmetric matrix A to\n* symmetric tridiagonal form by an orthogonal similarity\n* transformation Q' * A * Q, and returns the matrices V and W which are\n* needed to apply the transformation to the unreduced part of A.\n*\n* If UPLO = 'U', SLATRD reduces the last NB rows and columns of a\n* matrix, of which the upper triangle is supplied;\n* if UPLO = 'L', SLATRD reduces the first NB rows and columns of a\n* matrix, of which the lower triangle is supplied.",
"\n*\n* This is an auxiliary routine called by SSYTRD.",
"\n*\n* Arguments\n* =========\n*\n* UPLO (input) CHARACTER\n* Specifies whether the upper or lower triangular part of the\n* symmetric matrix A is stored:\n* = 'U': Upper triangular\n* = 'L': Lower triangular\n*\n* N (input) INTEGER\n* The order of the matrix A.\n*\n* NB (input) INTEGER\n* The number of rows and columns to be reduced.",
"\n*\n* A (input/output) REAL array, dimension (LDA,N)\n* On entry, the symmetric matrix A. If UPLO = 'U', the leading\n* n-by-n upper triangular part of A contains the upper\n* triangular part of the matrix A, and the strictly lower\n* triangular part of A is not referenced. ",
" If UPLO = 'L', the\n* leading n-by-n lower triangular part of A contains the lower\n* triangular part of the matrix A, and the strictly upper\n* triangular part of A is not referenced.",
"\n* On exit:\n* if UPLO = 'U', the last NB columns have been reduced to\n* tridiagonal form, with the diagonal elements overwriting\n* the diagonal elements of A; the elements above the diagonal\n* with the array TAU, represent the orthogonal matrix Q as a\n* product of elementary reflectors;\n* if UPLO = 'L', the first NB columns have been reduced to\n* tridiagonal form, with the diagonal elements overwriting\n* the diagonal elements of A; the elements below the diagonal\n* with the array TAU, represent the orthogonal matrix Q as a\n* product of elementary reflectors.",
"\n* See Further Details.",
"\n*\n* LDA (input) INTEGER\n* The leading dimension of the array A. LDA >= (1,N).",
"\n*\n* E (output) REAL array, dimension (N-1)\n* If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal\n* elements of the last NB columns of the reduced matrix;\n* if UPLO = 'L', E(1:nb) contains the subdiagonal elements of\n* the first NB columns of the reduced matrix.",
"\n*\n* TAU (output) REAL array, dimension (N-1)\n* The scalar factors of the elementary reflectors, stored in\n* TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.",
"\n* See Further Details.",
"\n*\n* W (output) REAL array, dimension (LDW,NB)\n* The n-by-nb matrix W required to update the unreduced part\n* of A.\n*\n* LDW (input) INTEGER\n* The leading dimension of the array W. LDW >= max(1,N).",
"\n*\n* Further Details\n* ===============\n*\n* If UPLO = 'U', the matrix Q is represented as a product of elementary\n* reflectors\n*\n* Q = H(n) H(n-1) . . . ",
"H(n-nb+1).",
"\n*\n* Each H(i) has the form\n*\n* H(i) = I - tau * v * v'\n*\n* where tau is a real scalar, and v is a real vector with\n* v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),\n* and tau in TAU(i-1).",
"\n*\n* If UPLO = 'L', the matrix Q is represented as a product of elementary\n* reflectors\n*\n* Q = H(1) H(2) . . . ",
"H(nb).",
"\n*\n* Each H(i) has the form\n*\n* H(i) = I - tau * v * v'\n*\n* where tau is a real scalar, and v is a real vector with\n* v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),\n* and tau in TAU(i).",
"\n*\n* The elements of the vectors v together form the n-by-nb matrix V\n* which is needed, with W, to apply the transformation to the unreduced\n* part of the matrix, using a symmetric rank-2k update of the form:\n* A := A - V*W' - W*V'.",
"\n*\n* The contents of A on exit are illustrated by the following examples\n* with n = 5 and nb = 2:\n*\n* if UPLO = 'U': if UPLO = 'L':\n*\n* ( a a a v4 v5 ) ( d )\n* ( a a v4 v5 ) ( 1 d )\n* ( a 1 v5 ) ( v1 1 a )\n* ( d 1 ) ( v1 v2 a a )\n* ( d ) ( v1 v2 a a a )\n*\n* where d denotes a diagonal element of the reduced matrix, a denotes\n* an element of the original matrix that is unchanged, and vi denotes\n* an element of the vector defining H(i).",
"\n*\n* =====================================================================\n*\n* .. Parameters ..\n REAL ZERO, ONE, HALF\n PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, HALF = 0.5E+0 )\n* ..\n* .. Local Scalars ..\n INTEGER I, IW\n REAL ALPHA\n* ..\n* .. External Subroutines ..\n EXTERNAL SAXPY, SGEMV, SLARFG, SSCAL, SSYMV\n* ..\n* .. External Functions ..\n LOGICAL LSAME\n REAL SDOT\n EXTERNAL LSAME, SDOT\n* ..\n* .. Intrinsic Functions ..\n INTRINSIC MIN\n* ..\n* .. Executable Statements ..\n*\n* Quick return if possible\n*\n IF( N.LE.0 )\n $ RETURN\n*\n IF( LSAME( UPLO, 'U' ) ) THEN\n*\n* Reduce last NB columns of upper triangle\n*\n DO 10 I = N, N - NB + 1, -1\n IW = I - N + NB\n IF( I.LT.N ) THEN\n*\n* Update A(1:i,i)\n*\n CALL SGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),\n $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )\n CALL SGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ),\n $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )\n END IF\n IF( I.GT.1 ) THEN\n*\n* Generate elementary reflector H(i) to annihilate\n* A(1:i-2,i)\n*\n CALL SLARFG( I-1, A( I-1, I ), A( 1, I ), 1, TAU( I-1 ) )\n E( I-1 ) = A( I-1, I )\n A( I-1, I ) = ONE\n*\n* Compute W(1:i-1,i)\n*\n CALL SSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,\n $ ZERO, W( 1, IW ), 1 )\n IF( I.LT.N ) THEN\n CALL SGEMV( 'Transpose', I-1, N-I, ONE, W( 1, IW+1 ),\n $ LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )\n CALL SGEMV( 'No transpose', I-1, N-I, -ONE,\n $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,\n $ W( 1, IW ), 1 )\n CALL SGEMV( 'Transpose', I-1, N-I, ONE, A( 1, I+1 ),\n $ LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )\n CALL SGEMV( 'No transpose', I-1, N-I, -ONE,\n $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,\n $ W( 1, IW ), 1 )\n END IF\n CALL SSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )\n ALPHA = -HALF*TAU( I-1 )*SDOT( I-1, W( 1, IW ), 1,\n $ A( 1, I ), 1 )\n CALL SAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )\n END IF\n*\n 10 CONTINUE\n ELSE\n*\n* Reduce first NB columns of lower triangle\n*\n DO 20 I = 1, NB\n*\n* Update A(i:n,i)\n*\n CALL SGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),\n $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )\n CALL SGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),\n $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )\n IF( I.LT.N ) THEN\n*\n* Generate elementary reflector H(i) to annihilate\n* A(i+2:n,i)\n*\n CALL SLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,\n $ TAU( I ) )\n E( I ) = A( I+1, I )\n A( I+1, I ) = ONE\n*\n* Compute W(i+1:n,i)\n*\n CALL SSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,\n $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )\n CALL SGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ), LDW,\n $ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )\n CALL SGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ),\n $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )\n CALL SGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ), LDA,\n $ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )\n CALL SGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ),\n $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )\n CALL SSCAL( N-I, TAU( I ), W( I+1, I ), 1 )\n ALPHA = -HALF*TAU( I )*SDOT( N-I, W( I+1, I ), 1,\n $ A( I+1, I ), 1 )\n CALL SAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 )\n END IF\n*\n 20 CONTINUE\n END IF\n*\n RETURN\n*\n* End of SLATRD\n*\n END\n"
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"Q:\n\nMotivic integration in positive characteristic: how much is known?",
"\n\nIt seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). ",
"My question is: how much can one prove over a positive characteristic field? ",
"What are the main difficulties in this setting; can one prove certain weaker versions of the main results of the theory (for example, using alterations instead of Hironaka's resolution of singularities)? ",
"Is there any text that treats these questions systematically?",
"\n\nA:\n\nThere are many theories of motivic integration. ",
"The first one, due to Kontsevich (on smooth varieties) and developed by Denef-Loeser (in general, Inventiones Math., ",
"1999) assumed originally that the characteristic was zero. ",
"Replacing arc schemes by Greenberg schemes, it has been generalized by Looijenga (Bourbaki seminar, Astérisque 276, 2002) to encompass varieties over $k[[t]]$, and then by Sebag (Bulletin SMF, 2004) to allow formal schemes over an arbitrary complete discrete valuation ring. ",
"A nice application is the definition by Loeser-Sebag (Duke Math. ",
"J., 2003) of the motivic Serre invariant of a rigid analytic space.",
"\nThis geometric theory is presented in this generality in a book in preparation by Nicaise, Sebag and myself.",
"\nMore recently, Cluckers and Loeser developed a more general theory (Inventiones Math., ",
"2008 and Annals of Math., ",
"2010) over henselian discretely valued fields of residual characteristic zero (rings such as $k[[t]]$, where the characteristic of $k$ is zero). ",
"The theory of Hrushovski and Kazhdan has a similar limitation.",
"\nCluckers and Loeser have then extended their theory to arbitrary henselian discretely valued fields of characteristic zero (Crelle, 2013).",
"\nWhile algebraic geometry works well in arbitrary characteristic, model theoretic inputs (e.g., Pas's theorem) presently impose such restrictions on the characteristic.",
"\n\n"
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"I tried to create a ViewHolder, but I get a NPE error when I try to click it.",
"\nstatic class ViewHolder {\n TextView camera;\n TextView players;\n TextView max_players;\n ImageView privata;\n Button Buton;\n}\n@Override\npublic View getView(int position, View convertView, ViewGroup parent) {\n\n String variabile[] = getItem(position).split(\"\\\\s+\");\n LayoutInflater linflater = LayoutInflater.from(getContext());\n View customView = linflater.inflate(R.layout.custom_row, parent, false);\n final ViewHolder holder = new ViewHolder();\n holder.camera = (TextView) customView.findViewById(R.id.",
"Nume);\n holder.players = (TextView) customView.findViewById(R.id.players);\n holder.max_players = (TextView) customView.findViewById(R.id.max_players);\n holder.privata = (ImageView) customView.findViewById(R.id.privata);\n holder.",
"Buton = (Button) customView.findViewById(R.id.",
"Buton);\n holder.camera.setText(variabile[0]);\n if (!",
"variabile[1].equals(\"true\")) {\n parola = false;\n holder.privata.setVisibility(View.",
"INVISIBLE);\n }\n holder.players.setText(variabile[2]);\n holder.max_players.setText(variabile[3]);\n room_id = variabile[4];\n nume = variabile[5];\n holder.",
"Buton.setOnClickListener(new View.",
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"Buton.setText(\"Iesi\");\n site = siteul + \"/join\";\n intra = false;\n } else {\n holder.",
"Buton.setText(\"Intra\");\n site = siteul + \"/leave\";\n intra = true;\n }\n new ATask().execute(site);\n }\n });\n return customView;\n}\n\nA:\n\nWhen using the ViewHolder pattern, you should check if the convertView in null or has been created before, in the getView method, and after that use setTag and getTag methods. ",
"like this :\n if (convertView == null) \n {\n LayoutInflater linflater = LayoutInflater.from(getContext());\n convertView = linflater.inflate(R.layout.your_list_item_view, parent, false);\n viewHolder.textView = (TextView)convertView.findViewById([the id]);\n .",
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"---\nabstract: 'We present a model which deepens into the role that normal scattering has on the thermal conductivity in semiconductor bulk, micro and nanoscale samples. ",
"Thermal conductivity as a function of the temperature undergoes a smooth transition from a kinetic to a collective regime that depends on the importance of normal scattering events. ",
"We demonstrate that in this transition, the key point to fit experimental data is changing the way to perform the average on the scattering rates. ",
"We apply the model to bulk Si with different isotopic compositions obtaining an accurate fit. ",
"Then we calculate the thermal conductivity of Si thin films and nanowires by only introducing the effective size as additional parameter. ",
"The model provides a better prediction of the thermal conductivity behavior valid for all temperatures and sizes above 30 nm with a single expression. ",
"Avoiding the introduction of confinement or quantum effects, the model permits to establish the limit of classical theories in the study of the thermal conductivity in nanoscopic systems.'",
"\nauthor:\n- 'C. de Tomas'\n- 'A. Cantarero'\n- 'A. F. Lopeandia'\n- 'F. X. Alvarez'\ntitle: From kinetic to collective behavior in thermal transport on semiconductors and semiconductor nanostructures\n---\n\nIntroduction {#intro}\n============\n\nA general model, able to explain the thermal conductivity $\\kappa$ in macro-, micro- and nanostructured systems is still an open challenge. ",
"Experimental measurements on low-dimensional semiconductors [@Asheghi1998; @Li2003] have shown a drastic size-dependent reduction of $\\kappa$ as compared to bulk values.[@Inyushkin2004; @Glassbrenner1964] A great effort has been devoted in the last years to develop a general model which provides an accurate understanding of this reduction.[@Chantrenne2005; @Mingo2003; @Kazan2010; @Balandin2001; @Martin2009; @Morelli2002] At present, we can confirm that, when the size of the samples is reduced, classical boundary effects are expected due to the existence of a surface or an interface. ",
"Below some nanometers phonon confinement may also influence thermal transport through the modification of the dispersion relations.",
"\n\nHowever, it is still under debate which are the most important effects at the different length scales since most of the proposed models do not agree even in the origin of the reduction of the thermal conductivity, whether it is due to a change in the relaxation times, a confinement or a quantum effect, especially within the range of $10-100$ nanometers. ",
"In order to obtain a thermal transport model valid at all ranges of sizes and temperatures, it is necessary to have some certainty about the limits of applicability of the classical approaches without the inclusion of the mentioned changes and the size dimensions where those are strictly necessary.",
"\n\nRecent works have focused their attention on the calculation of phonon scattering rates by *ab initio* techniques.[@Broido2007; @Broido2010; @Broido2012; @Lindsay2013; @Fugallo2013] These works suggest that the main reason for the poor adjustment of current theories arises from the use of empirical potentials with adjustable parameters or the use of classical expressions for the relaxation times. ",
"Based on *ab initio* techniques, they solve numerically the Boltzmann transport equation (BTE) to obtain the scattering times and predict the thermal conductivity. ",
"In the last years, the thermal conductivity of several materials has been calculated (see Ref. ",
"and references therein). ",
"In these works, the theoretical predictions agree very well with the experiment in particular intervals of sizes and temperatures. ",
"Very recently, Fugallo *et al.* [",
"@Fugallo2013], also using *ab initio* techniques, calculate the thermal conductivity of bulk diamond and isotopically enriched diamond by solving the BTE using the variational principle and the conjugate gradient scheme. ",
"They introduced the scattering due to boundary effects with a shape factor to fit the low temperature region. ",
"The Mathiessen rule is used in both cases to account for the different scattering mechanisms.",
"\n\nIn spite of these advances, the models based on *ab initio* techniques looses some of the thermodynamics involved in the heat transport mechanisms, hidden behind the numerical complexity of the models. ",
"At this stage, a phenomenological model is always desirable when the physical processes can be clearly described. ",
"This perspective also aimed the work by Allen[@Allen2013], where the widely used Callaway model[@Callaway1958] is improved by a more rigorous treatment of phonon-phonon scattering, paying particular attention to the introduction of the normal scattering relaxation time into the expression of the lattice thermal conductivity.",
"\n\nIn the last decade, some authors have suggested that in order to predice nanoscale transport parameters, memory and nonlocalities had to be included [@Xavi2007; @EIT1993] in the expressions. ",
"In this line of thought, here we demonstrate that some issues appearing when fitting thermal conductivity data are not related to the particular expression used for the relaxation times but with the way their thermodynamic averages are calculated. ",
"We show, from an approach based on Guyer and Krumhansl model, [@Krumhansl1965; @Guyer1966; @Guyer1966a] that a more appropriate equation for $\\kappa$ can be obtained. ",
"This equation will provide a new insight into the underlying physics of thermal transport. ",
"It introduces a thermodynamic perspective that allows to understand the differences in phonon behavior in terms of the mixing rate of the different phonon-phonon processes. ",
"Our proposal is in good agreement with experimental data on bulk silicon,[@Inyushkin2004] thin films (TFs),[@Asheghi1998] and nanowires (NWs),[@Li2003] with characteristic sizes above 30 nm. ",
"We show that confinement or quantum effects are not necessary to understand the lattice thermal transport above these sizes and that the difficulty of prediction at the nanoscale seems to be deeply related to the thermodynamic treatment of phonon-phonon interactions. ",
"At the same time, this allows to establish a lower limit for classical models, where bulk properties are enough to understand the phenomenology. ",
"Only below this limit, of the order of a few tens of nanometers, confinement effects may play a role.[@Srivastava2007]\n\nApproaches to solve the Boltzmann transport equation\n====================================================\n\nBTE is the usual starting point in all thermal conductivity works. ",
"Its mathematical form and the physical interpretation of its terms in thermal transport applications have been widely discussed in the literature[@Peierls1955; @Ziman1979]. ",
"Summarizing, when a small temperature difference $\\delta T$ is applied on a system, the phonon distribution $f_{\\bm{q}}$ moves from equilibrium at a linear rate. ",
"On the other hand, collisions turn the phonon distribution back to equilibrium at a rate that depends on the scattering transition rate. ",
"The BTE allows to obtain the resulting phonon distribution function by relating both rates $$\\label{boltz_0}\n\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{drift}}}= \\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scatt}}}.$$ Unfortunately, the analytical solution of Eq. (",
"\\[boltz\\_0\\]) is unknown. ",
"Two possible alternatives are: i) to solve the full equation numerically or ii) to derive some simplified expression replacing part of Eq. (",
"\\[boltz\\_0\\]) and solve it analytically.",
"\n\nThe computational power nowadays allows the numerical solution of the BTE in combination with density functional theory obtaining remarkable results in particular regions of temperature. ",
"Specifically, for silicon, comparison with natural and isotopically enriched bulk samples has been obtained in the \\[50-350\\] K range [@Ward2009], but for low temperatures, the grid of $q$ points needed in this approach is out of the calculational capability [@Broido2012]. ",
"In order to improve this, Fugallo et al. [",
"@Fugallo2013] combined *ab initio* with a phenomenological expression with a fitting parameter for the boundary scattering for bulk diamond. ",
"In reduced size samples, predictions for wires have not been able to be compared with experiments since, in words of the authors, they provide larger values of $\\kappa$ [@Broido2012]. ",
"To date this kind of approach has not been able to obtain a single solution valid at all ranges of temperatures and sample sizes for this material, demonstrating that phenomenological approaches are still necessary.",
"\n\nHere we develop an approach that allows to distinguish between two different regimes and shows that the difficulty for obtaining a global solution lies in the fact that each regime happens at different temperature intervals. ",
"In this section we define the terms that will be used in our model to obtain the final expression of the thermal conductivity.",
"\n\nIn equilibrium it can be easily demonstrated that phonons follow the Bose-Einstein distribution function $$\\label{dist_equil_1}\nf^{0}_{\\bm{q}}=\\frac{1}{e^{\\hbar \\omega_{\\bm{q}}/k_{B}T}-1}\\quad,$$ where $\\hbar\\omega_{\\bm{q}}\\equiv\\varepsilon_{\\bm q}$ is the energy of the phonon mode ($\\nu$,$\\bm{q}$) (the branch $\\nu$ will be omitted for simplicity), $T$ the absolute temperature and $k_{B}$ the Boltzmann constant.",
"\n\nIf a temperature difference is applied on the system $\\delta T$, an asymmetry in $f_{\\bm{q}}$ will be generated in the direction of the resulting gradient $\\nabla T$. In general, the final form of the distribution can be very complex, but under small $\\delta T$ the deviations from equilibrium are expected to be small. ",
"In that case, we can expand $f_{\\bm{q}}$ and keep the first term in the expansion: $$\\label{aprox_non_eq_dist}\nf_{\\bm{q}}\\simeq f^{0}_{\\bm{q}} + \\frac{\\partial f_{\\bm{q}}}{\\partial \\varepsilon_{\\bm{q}}}\\delta \\varepsilon\\simeq\nf^{0}_{\\bm{q}} + \\frac{\\partial f^{0}_{\\bm{q}}}{\\partial \\varepsilon_{\\bm{q}}}\\Phi_{\\bm{q}} = f^{0}_{\\bm{q}} + \\frac{f^{0}_{\\bm{q}}(f^{0}_{\\bm{q}}+1)}{k_{B}T}\\Phi_{\\bm{q}}\\quad ,$$ where $\\Phi_{\\bm{q}}$ is a smooth function of the energy and temperature whose precise form depends on the scattering processes. ",
"Expressed in these terms, solving BTE is reduced to obtain an expression for $\\Phi_{\\bm{q}}$, which will lead to a thermal conductivity equation. ",
"The approach used to solve the problem will depend ultimately on the expected form of $\\Phi_{\\bm{q}}$.\n\nThere are roughly two main approaches to solve BTE analytically: the kinetic methods (KM) and the variational methods (VM). ",
"KM can be applied when the distribution function is expected to be very close to equilibrium. ",
"In this case, the collision term is usually simplified by assuming that it is proportional to the inverse of a relaxation time (relaxation time approximation (RTA)), depending only on the values of a single mode. ",
"Finding relaxation times for reduced regions of temperature and size is not difficult. ",
"The problem appears if one wants to extend the region of applicability to wider intervals using the same KM approach with the same RTA expression. ",
"In the last decades, the miniaturization has worsened this situation, showing dramatic divergences between KM-RTA predictions and the experimental results when bulk and nanoscale samples are simulated with the same relaxation time expressions.",
"\n\nIn contrast, when the system is not so close to equilibrium, VM provides a better way to solve BTE. ",
"In general the collision terms in VM cannot be expressed analytically, instead they have to be obtained by integration using a trial function. ",
"This trial function should be close to the actual solution to have a good convergence. ",
"The main drawback is that this function is not necessarily the same in all temperature ranges. ",
"In conclusion, this approach is only useful in regions where the form of the phonon distribution is known to some extent.",
"\n\nAlthough thermal conductivity obtained within the KM and VM seem to be disconnected, from thermodynamic reasoning we will demonstrate that both can be derived from the balance of entropy production. ",
"The main difference between both approaches resides in the way this balance is performed. ",
"Starting from this point, it is easy to demonstrate that a general expression for thermal conductivity can be obtained by combining the distribution function in these two extreme situations: the first one where resistive processes are dominant and equilibrium can be rapidly achieved (related to KM) and the second one where although equilibrium cannot be easily reached, conservation of momentum in collisions allows us to determine analytically the scattering term (related to VM).",
"\n\nTo obtain $\\kappa$ in each limit, an expression for the scattering term in Eq. (",
"\\[boltz\\_0\\]) is needed. ",
"In KM this is usually done by the RTA approach, but in VM the expected form of the distribution function do not provide a simple expression. ",
"In the particular case when normal collisions are dominant, we suggest that the same RTA expression can be used, leaving the difference between approaches only in the way to perform the thermodynamic averages with this relaxation times.",
"\n\nResistive vs Normal scattering (equilibrium vs non-equilibrium) {#resistive_normal}\n---------------------------------------------------------------\n\nAs indicated before, the expected form of $\\Phi_{\\bm{q}}$ will determine the choice between a KM or a VM approach. ",
"The calculation of the scattering rates depends on it and, at the same time this depends on which scattering mechanism is dominating the system. ",
"Determining the dominating mechanism is thus the first important question to solve.",
"\n\nPhonons can relax by different mechanisms, colliding with boundaries, impurities, electrons and between them. ",
"All these mechanisms are resistive except some part of the phonon-phonon collisions. ",
"Two phonons with wave number and energy $(\\bm q,\\omega_{\\bm q})$ and $(\\bm q',\\omega_{\\bm q'})$ can scatter and produce, as a result, a new phonon $(\\bm q'',\\omega_{\\bm q''})$ (or vice-versa). ",
"In all events, energy must be conserved, but the wave number or quasi-momentum can be lost due to the interaction with the whole lattice. ",
"The equation $$\\bm q+\\bm q'=\\bm q''+\\bm G,$$ where $\\bm G$ is a reciprocal lattice vector, expresses the fact that the total lattice can acquire an amount of momentum $\\bm G$ because the resultant phonon is reflected outside of the first Brillouin zone (BZ).[@Ziman1979] If the quasi-momentum is conserved ($\\bm G=0$) the scattering processes are called normal or N-processes, while in the general case ($\\bm G\\neq 0$) they are called Umklapp or U-processes. ",
"Regarding the dominance of the N-processes two limiting behaviors can be considered:\n\ni\\) When resistive collisions are dominant and N-processes are negligible, momentum will be completely dissipated and its average value is zero. ",
"The only way to move the phonon distribution from equilibrium is by changing its temperature. ",
"In that case, the distribution function takes the form $$\\label{dist_rta}\nf_{\\bm{q}}=\\frac{1}{e^{\\hbar \\omega_{\\bm{q}}/k_{B}(T+\\delta T)}-1}\\approx\n\\frac{1}{e^{\\hbar \\omega_{\\bm{q}}/k_{B}T}e^{1-\\delta T/T}-1}.$$ Comparing with Eq. (",
"\\[aprox\\_non\\_eq\\_dist\\]) an expression for $ \\Phi_{\\bm{q}}$ can be obtained $$\\label{phi_for_rta}\n \\Phi_{\\bm{q}}= \\hbar \\omega_{\\bm{q}} \\frac{\\delta T}{T}.$$ In this situation KM is the most suitable approach to use.",
"\n\nii\\) When N-processes are dominant, the system will not be able to relax the momentum to zero (the quasi-momentum is conserved) and a displacement $\\bm u$ of the distribution function in the direction of the thermal gradient is expected. ",
"The distribution function takes the form [@Krumhansl1965] $$\\label{dist_vm}\nf_{\\bm{q}}=\\frac{1}{e^{(\\hbar \\omega_{\\bm{q}}-\\bm{u}\\cdot {\\bm{q}})/k_{B}T}-1}$$ which is in a non-equilibrium situation. ",
"Then, $\\Phi_{\\bm{q}}$ takes the form $$\\label{phi_for_vm}\n \\Phi_{\\bm{q}}=\\bm{u}\\cdot \\bm{q}$$ In this case the VM approach must be used.",
"\n\nSumming up, Eqs. (",
"\\[phi\\_for\\_rta\\]) and (\\[phi\\_for\\_vm\\]) are the two forms of $ \\Phi_{\\bm{q}}$ expected for the distribution function in each approach, KM and VM respectively, corresponding to two extreme situations described above. ",
"Next, we will use both expressions of $ \\Phi_{\\bm{q}}$ to show that in some situations, they yield equivalent expressions for the relaxation times.",
"\n\nDefining Scattering rates\n-------------------------\n\nOnce we have determined both expressions for $\\Phi_{\\bm{q}}$ in the two limiting cases, we can use them to determine the collision term in Eq. (",
"\\[boltz\\_0\\]) in each case. ",
"This depends on the transition probabilities and the form of the distribution functions. ",
"In general, the collision term in the Boltzmann equation can be written, for elastic scattering, as $$\\label{collision_1}\n\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}}=\\int \\frac{\\Phi_{\\bm{q}}-\\Phi_{\\bm{q}'}}{k_{B}T} P_{\\bm{q}}^{\\bm{q}'} d\\bm{q}'.$$ where $P_{\\bm{q}}^{\\bm{q'}}$ are the scattering transition rates from mode $\\bm{q}$ to $\\bm{q'}$ when the distribution functions correspond to equilibrium.",
"\n\nThe integral (\\[collision\\_1\\]) is expressing the fact that relaxation process in an out of equilibrium is modified by terms $\\Phi_{\\bm{q}}-\\Phi_{\\bm{q'}}$, *i. e.* depending on the displacement with respect to equilibrium of the different colliding particles. ",
"Expression (\\[collision\\_1\\]) can be generalized for an arbitrary number of colliding particles:\n\n$$\\label{collision_2}\n\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}} = \\frac{1}{k_BT}\n\\int \\left[\\Phi_{\\bm{q}} + \\displaystyle\\sum_{i=1}^{n}\\Phi_{\\bm{q}_{i}}-\\displaystyle\\sum_{j=1}^{m}\n\\Phi_{\\bm{q}_{j}}\\right] P_{\\bm{q}\\bm{q}_{1}...\\bm{q}_{n}}^{\\bm{q}'_{1}..\\bm{q}'_{m}}\n\\prod_{i=1\\atop j=1}^{m\\atop n} d\\bm{q}_{i} d\\bm{q}'_{j}$$\n\nwhere $\\bm{q}$ collides with $\\left\\{\\bm{q}_{i}\\right\\}$ giving as a result the modes $\\left\\{\\bm{q}'_{j}\\right\\}$. Expression (\\[collision\\_2\\]) shows the main complexity of solving the BTE equation. ",
"The scattering term requires the actual distribution function inside an integral expression establishing BTE as an integro-differential equation. ",
"One can use a numerical approach to solve it but other approximations can also be employed. ",
"These are usually based in the fact that the distribution used in the integral does not modify significantly the final result in some limiting situations. ",
"In RTA we assume that the system is close enough to equilibrium that the differences between using the actual form of the distribution or the equilibrium form in the collision integral (\\[collision\\_2\\]) is not significant.",
"This is like saying that the only mode out of equilibrium is the one with wave number $\\bm{q}$ and that the remaining modes rest in equilibrium. ",
"Thus $$\\label{condition_1}\n\\Phi_{\\bm{q}_{i}}=\\Phi_{\\bm{q}'_{j}}=0$$ for all $\\bm{q}_{i}\\neq\\bm{q}$ and $\\bm{q}'_{j}\\neq\\bm{q}$. In this case, $$\\label{collision_3}\n\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}} = \\frac{\\Phi_{\\bm{q}}}{k_{B}T}\n\\int P_{\\bm{q}\\bm{q}_{1}...\\bm{q}_{n}}^{\\bm{q}'_{1}..\\bm{q}'_{m}}\\prod_{i=1\\atop j=1}^{m\\atop n} d\\bm{q}_{i} d\\bm{q}'_{j}$$ If we substitute Eq. (",
"\\[aprox\\_non\\_eq\\_dist\\]) in (\\[collision\\_3\\]) we have $$\\label{collision_4}\n\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}} = \\frac{f_{\\bm{q}}-f^0_{\\bm{q}}}{f_{\\bm{q}}^{0}(f_{\\bm{q}}^{0}+1)}\n\\int P_{\\bm{q}\\bm{q}_{1}...\\bm{q}_{n}}^{\\bm{q}'_{1}..\\bm{q}'_{m}}\\prod_{i=1\\atop j=1}^{m\\atop n} d\\bm{q}_{i} d\\bm{q}'_{j} \\quad .$$\n\nThus, we can define the relaxation time $\\tau_{\\bm{q}}$ of mode $\\bm{q}$ as $$\\label{collision_5}\n\\frac{1}{\\tau_{\\bm{q}}} = \\frac{1}{f_{\\bm{q}}^{0}(f_{\\bm{q}}^{0}+1)} \\int P_{\\bm{q}\\bm{q}_{1}...\\bm{q}_{n}}^{\\bm{q}'_{1}..\\bm{q}'_{m}} \\prod_{i=1\\atop j=1}^{m\\atop n} d\\bm{q}_{i} d\\bm{q}'_{j}$$ and so, we obtain the BTE solution in the well known RTA approach $$\\label{RTA}\n\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}} = \\frac{f_{\\bm{q}}-f_{\\bm{q}}^{0}}{\\tau_{\\bm{q}}} \\quad .$$\n\nWe can make a similar assumption when normal scattering is the dominant relaxation process. ",
"The only change is that the distribution function where the actual distribution function will relax is that of Eqs. (",
"\\[dist\\_vm\\])-(\\[phi\\_for\\_vm\\]). ",
"In that case, condition (\\[condition\\_1\\]) cannot be fulfilled locally by each mode, but it can be demonstrated that in the linear regime not much error is made in Eq. (",
"\\[collision\\_2\\])[@Ziman1979] if we consider that $$\\label{collision_6}\n\\int \\Phi_{\\bm{q}_{i}} P_{\\bm{q}\\bm{q}_{1}...\\bm{q}_{n}}^{\\bm{q}'_{1}..\\bm{q}'_{m}} d\\bm{q}_{i}=\n\\int \\Phi_{\\bm{q}'_{j}} P_{\\bm{q}\\bm{q}_{1}...\\bm{q}_{n}}^{\\bm{q}'_{1}..\\bm{q}'_{m}} d\\bm{q}_{i}=0 \\qquad\\forall i,j$$ This condition leads to the same result as that obtained near equilibrium (\\[collision\\_4\\])-(\\[collision\\_5\\]), since condition (\\[collision\\_6\\]) is equivalent to condition (\\[condition\\_1\\]). ",
"Thus, we can use the same expression for the scattering rates in both limiting situations, near equilibrium and in non-equilibrium, despite of the very different nature of the two situations. ",
"Note that by using Eqs. (",
"\\[dist\\_rta\\])-(\\[phi\\_for\\_rta\\]) and (\\[dist\\_vm\\])-(\\[phi\\_for\\_vm\\]) in eq. (",
"\\[collision\\_6\\]) we are not stating that resistive processes are suppressed. ",
"In fact $P_{\\bm{q}\\bm{q}_{1}...\\bm{q}_{n}}^{\\bm{q}'_{1}..\\bm{q}'_{m}}$ are the transition rates for all the resistive scattering processes. ",
"We are only considering that collision integral (\\[collision\\_6\\]) does not change significantly when one uses the actual form of the distribution function or the proposed approximations in the corresponding regimes.",
"\n\nIn RTA in the special case when all the resistive terms are absolutely negligible, either KM and VM expression will give an infinite thermal conductivity as expected. ",
"In the following section we apply this result to obtain an expression for the thermal conductivity under each situation in the case where non-negligible resistive terms are present. ",
"The approximations here proposed will allow us to calculate two well differentiate regimes of behavior in the thermal transport: the kinetic and the collective regime.",
"\n\nThermal conductivity regimes\n============================\n\nHere we propose to derive thermal conductivity from the balance of entropy as obtained by Ziman[@Ziman1979]. ",
"The reason for this choice lies in the nature of normal scattering. ",
"Entropy generation is related to resistive collisions and normal scattering is not resistive. ",
"It is logical to think that entropy production can be modified when these kind of collisions are dominant. ",
"In this section we analyze these differences.",
"\n\nIn this deduction thermal conductivity is obtained from the equality of entropy production calculated from the drift and the collision terms in Eq. (",
"\\[boltz\\_0\\]). ",
"The collision term is obtained under a microscopic formalism, and the drift term is expressed in thermodynamic variables.",
"\n\nThe key point to notice is that N-processes, despite of being non-resistive, mix the different modes, affecting the balance between drift and collisions. ",
"If N-processes are not important and mode mixing is low, entropy balance should be fulfilled individually by each mode, that is, locally in momentum space. ",
"This leads to the thermal conductivity in the *kinetic regime*. ",
"On the other hand, when mode mixing is high (N-processes dominate) the entropy balance should be achieved globally, in this case we obtain the thermal conductivity in the *collective regime*. ",
"Depending on the intensity of the normal collisions we should select the local or the global version for the entropy production balance. ",
"Next, we detail both regimes of behavior and obtain the corresponding thermal conductivity contribution.",
"\n\n\\\n\nKinetic regime\n--------------\n\nEntropy of a distribution of bosons is $$\\label{entropy}\n\\frac{s_{\\bm{q}}}{k_{B}}= f_{\\bm{q}} \\ln f_{\\bm{q}} - (f_{\\bm{q}}-1) \\ln (f_{\\bm{q}}-1) \\quad .$$ The variation of entropy can be obtained from Eqs. (",
"\\[entropy\\]) and (\\[aprox\\_non\\_eq\\_dist\\]). ",
"If we take only linear terms in $\\Phi_{\\bm{q}}$ this can be written as [@Ziman1979] $$\\label{entropy_production_local}\n\\left.\\dot{s}_{\\bm{q}}\\right|_{\\textrm{\\tiny{scat}}}=\\left.\\frac{\\partial s_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}} = \\frac{\\Phi_{\\bm{q}}}{T}\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}},$$ Thermodynamically, the entropy variation can be also written in terms of the heat flux $$\\label{entropy_drift_def}\n\\left.\\dot{s}_{\\bm{q}}\\right|_{\\textrm{\\tiny{drift}}}=\\left.\\frac{\\partial s_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{drift}}} = {\\bm j}_{\\bm{q}}\\cdot \\nabla \\left(\\frac{1}{T}\\right) =\n\\frac{{\\bm j}^{2}_{\\bm{q}}}{\\kappa_{\\bm{q}} T^2}$$ where the heat flux of mode $\\bm q$ is $$\\label{flux_micro}\n{\\bm j}_{\\bm{q}}= \\hbar \\omega_{\\bm q}\\bm{v}_{g} (f_{\\bm{q}}-f^{0}_{\\bm{q}}) = \\hbar \\omega_{\\bm q} \\bm{v}_{g} f_{\\bm{q}}^{0}(f_{\\bm{q}}^{0}+1)\\frac{ \\Phi_{\\bm{q}}}{k_BT}$$ and we have used the fact that $\\bm{j}_{\\bm{q}}=-\\kappa_{\\bm{q}} \\nabla T$, where $\\kappa_{\\bm{q}}$ is the thermal conductivity of mode $\\bm{q}$, and $\\bm{v}_{g}$ is the group velocity.",
"\n\nEquating (\\[entropy\\_drift\\_def\\]) and (\\[entropy\\_production\\_local\\]) leads to an expression giving the thermal conductivity of each mode $$\\label{conductivity_mode_kin}\n\\kappa_{\\bm{q}}=\\frac{{\\bm j}^{2}_{\\bm{q}}}{T\\Phi_{\\bm{q}}\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}}}.$$ Integrating (\\[conductivity\\_mode\\_kin\\]) over all modes yields total thermal conductivity in this kinetic regime $$\\kappa_{\\rm{kin}}=\\int\\kappa_{\\bm{q}}d{\\bm{q}}=\\int\\frac{{\\bm j}^{2}_{\\bm{q}}}{T\\Phi_{\\bm{q}}\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}}}d{\\bm{q}}$$ and if we substitute Eq. (",
"\\[flux\\_micro\\]) we finally obtain $$\\label{conductivity_kin}\n\\kappa_{\\rm{kin}}=\\int\\frac{\\left[\\hbar \\omega_{\\bm q} \\bm{v}_{g} f_{\\bm{q}}^{0}(f_{\\bm{q}}^{0}+1)\\frac{ \\Phi_{\\bm{q}}}{k_BT}\\right]^2}{T\\Phi_{\\bm{q}}\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}}}d{\\bm{q}}$$\n\nCollective regime\n-----------------\n\nIn the second limiting case, phonons behave as a collectivity and each mode do not contribute to the entropy production individually but collectively. ",
"In this case the balance of entropy should be achieved globally and integration should be performed before equating terms. ",
"Thus, the total entropy production is on one side $$\\label{entropy_global_scat}\n \\left.\\dot{s}_{\\rm{tot}}\\right|_{\\textrm{\\tiny{scat}}} =\\int \\left. ",
"\\dot{s}_{\\bm{q}}\\right|_{\\textrm{\\tiny{scat}}}d{\\bm{q}} = \\int \\frac{\\Phi_{\\bm{q}}}{T}\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}} d{\\bm{q}}$$ and on the other we must account for a total heat flux, giving $$\\left.\\dot{s}_{\\rm{tot}}\\right|_{\\textrm{\\tiny{drift}}} = {\\bm j^2_{\\rm{tot}}} \\cdot \\nabla \\left(\\frac{1}{T}\\right).$$ Using the Fourier’s law ${\\bm j_{\\rm{tot}}} = -\\kappa\\nabla T$, we obtain $$\\label{entropy_global_drift}\n\\left. ",
"\\dot{s}_{\\rm{tot}} \\right|_{\\textrm{\\tiny{drift}}} = \\frac{\\bm j^2_{\\rm{tot}}}{\\kappa T^2}.$$ being $\\kappa$ the global thermal conductivity achieved in this regime. ",
"We denote it as $\\kappa_{\\rm{coll}}$ and we obtain its expression by equating (\\[entropy\\_global\\_scat\\]) and (\\[entropy\\_global\\_drift\\]) $$\\label{conductivity_global}\n\\kappa_{\\rm{coll}}=\\frac{\\bm j^2_{\\rm{tot}}}{T^2\\int \\frac{\\Phi_{\\bm{q}}}{T}\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}}d{\\bm{q}}}$$ where the total heat flux is $$\\label{heat_flux_collect}\n{\\bm j_{\\rm{tot}}}=\\int {\\bm j}_{\\bm{q}}d{\\bm{q}}=\\int \\hbar \\omega_{\\bm q} \\bm{v}_{g} f_{\\bm{q}}^{0}(f_{\\bm{q}}^{0}+1)\\frac{ \\Phi_{\\bm{q}}}{T}d{\\bm{q}}$$\n\nBy substituting this expression in Eq. (",
"\\[conductivity\\_global\\]), we have\n\n$$\\label{conductivity_global2}\n\\kappa_{\\rm{coll}}=\\frac{\\left[\\int \\hbar \\omega_{\\bm q} \\bm{v}_{g} f_{\\bm{q}}^{0}(f_{\\bm{q}}^{0}+1)\\frac{ \\Phi_{\\bm{q}}}{k_BT}d{\\bm{q}} \\right]^2}{T^2\\int \\frac{\\Phi_{\\bm{q}}}{T}\\left.\\frac{\\partial f_{\\bm{q}}}{\\partial t}\\right|_{\\textrm{\\tiny{scat}}} d{\\bm{q}} }$$\n\nThis new regime relies on a thermodynamic basis, and it can not be deduced from a framework where normal scattering is treated as a resistive mechanism like in Callaway model. ",
"After deducing the expression of the thermal conductivity in each regime, we need to choose a magnitude able to determine if we are in the local or global behavior. ",
"Secondly in order to calculate the integrals in (\\[conductivity\\_kin\\]) and (\\[conductivity\\_global2\\]), we need some expressions for the collision terms. ",
"This will be done in the next section.",
"\n\nThermal conductivity in terms of frequency and relaxation times {#sec:thermal_conduct}\n===============================================================\n\nWe are now able to calculate the thermal conductivity from Eq. (",
"\\[conductivity\\_kin\\]) for the kinetic regime and from Eq. (",
"\\[conductivity\\_global2\\]) for the collective regime. ",
"In order to obtain numerical results, first we need to express them in terms of the equilibrium distribution function and the relaxation times. ",
"Using (\\[aprox\\_non\\_eq\\_dist\\])-(\\[RTA\\]) in Eq. (",
"\\[conductivity\\_kin\\]), $\\kappa_{\\rm{kin}}$ can be rewritten as $$\\label{conductivity_kin_2}\n\\kappa_{\\rm{kin}}=\\int \\hbar \\omega_{\\bm{q}} \\tau_{\\bm{q}} \\bm{v}_g^{2} \\frac{\\partial f_{\\bm{q}}^{0}}{\\partial T} d{\\bm{q}}$$ which is the classical KM expression for the thermal conductivity. ",
"Here and onward we have omitted the index for the phonon branch in the integrals for the shake of simplicity.",
"\n\nFor (\\[conductivity\\_global2\\]), one can make the same substitutions to obtain $$\\label{conductivity_global_3}\n\\kappa_{\\rm{coll}}=\\frac{\\left(\\int \\Phi_{\\bm{q}} \\bm{v}_{g} \\frac{\\partial f_{\\bm{q}}^{0}}{\\partial T} d{\\bm{q}}\\right)^{2}}{\\int \\frac{\\Phi_{\\bm{q}}^{2}}{\\hbar \\omega_{\\bm{q}}}\\frac{1}{\\tau_{\\bm{q}}} \\frac{\\partial f_{\\bm{q}}^{0}}{\\partial T} d{\\bm{q}} } \\quad.$$\n\n$\\kappa_{\\rm{kin}}$ and $\\kappa_{\\rm{coll}}$ can be re-expressed in terms of frequency to simplify the integration in isotropic materials. ",
"This is done by the substitution $d{\\bm{q}}\\rightarrow D_{\\omega}d\\omega$ , being $D_{\\omega}$ the density of states (DOS), and integrating the angular part. ",
"For the kinetic regime this leads to the expression $$\\label{conductivity_kin_dos}\n\\kappa_{\\rm{kin}}=\\frac{1}{3}\\int \\hbar \\omega \\tau_{\\omega} v_g^{2} \\frac{\\partial f^0_{\\omega}}{\\partial T} D_{\\omega}d\\omega$$ where now the frequency dependence is indicated with the subindex (in the group velocity the subindex is omitted for simplicity), and for the collective regime $$\\label{conductivity_coll_dos}\n\\kappa_{\\mathrm{coll}}=\\frac{1}{3}\\frac{\\left(\\int v_{g} q_{\\omega} \\frac{\\partial f^0_{\\omega}}{\\partial T}D_{\\omega}d\\omega\\right)^{2}}{\\int \\frac{q^2_{\\omega}}{\\hbar \\omega}\\frac{1}{\\tau_{\\omega}} \\frac{\\partial f^0_{\\omega}}{\\partial T}D_{\\omega}d\\omega}$$ where we have used the explicit form (\\[phi\\_for\\_vm\\]) to express $\\Phi_{\\bm{q}}$ in terms of the wave vector $q_{\\omega}$. The only question to be addressed in Eq. (",
"\\[conductivity\\_coll\\_dos\\]) is that in order to maintain isotropy, $q^2_{\\omega}$ should be a frequency averaged value. ",
"This does not lead to large variations in isotropic materials.",
"\n\nAs we have already pointed, in both expression (\\[conductivity\\_kin\\_dos\\]) and (\\[conductivity\\_coll\\_dos\\]) $\\tau_{\\omega}$ is the same and accounts for the total relaxation time contributing to thermal resistance. ",
"Then, we denominate it $\\tau_{R_\\omega}$. Finally, we need a magnitude which accounts for the kind of regime the phonon distribution is undergoing at the different temperatures. ",
"As we have commented, this is determined by the degree of mixing between modes. ",
"Since this is related to the dominance of normal with respect to resistive processes, a switching factor weighting the relative importance of these processes should be used. ",
"This factor can be calculated from a matrix representation[@Guyer1966a] $$\\label{sigma}\n\\Sigma\\equiv\\frac{1}{1+\\frac{<\\tau_N>}{<\\tau_R>}}$$ where $\\tau_N$ is the relaxation time due to N-processes and $\\tau_R$ is the relaxation time due to resistive processes. ",
"Both relaxation times $\\tau_N$ and $\\tau_R$ are averaged over all modes. ",
"This is calculated as $$\\label{k_kin_int}\n\\langle \\tau_{i}\\rangle=\\frac{\\int \\hbar \\omega \\tau_{i_\\omega} \\frac{\\partial f^0_{\\omega}}{\\partial T}d\\bm{q}}{\\int \\hbar \\omega \\frac{\\partial f^0_{\\omega}}{\\partial T}d\\bm{q}}$$ with subindex $i$ indicating $N$ or $R$.\n\nThe general expression of the thermal conductivity must include this switching factor to account for all the intermediate regimes between the limiting regimes, *i*. *",
"e*. ",
"from kinetic to collective regime. ",
"Thus, $$\\label{kappa_tot}\n \\kappa=\\kappa_{\\mathrm{kin}}(1-\\Sigma)+ \\kappa_{\\mathrm{coll}}\\Sigma$$\n\nIf we are in the kinetic (unmixed-mode) limit $\\tau_N>>\\tau_R$ then $\\Sigma \\rightarrow 0$ and $\\kappa\\rightarrow\\kappa_{\\mathrm{kin}}$. If we are in the collective (mixed-mode) limit $\\tau_N<<\\tau_R$ then $\\Sigma \\rightarrow 1$ and $\\kappa\\rightarrow\\kappa_{\\mathrm{coll}}$.\n\nDifferent phenomenological behavior can be deduced from the mathematical difference in performing the averages in (\\[conductivity\\_kin\\_dos\\]) and (\\[conductivity\\_coll\\_dos\\]). ",
"This differences are equivalent to add resistivities in serial or parallel, if we interpret the scattering events on a particular mode as a resistance. ",
"This can give physical insight in order to interpret the thermal conductivity behavior in the different regimes. ",
"From Eq. (",
"\\[kappa\\_tot\\]) it can be deduced why all models based on a single approach (KM or VM) fail when extended to a global model in a large range of temperatures. ",
"In this extension they are used in an approximation where they are not supposed to be valid. ",
"With this Eq. (",
"\\[kappa\\_tot\\]), the behavior change is included in the model, extending its applicability to the whole temperature range. ",
"Another remarkable difference is the way to include size effects in both expressions. ",
"This is discussed on the next section.",
"\n\nSize-effects on the kinetic and collective terms\n------------------------------------------------\n\nIn an infinite semiconductor sample at near room temperature one can consider that only impurities scattering and umklapp scattering participate significantly, then by means of the Mathiessen’s rule $$\\label{mat_bulk}\n\\tau^{-1}_{R_\\omega}=\\tau^{-1}_{I_\\omega}+\\tau^{-1}_{U_\\omega} \\quad.$$\n\nRelaxation times allow to calculate a related term, the phonon mean free path $\\ell$, that is the product between the relaxation time of a mode and its group velocity $\\ell=v_g\\tau$. If the dimension of the system is finite and the temperature is low, intrinsic mean free paths can be larger than the size of the system. ",
"In this case, boundary effects need to be included.",
"\n\nIn the kinetic regime of the thermal conductivity, as the phonons behave individually, each mode could experience independently a scattering with the boundary. ",
"Then, an extra term considering this effect should be included in the kinetic term of Eq. (",
"\\[kappa\\_tot\\]) by using the Mathiessen’s rule in combination with the intrinsic events, this is $\\tau_{B_\\omega}$ the relaxation time due to boundary scattering $$\\label{mat_boundary}\n\\tau^{-1}_{R_\\omega}=\\tau^{-1}_{I_\\omega}+\\tau^{-1}_{U_\\omega}+\\tau^{-1}_{B_\\omega} \\quad .$$\n\nHowever, in the collective term some caution has to be taken. ",
"In this regime a scattering rate is a quantity describing the distribution globally. ",
"In other words, one cannot assume an extra scattering term in each mode independently because the boundary is noticed by the whole phonon collectivity. ",
"Thermodynamically, this is the same situation as flow on a pipe. ",
"Carriers in the center of the pipe notice the boundary not by themselves but through the collisions with the rest of the particles. ",
"The net effect on the flow is the reduction of the flow on the surface. ",
"The usual solution for this situation is to assume that the flow on the surface is zero. ",
"This is feasible if surfaces are rough enough. ",
"Once imposed this extra assumption, a geometrical factor $F$ depending on the roughness and the transversal size of the system should be included in the collective term of Eq. (",
"\\[kappa\\_tot\\]). ",
"In the work by Guyer and Krumhansl[@Guyer1966a] this factor is calculated for a cylindrical shape. ",
"In order to generalize the geometrical factor to account for several geometries and so extend the range of validity of the collective term from bulk to small size samples, we used an expression derived in a previous work[@Xavi2007] $$F(L_{\\rm eff})=\\frac{1}{2\\pi^2}\\frac{L_{\\rm eff}^2}{\\ell^2}\\left(\\sqrt{1+4\\pi^2\\frac{\\ell^2}{L_{\\rm eff}^2}}-1\\right)\\label{factor_F}\\quad ,$$ being $\\ell$ the phonon mean free path and $L_{\\rm eff}$ is the effective length of the system. ",
"By geometrical considerations it can be deduced[@Ziman1979; @Zhang2007] that $L_{\\mathrm{eff}}=d$ for nanowires of diameter $d$, $L_{\\mathrm{eff}}=\\sqrt{\\pi/2}L$ for square wires of size $L$ and $L_{\\mathrm{eff}}=2.25h$ for thin layers of thickness $h$. Expression (\\[factor\\_F\\]) was obtained in the framework of the Extended Irreversible Thermodynamics[@EIT1993] and includes in its derivation higher order terms into the BTE expansion, which can be important when the size of the samples are of the order of the phonon mean free path and it has some advantages: it is analytical, it can be used for different geometries and it takes automatically into consideration the degree of non-equilibrium present in the sample depending on the normal and resistive relaxation times. ",
"Regarding the mean free path, from the works by Alvarez *et al.*[@Xavi2007] and Guyer-Krumhansl[@Guyer1966a] it can be easily deduced that $\\ell=v_g\\sqrt{\\langle\\tau_N\\rangle\\langle\\tau_R^{-1}\\rangle^{-1}}$, reminding that mean relaxation times are calculated from Eq. (",
"\\[k\\_kin\\_int\\]).",
"\n\nFinally, the thermal conductivity for small size samples would be $$\\label{kappa_final}\n\\kappa=\\kappa_{\\rm{kin}}(1-\\Sigma)+\\kappa_{\\rm{coll}}\\Sigma F(L_{\\rm eff})$$ Note that if $\\ell/L_{\\mathrm{eff}}\\rightarrow 0$ ($\\ell \\ll L_{\\mathrm{eff}}$), then $F(L_{\\rm eff})\\rightarrow 1$ and we recover Eq. (",
"\\[kappa\\_tot\\]). ",
"In the opposite limit, $\\ell/L_{\\mathrm{eff}}\\rightarrow \\infty$ ($\\ell \\gg L_{\\mathrm{eff}}$), $F\\sim L_{\\mathrm{eff}}/\\pi\\ell\\rightarrow 0$.\n\nNext, we will test the validity of our model by applying it on different silicon samples since Si is a well-characterized semiconductor in the literature. ",
"This requires to calculate previously its dispersion relations, DOS, and relaxation times.",
"\n\nSilicon dispersion relations and density of states\n==================================================\n\nThe bond charge model (BCM) proposed by Weber [@Weber1974] provides accurate and complete dispersion relation for group IV semiconductors, III-V and II-VI compounds, and they can be obtained with a minimum set of force constants, actually 4 parameters for Si, and 5 in the case of III-V or II-VI compounds.[@Camacho1999] Furthermore, the BCM reproduces very well the transversal acoustic phonon branches close to the border of the Brillouin zone while other models with much more parameters are not able to do it. ",
"The use of the complete dispersion relation includes the role of optical phonons on $\\kappa$, neglected in the Debye approximation.",
"\n\n\\\n\nIn Fig. ",
"\\[fig:RD\\] we show the dispersion relations and DOS we have computed with the BCM. ",
"The dispersion relations fit very well the neutron experimental data,[@Nilsson1972] as the *ab initio* calculations performed by Ward and Broido.[@Broido2010] The DOS calculation agrees also very well with the literature.",
"\n\nSilicon relaxation times: dependence with frequency and temperature\n===================================================================\n\nExpressions for the relaxation times are also needed to compare with data. ",
"We have chosen simple expressions in order to show that even in this case they lead to remarkable predictions. ",
"The use of more accurate expressions obtained for example by *ab initio* calculations will lead for sure to better fits.",
"\n\nIn the case of impurity scattering (or mass defect), we use the expression $$\\label{tauI}\n\\tau^{-1}_{I_\\omega}=\\frac{\\pi}{6}V\\Gamma\\omega^2 D_{\\omega}\\quad ,$$ being $V$ the atomic volume and $\\Gamma= \\sum_i f_i\\left(\\Delta M/M\\right)^2 $ the mass-fluctuation factor, with $f_i$ the isotopic fraction. ",
"This expression is given by Tamura[@Tamura1983] and it is obtained from second-order perturbation theory for diamond-like materials. ",
"This general expression recovers the conventional expression given by Klemens[@Klemens1955] under Debye model conditions , *i.e.* $\\tau^{-1}_{I_\\omega}=A\\omega^{4}$ with $A=V\\Gamma/(4\\pi v_{g}^{3})$. The advantage of Eq. (",
"\\[tauI\\]) is that gives us a calculated relaxation time, with no fitting parameters.",
"\n\nFor boundary scattering, we use the usual expression[@Casimir1938; @Berman1953; @Berman1955] $$\\tau^{-1}_{B_\\omega}=\\frac{v_{g}}{L_{\\mathrm{eff}}}\\quad ,\n\\label{tauB}$$ where $L_{\\mathrm{eff}}$ is the effective length of the sample and $v_{g}$ again the group velocity calculated from the dispersion relations.",
"\n\nThe relaxation times for N- and U-processes will be taken, in the intermediate temperature range, from those provided by Ward and Broido,[@Broido2010] which fit their *ab initio* calculations. ",
"We have modified their expressions of $\\tau_{U}$ and $\\tau_{N}$ in order to extend them to the low and high temperature regimes, respectively. ",
"As shown by Herring,[@Herring1954] N-scattering at low temperatures must be of the form $\\omega^{n} T^{5-n}$, $n$ being an integer, while at the high temperature region it should follow a $T^{-1}$ law. ",
"Since the expression provided by Ward and Broido does not follow the right temperature dependence at high temperatures, we have included the additional term $1/(B' _{N}T)$. In this way, the expression will be valid in the whole temperature range $$\\tau_{N_\\omega}=\\frac{1}{B' _{N}T}+\\frac{1}{B_{N}T^{3}\\omega^2[1-\\exp(-3T/\\Theta_D)]}\\quad .",
"\n\\label{tauN}$$ where $\\Theta_D$ is the Debye temperature. ",
"Concerning U-processes, following the argument provided by Ziman,[@Ziman1979] at low temperatures the scattering of two phonons with wave vectors $\\bm q_1$ and $\\bm q_2$ cannot provide $\\bm q_3+\\bm G$, with $\\bm G\\neq 0$, since low temperature means low energy or low $\\bm q_i$. In other words, U-processes are not possible at low temperature. ",
"We have established a temperature limit assuming that, for $\\bm q_U=2\\pi/3a$ ($a$ being the lattice parameter of Si), $1/3$ the limit of the Brillouin zone, the probability of U-processes decreases exponentially. ",
"The temperature limit $\\Theta_U$ is calculated through the expression $\\hbar\\omega_{\\bm q_U}\\approx k_B\\Theta_U$. For Si we obtain $\\Theta_U\\approx 100$ K. The final expression for U-processes is: $$\\tau_{U_\\omega}=\\frac{\\exp(\\Theta_U/T)}{B_U \\omega^4T[1-\\exp(-3T/\\Theta_D)]}\\quad .",
"\n\\label{tauU}$$ At high enough temperatures, the numerator of Eq. (",
"\\[tauU\\]) is 1 and we recover Ward and Broido’s expression.",
"\n\nResults and Discussion\n======================\n\nIn the following, we compare our predictions with experimental data on silicon samples of different sizes at a large temperature range. ",
"For calculations we use Eq. (",
"\\[kappa\\_final\\]) where (\\[conductivity\\_kin\\_dos\\]) and (\\[conductivity\\_coll\\_dos\\]) are the corresponding kinetic and collective terms, (\\[sigma\\]) is used for the switching factor and (\\[factor\\_F\\]) for the form factor. ",
"In all these expressions we use expressions (\\[tauI\\])-(\\[tauU\\]) for the relaxation times. ",
"Group velocities are always calculated from dispersion relations. ",
"Results using Eq. (",
"\\[kappa\\_final\\]) and (\\[tauI\\])-(\\[tauU\\]) are plotted in Fig. ",
"\\[fig:natural\\_bulk\\_conductivity\\] and compared to data from the work by Inyushkin *et al.*[@Inyushkin2004] for natural and enriched 99.983$\\%$ $^{28}$Si ($^{\\rm na}$Si and $^{\\rm iso}$Si). ",
"The parameters $B_N, B_{N}$ and $B_U$ appearing in the phonon-phonon relaxation times (\\[tauN\\])-(\\[tauU\\]) are obtained by fitting $^{\\rm na}$Si sample and their values are shown in Table \\[table:parameters\\]. ",
"The same values are used for the enriched sample. ",
"The remaining scattering rates (\\[tauI\\])-(\\[tauB\\]) are free of adjustable parameters. ",
"Both samples ($^{\\rm na}$Si and $^{\\rm iso}$Si) are reported to have the same effective size $L_{\\rm eff}=2.8$ mm. ",
"We have used a mass-fluctuation factor of $\\Gamma_{^{\\rm na}\\rm Si }=2.01\\times10^{-4}$ and $\\Gamma_{^{\\rm iso}\\rm Si}=\\Gamma_{^{\\rm na}\\rm Si}/625=3.2\\times10^{-7}$ respectively. ",
"Note that the position of the peak for both $^{\\rm na}$Si is correctly fitted and for $^{\\rm iso}$Si is correctly predicted (solid lines overlap experimental points in the plot) being the only change between both samples the calculated mass-fluctuation factor. ",
"This is a proof of the consistency of our model and confirms the prediction given by Inyushkin *et al*.[@Inyushkin2004]. ",
"In the following subsection we have done the same test for Callaway and pure RTA models, obtaining worse results (see Fig. ",
"\\[fig:comparison\\]).",
"\n\nIn Fig. ",
"\\[fig:natural\\_bulk\\_conductivity\\] we also show the limiting curves corresponding to kinetic $\\kappa_{\\rm kin}$ and collective regime $\\kappa_{\\rm coll}$ for $^{\\rm na}$Si according to Eqs. (",
"\\[conductivity\\_kin\\_dos\\]) and (\\[conductivity\\_coll\\_dos\\]) respectively. ",
"It can be seen that in the low temperature range the sample is entirely in the kinetic regime, since boundary is expected to dominate over normal scattering. ",
"$\\kappa$ tends to the collective regime as temperature rises and N-processes begin to be dominant. ",
"In the collective regime all the phonons notice the scattering events suffered by the rest of the collectivity, thus the thermal conductivity is significantly lower than in the kinetic regime. ",
"At this point one can notice the first important implication of the present formulation. ",
"Both limits contain only resistive terms in their integrals, but $\\kappa_{\\rm coll}$ is less conductive than $\\kappa_{\\rm kin}$. This seems to be in contradiction with the fact that $\\kappa_{\\rm coll}$ is governed by normal scatterings and this has a non-resistive nature. ",
"Actually the ability of N-processes at distributing the energy between modes enhances the resistive character of the rest of the scattering mechanisms. ",
"This physics can be understood thanks to the different mathematical treatment of the relaxation times inside the integrals, interpreted in terms of serial and parallel resistivities in Sec. ",
"\\[sec:thermal\\_conduct\\]. ",
"Our model allows to understand this unlike Callaway model where normal scattering is considered inside the resistive integrals. ",
"Another remarkable behavior is the dominance of normal scattering even at room temperature. ",
"One can expect umklapp processes to dominate at high temperatures, but it can be seen that is not the case of bulk silicon at room temperature. ",
"It can be observed in Fig. ",
"\\[fig:natural\\_bulk\\_conductivity\\] the curves seem to suggest a change in the tendency at high temperature regime. ",
"$\\kappa$ seems to tend to a more kinetic behavior at very high temperature. ",
"The temperature range where kinetic regime happens at high temperature will depend on the height of the dispersion relations that eventually determines the importance of umklapp respect to normal scattering.",
"\n\nThe transition from one regime to another is determined by $\\Sigma$, shown in Fig. ",
"\\[fig:sigma\\]. ",
"At very low temperatures, the boundary scattering present in $\\tau_{R}$ behaves as $\\tau_{B}\\sim L_{\\rm eff}\\ll\\tau_{N}$ and yields $\\Sigma=0$, we are clearly in the kinetic regime $\\kappa\\sim \\kappa_{\\rm kin}$. At room temperature we can easily calculate the ratio of $\\tau_N/\\tau_U$ (neglecting all other scattering mechanisms) and realize that it is of the order of 0.1, thus $\\Sigma\\approx 1$ (actually $\\Sigma= 0.9$) and we are in the collective regime, $\\kappa\\sim \\kappa_{\\rm coll}$. It can be observed in Fig. ",
"\\[fig:sigma\\] that for $^{\\rm iso}$Si the transition to the collective regime is sharper than for $^{\\rm na}$Si. ",
"This is due to the fact that, for these samples, the transition happens in the region of impurity scattering dominance.",
"\n\nResults for silicon TFs and NWs are shown in Figs. ",
"\\[fig:conductivityTF\\] and \\[fig:conductivityNW\\] respectively. ",
"TFs are those from the work by Asheghi *et al.*[@Asheghi1998], with thicknesses of $h=1.6 \\mu$m, 830 nm, 420 nm, 100 nm and 30 nm. ",
"Their respective effective lengths are thus $L_{\\rm eff}=3.6 \\mu$m, 1.87 $\\mu$m, 945 nm, 225 nm and 67.5 nm. ",
"NWs are those from the work by Li *et al.*[@Li2003] with diameters $d=$ 115 nm, 56 nm, 37 nm and 22nm. ",
"In this case $L_{\\rm eff}$ is equivalent to the diameters. ",
"The rest of the parameters remain the same as in the case of $^{\\rm na}$Si bulk. ",
"It can be observed that all curves are in good agreement with the experimental data with the exception of the thinnest NW (22 nm) and in some intermediate temperature region for the 37 nm NW. ",
"Note that all these samples may contain a certain concentration of impurities due to fabrication process[@Asheghi1998], but we have maintained the mass-fluctuation factor $\\Gamma_{^{\\rm na}\\rm Si}$ for all the nanoscale samples because there is no reported data about this question. ",
"From the plots we can confirm that Eq. (",
"\\[kappa\\_final\\]) is able to correctly describe thermal conductivity behavior for general geometries and sizes without the inclusion of confinement effects above an effective size of 30 nm.",
"\n\nFurthermore, it is clear from Fig. ",
"\\[fig:sigma\\] that the smaller $L_{\\rm eff}$ the more kinetic $\\kappa$ is. ",
"This is reasonable and expected, since at reduced sizes boundary scattering rate should contribute the most to thermal resistance not only at low temperature but also at room temperature. ",
"The size effects are illustrated through the form factor $F(L_{\\rm eff})$ plotted in Fig. ",
"\\[fig:factorF\\].",
"\n\n![(",
"Color online) Total thermal conductivity as a function of temperature in a double logarithmic plot for $^{\\rm na}$Si and $^{\\rm iso}$Si, as a result of fitting Eq. (",
"\\[kappa\\_final\\]) to experimental data from Inyushkin *et al.* [",
"@Inyushkin2004], with fitting parameters shown in Table \\[table:parameters\\]. ",
"Kinetic and collective thermal conductivity regimes for $^{\\rm na}$Si are also plotted in dashed lines.[]{data-label=\"fig:natural_bulk_conductivity\"}](bulks.eps){width=\"45.00000%\"}\n\n -------------------------- ----------------------- ---------------------- -----------------------------\n $B_U$ (s$^3$K$^{-1}$) $B_{N}$ (sK$^{-3}$) $B'_{N}$ (s$^{-1}$K$^{-1}$)\n \\[0.5ex\\] This model $2.8\\times 10^{-46}$ $3.9\\times 10^{-23}$ $4.0\\times 10^8$\n \\[0.5ex\\] Callaway model $1.4\\times 10^{-46}$ $3.5\\times 10^{-24}$ $1.0\\times 10^7$\n \\[0.5ex\\] standard RTA $1.9\\times 10^{-45}$ $9.3\\times 10^{-23}$ $3.2\\times 10^5$\n \\[0.5ex\\] \n -------------------------- ----------------------- ---------------------- -----------------------------\n\n : Fitting parameters for $^{\\rm na}$Si bulk\n\n\\[table:parameters\\]\n\n![(",
"Color online) Thermal conductivity of different thicknesses (1.6 $\\mu$m, 830 nm, 420 nm, 100 nm and 30 nm) Si thin films as a function of temperature in a double logarithmic plot. ",
"Model predictions are shown in lines according to the legend. ",
"Experimental data[@Asheghi1998] are shown in symbols also according to the legend. ",
"$^{na}$Si bulk thermal conductivity is plotted for reference (black dashed line).[]{data-label=\"fig:conductivityTF\"}](k_TF.eps){width=\"45.00000%\"}\n\n![(",
"Color online) Thermal conductivity of different diameter Si nanowires as a function of temperature in a double logarithmic plot. ",
"Model predictions for different diameters (115 nm, 56 nm, 37 nm and 22 nm) are shown in lines according to the legend. ",
"Experimental data[@Li2003] are shown in symbols also according to the legend. ",
"$^{na}$Si bulk thermal conductivity is plotted for reference (black solid line).[]{data-label=\"fig:conductivityNW\"}](k_NW.eps){width=\"45.00000%\"}\n\n![(",
"Color online) Switching factor $\\Sigma$ as a function of temperature in a semilogarithmic plot for $^{\\rm na}$Si and $^{\\rm iso}$Si bulk, 830nm TF and 115nm NW.[]{data-label=\"fig:sigma\"}](Sigma_multi.eps){width=\"45.00000%\"}\n\n![(",
"Color online) Geometric factor $F(L_{\\rm eff})$ as a function of temperature in a semilogarithmic plot for $^{\\rm na}$Si and $^{\\rm iso}$Si bulk, 830nm TF and 115nm NW.[]{data-label=\"fig:factorF\"}](factorG_multi.eps){width=\"45.00000%\"}\n\nIt can be noticed from the plots that if one hopes to fit the experimental values with a pure kinetic expression, extra thermal resistivity should be added in room and high temperature regions to reduce the predicted values. ",
"The presence of the collective term in our Eq. (",
"\\[kappa\\_final\\]) makes unnecessary this adjustment. ",
"On the contrary, our model explains why VM models should give poor results at low temperatures. ",
"The boundary term leads the system to a kinetic regime at these temperatures, raising the thermal conductivity.",
"\n\nObviously, our phenomenological expressions for the relaxation times cannot be used to obtain an extremely accurate fit. ",
"Further improvements of the model can be achieved by a more precise treatment of scattering times through *ab initio* techniques, but we have demonstrated that some issues related to relaxation times come from their incorrect averaging. ",
"We can conclude that an appropriate treatment of the N-processes makes unnecessary the introduction of new terms in the expression of $\\kappa$. Probably rough surfaces[@Hochbaum2008] would need additional considerations to improve the fit but this is out of the scope of the present work.",
"\n\nComparison with other models\n----------------------------\n\n![(",
"Color online) Thermal conductivity $\\kappa$ as a function of temperature for $^{\\rm na}$Si and $^{\\rm iso}$Si bulk, and 115nm NW. ",
"Best fit provided by standard RTA and Callaway models against this work model.[]{data-label=\"fig:comparison\"}](GKvsC.eps){width=\"45.00000%\"}\n\nIn order to show the improvement of our model over standard RTA and Callaway model[@Callaway1958], we compare our results with those obtained with these usual approaches. ",
"The procedure we have followed to fit $^{\\rm na}$Si is the same as in our approach. ",
"The same relaxation times equations (\\[tauB\\])-(\\[tauU\\]) are used in the three approaches to highlight only the models accuracy. ",
"The values of the fitting parameters that provide the best results for $^{\\rm na}$Si in each approach are shown in Table \\[table:parameters\\]. ",
"Then, to test the prediction capability, we have changed the mass-fluctuation factor for ($^{\\rm iso}$Si and the effective size for the 115nm NW. ",
"Results are provided in Fig. ",
"\\[fig:comparison\\].",
"\n\nAs expected, RTA reproduces very well $^{\\rm na}$Si in the low temperature range, but from $T>200K$ begins to diverge from experimental data. ",
"However it underpredicts the $^{\\rm iso}$Si peak and from this point forward. ",
"At the nanoscale it also fails in the prediction as shown in the plot. ",
"On the other hand, although Callaway model is able to reproduce correctly $^{\\rm na}$Si sample, it overpredicts $^{\\rm iso}$Si and the 115 nm nanowire.",
"\n\nIn the literature we can find two kind of approaches. ",
"Firstly we have models that focus on the fitting to natural and isotopically enriched bulks in the whole or partial temperature range, but they are not proved at the nanoscale.[@Morelli2002] On the opposite way, we can find models focused on the fitting to the nanoscale giving a good agreement with measurements but they are not proved at reproducing other isotopic composition bulks.[@Mingo2003; @Chantrenne2005] Since providing a good fit at the peak region for both bulks is very difficult, most of the published models do not show the corresponding temperature interval. ",
"Normally they show fits and predictions from $T>50K$. With 4 simple and representative scattering events (boundary, impurities, normal and umklapp) our model is able to provide a very satisfactory fit from the macro to the nanoscale in the whole range of temperatures.",
"\n\n![(",
"Color online) Thermal conductivity of all the silicon samples studied in this work (Bulk, thin films and nanowires). ",
"It can be seen that a very good global agreement is obtained at all ranges of size and temperature.[]{data-label=\"fig:conductivity_tot\"}](k_total.eps){width=\"\\columnwidth\"}\n\nIn Fig. ",
"\\[fig:conductivity\\_tot\\] we show the global prediction achieved by our model, with this plot one can notice in a single view how the thermal conductivity works for the complete set of different size, shape and composition Si samples in the \\[1-1000\\]K temperature interval.",
"\n\nConclusions\n===========\n\nThis work shows that the key point for an accurate description of the thermal conductivity in the whole range of temperatures is taking into account the effect of normal processes on the phonon collective behavior. ",
"As a consequence two well differentiated thermal transport regimes are studied for the first time, kinetic and collective, depending on the relative importance of normal processes.",
"\n\nThe proposed model gives an expression of $\\kappa$ valid for all ranges of temperatures. ",
"This expression is obtained by combining the VM and RTA approaches including a switching factor that determines the transport regime in terms of the normal and resistive mean scattering times. ",
"In these regimes, differences in the phonon averaging and in the way to account for the boundary effects are considered.",
"\n\nWe have also included higher-order non-equilibrium effects through an analytical function $F(L_{\\rm eff})$ to generalize the model to any kind of sample depending on its geometry and characteristic size. ",
"The obtained results agree very well with experimental measurements of different Si samples of characteristic length above 30 nm, proving that above this size quantum confinement effects are not necessary to explain thermal transport.",
"\n\nThe authors acknowledge financial support from projects CSD2010-00044, FIS2012-32099, MAT2012-33483, and 2009-SGR00164, and from a Marie Curie Reintegration Grant. ",
"The authors thank Profs. ",
"D. Jou, J. Camacho and J. Bafaluy for fruitful discussions and M. M. de Lima Jr for a critical reading of the manuscript. ",
"Thanks are also given to the Red Española de Supercomputación providing access to the supercomputer TIRANT.",
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"Readers' comments\n\nThe issue is not a lack of US influence in Egypt, but that such influence has not been used properly, whether with respect to Mubarak, Morsi or the new \"civilian\" government (aka the military). ",
"Instead of viewing Egypt through the vague prism of grandiose geopolitics, the United States should be treating Egypt for what it is -- a developing country with 85 million individuals situated on the shores of the Mediterranean and Red Seas. ",
"Egypt is not an army, not a religion and not a transient ideology. ",
"Use US leverage to open Egypt up to its potential - back the IMF loan, forge a free trade agreement and encourage more non-political exchanges. ",
"The rest will take care of itself. ",
"http://balasticman.blogspot.it/2013/07/now-what-to-do-with-egypt.html\n\nThere is an insanity in America that if you only give enough guns to everyone, that there will be peace in the world and no crime or tyranny in the homeland.",
"\n\nPresident Obama is the most competent Commander in Chief I've known; a President whose courage to remain silent before his accusers while contemplating the advice of his commanders, balancing the military cost against political reality, and always putting country before politics. ",
"He is respected by those he respects for the bravery, duty and honor in serving this nation faithfully.",
"\n\nI fought in Vietnam, 5th Special Forces, Detachment B-36, 3rd Mobile Strike Force, and trust President Obama to navigate and avoid the Middle East religious war quagmire given our national interests and to resist the pressure from the trigger-happy conservatives who want to engage militarily in every conflict.",
"\n\ndslaby,\n.",
"\nYou are the proof that some, at home and abroad, applaud Mr Obama’s caution as Egypt fumbles its way towards greater democracy. ",
"The world appreciates that you fought in Vietnam, 5th Special Forces, Detachment B-36, 3rd Mobile Strike Force. ",
"Unfortunately, that distinction does not entitle you to speak for those whose experiences are more recent or geographically relevant.",
"\n.",
"\nWhat on earth do \"trigger-happy (2nd amendment) conservatives\" have to do with Obama's obvious foreign policy blunders? ",
"At this point I feel sorry for U.S. Secretary of State John Kerry, also a Viet Nam vet. ",
"I don't know how he'll ever repair the damage done by the most competent Commander in Chief you've ever known.",
"\n\nI would disagree regarding the relevance: Vietnam was mostly about reassuring Europe that America would keep its defense commitment in case of Soviet invasion. ",
"The recent experiences in South Asia and Middle East also reflect the strategies such as counter-insurgency and strategic national interests defined in oil security. ",
"I'm interested in your identifying the \"obvious\" foreign policy blunders of President Obama.",
"\n\nSometimes the U.S. is blamed for doing something or not doing something. ",
"In Egypt it is blamed no matter what.",
"\n\nWe've spent 50 years trying to get people in this part of the world to not kill each other. ",
"Why? ",
"Oil. ",
"The dithering that enrages me is all those years of doing nothing about U.S. foreign oil dependence.",
"\n\nAll that had to happen is for a tax to keep oil prices as high, adjusted for inflation, as they are now. ",
"High enough to encourage domestic production, conservation, and lifestyle changes and alternatives. ",
"The particular mix would be determined by circumstances and the market.",
"\n\nThat is what is happening now. ",
"That is what started to happen twice before, but then a plunge in oil prices and the lure of cheap and easy brought a stop to it, leaving a part of the world that blames us for all their problems no matter what with a boot on our economic throat.",
"\n\nWhat I want Obama to do is institute a tax on oil and related products, produced or imported, that rises as the price falls, keeping the price where it is now plus inflation. ",
"That's the right response to all of this.",
"\n\nMr Obama has done well to 'dither' over the Arab Spring. ",
"Democracy should come to the Middle East through the will and toil of its people, not from the guns of the American military. ",
"Only then will it be lasting\n\nYou think maybe between doing nothing and guns there might be a middle ground Just Passing Through? ",
"Like having a coherent policy for example. ",
"I love the black and white thinking. ",
"Maybe the world is in fact a shade of gray. ",
"You think?",
"\n\nHmm. ",
"I did phrase that rather badly it. ",
"Of course I do think there is a middle ground between that but what can America do anyway? ",
"One of the top posts said it better that aiding one side only leads to the other hating America. ",
"What would you consider a coherent policy for this mess?",
"\n\nWhat would you call this policy? \"",
"U.S. plan for what we think another country should do during a period of uncertain transition, where it's not clear who we should side with, and what exactly the benefits of having this coherent policy are.\"",
"\nThe world is not just a shade of gray, it is infinite shades of gray with countless blends of white and black. ",
"Maybe we are all dreamers.",
"\n\nEgypt's military coup was planned. ",
"Months before the coup, Egypt's military generals and opposition leaders were meeting at the Navy Club to conspire the coup. ",
"The plan was that if opposition leaders create enough protests, the generals will bring down the Morsi presidency. ",
"The frequent power-blackouts and petrol shortages were part of the coup to make people angry at Morsi and protest. ",
"Power-blackouts and petrol shortage disappeared immediately after the coup. ",
"This is a clear sign of coup conspiracy.",
"\nSome argue that the coup was planned by America and ordered to the Egyptian military. ",
"After all, Egyptian military is controlled by America. ",
"After all, the coup was Americanplot!",
"\n\nThe more there is no proof of a conspiracy, the more it must be a conspiracy, right. ",
"I am glad we have such great analysis from someone who so clearly is in the know on all the events here. ",
"Thanks for sharing your great knowledge.",
"\n\nThe US is a democracy, with a large well organised Jewish lobby, who denounce anything the US does that is against the policies of Isreal. ",
"There is no comparable lobby for the arabs or muslims. ",
"So it's not so strange and unfathomable.",
"\n\nDespite this the US does provide significant aid to the muslim world, of which Egypt is a major recipient. ",
"In neither case does this aid seem to 'buy' that country. ",
"Isreal still bulldozes and builds its own communities on the proposed basis for a state of Palestine and most muslims still blame the US for all their problems. ",
"Hey ho.",
"\n\nCall me a dreamer, and consider this post fairly useless, but I will continue nonetheless. ",
"Could we ban military aid between countries? ",
"What good does it accomplish for a nation to help arm another nation? ",
"How many schools, roads, sewers, or bridges could be built with the billions that are sent as military aid to poverty plagued countries. ",
"Well, we could lose jobs at U.S. defense contracting companies, but maybe we could pay people to build desks, bulldozers, effluent pumps, or rivets. ",
"The world would be a better place. ",
"If a country needs military aid, let them apply to the UN or some other worthy non-nation institution, and pay for peacekeepers or a little hardware as determined necessary for their particular need.",
"\nMaybe I'm ranting, and I'm not sure if I am adding positive intelligent commentary or not-so I'll stop now.",
"\n\nEgyptian government and civilians' attitudes toward the U.S. do not really matter, they can howl anti-America slogans all day and night, as long as America can buy influence over the Egyptian military. ",
"The sad thing is, as I said before, only $1.5 billion is enough to buy sufficient influence over the Egyptian military. ",
"With the Egyptian military under control, the U.S. can change the civilian leadership at will through the military, and install whoever that will satisfy America's demands, just like what they did to the Morsi government - democracy be damned. ",
"Do not be fooled by the anti-America slogans from either the Egyptian military or government, they are meant for domestic audience only to garner their support. ",
"Behind the scene, the Egyptian military had their balls firmly in the hands of America.",
"\n\nActually, I think the Egyptian military cares less about what the U.S. government thinks and more about protecting its interests in the Egyptian economy and political sphere. ",
"Al-Sisi and company most likely want to play the neutral kingmaker to ensure no one will disrupt the comfortable economic base Egyptian military officers have created for themselves.",
"\n\nExplain why our tax dollars are going to arm an unstable country with a violent christophobic and anti-semitic majority? ",
"What possible good can come from this insane policy? ",
"Will it make them \"Like us\"?",
"\n\nWell, Egypt is arguably the most powerful, and most populated, country in the Middle East, and the first to sign a peace treaty and recognize Israel's right to exist. ",
"Without Egyptian military help, it's virtually impossible for the other countries there to successfully invade Israel or otherwise threaten it's existence without using nuclear weapons, which none of them currently have. ",
"Egypt was stable under a dictatorship friendly to the US for decades, it's only recently become unstable.",
"\n\nIn reality, we help Egypt to help Israel. ",
"This's why Obama and crew were hesitant to welcome a democratic revolution there, because the islamists who will invariably win those elections might drop Egypt's support for the treaty.",
"\n\nGeopolitics and billions in aid aren't given to people for them to \"like us\" - it has very real and tangible uses, and countries don't always have the luxury of only being buddies with others that completely agree on every issue.",
"\n\nInvade Israel? ",
"With help from the Egyptians? ",
"Now that is a very funny joke. ",
"The US will fight it's foreign wars on Israeli soil before it fights them in US soil. ",
"The Israelis may be cunning and scheming, but the Anglo is too much for them.",
"\n\nAmong the largest regular transfers of cash in World Diplomacy is America's annual $1.5 Billion tribute paid to Egypt.",
"\n***Remember the Marshall Plan to reconstruct Western Europe after WWII was $1 Billion in TOTAL.",
"\n\nIT is nominally paid as ransom to maintain the fragile Israeli-Arab Peace:\nan extortion fee for Arabs to not attack.",
"\n\nIt has become an Egyptian entitlement that they expect with NO strings attached.",
"\nToday the Egyptian Army accounts for 30% of Egyptian GDP.",
"\nThe American Aid has a lot to do with it.",
"\n\nCaesar himself could not get such generous tributes.",
"\nEgypt is the New Rome.",
"\n\n\"Caesar himself could not get such generous tributes.",
"\nEgypt is the New Rome.\"",
"\n\nYou've written some non-sense before, but this tops it. ",
"Comparing the rebuilding of Western Europe to military aid to Egypt is just absurd. ",
"The Marshall Plan was to REBUILD, not to develop. ",
"Very different concepts. ",
"Western Europe already had all of their institutions in place, we just literally had to rebuild the buildings destroyed. ",
"Aid to Egypt and other nations is meant to either stabilize or develop institutions that are either very corrupt and need an overhaul or to develop them where they are lacking.",
"\n\nThere are plenty of strings attached to the deal. ",
"These are not grants we give them, it is aid in which they have to use by buying US military equipment or training. ",
"Most of that money comes right back to us. ",
"Therefore, it is more of a subsidiary to our own military industry than anything else.",
"\n\nPlease be more careful when posting. ",
"There is already so much non-sense in the world, and we really don't need any more of it.",
"\n\nWhen a country is in the middle of a revolution/civil war, the last thing any foreign power should do it is to influence the outcome. ",
"Because when the people are 50/50 split down the middle, you will end up with half of the populating hating you no matter which side you support. ",
"It happened in Russia to those who backed the Whites, happened in Iran to all who backed the Shah and it is happening now to the fools who backed the Brotherhood.",
"\n\nIt really blows my mind that a post can shift from talking about how people blame the US to US influence. ",
"The problem for me is that talking about the US and what it can or can't do is a distraction from the reality that Egypt is a large country with nearly 100 million people and that Egyptians make their decisions about their country.",
"\n\nIn other words, show me a country where another country can come in and say, \"Gee, you really need to do this\" with an implicit or else in there. ",
"I can't think of a country. ",
"We used to have examples when the Soviets controlled what was said and done in E. Europe but no one tells Denmark what to do or Guyana or Ecuador or Laos ... so why exactly would Egypt be so susceptible to outside influence when it is a huge place with a vast population?",
"\n\nI think a much more rational and intelligent approach would be to look at the statements and note they are attempts to blame outsiders. ",
"This happens in the US and in every country: it's someone else's fault and look, the other side is aligned with x. Fill that x in with what you want. ",
"Say for example that you tar Democrats by saying they want illegals to flood the country and vote. ",
"Same kind of blame game in which you identify an outsider and say the other side is tied to them. ",
"At least in Egypt there is a connection between the military and the US.",
"\n\nIt gets to be nonsensical. ",
"I recently watched a sermon from Damascus in which their civil war was the fault of Jews.",
"\n\nThat's the beauty of it. ",
"You don't actually need to find a connection. ",
"You can simply proclaim the Jews responsible for your pancakes, and it will be so. ",
"Jews also made me wake up late this morning, forcing me to skip breakfast. ",
"Be lucky you even had burnt pancakes. ",
"The Jews didn't afford me that luxury.",
"\n\nThis post shifts from the realities on the ground in today's Cairo to the distant memory of a speech given there in 2009 (i.e. the American president simply lacks the leverage to influence events in Egypt).",
"\n\n1. ",
"America does not wield as much clout in Egypt (or in other parts of the world) as it did in the past because it is cash-strapped. ",
"America’s billion-ish military aid to Egypt, although on an annual basis, will appear small in Cairo compared to a one-time 12-billion windfall that’s expected from the Gulf monarchies this year. ",
"One might say that the hallmark of an aging, traditional power-broker is the inability to pony up large sums of money in a short period of time because of all other commitments that it has, while the rising powers have the exact opposite attribute. ",
"This was the same with America as a young superpower 70 years ago.",
"\n\n2. ",
"Having said that, America’s influence also obviously suffers from its unpopularity, whether deserved or not. ",
"Respect is the ultimate currency.",
"\n\n3. ",
"The rush, by Egypt’s military government and its Islamist opposition, to claim America as their enemy presents a perverse opportunity for Washington. ",
"A vocal, yet non-tangible American support for either one of these factions can instantly demolish their nationalist credentials and make them lose power. ",
"Has this possibility been considered in America’s foreign policy circles? ",
"Stir some s**t up, poison enemies with your friendship.",
"\n\nRe: point 3\nYou can be certain that people in power are aware of that option. ",
"The conundrum is, though, whether the poisoning of one side is worth the inevitable helping of the other side? ",
"I am guessing that people with empirical knowledge about the Egyptian factions at play - and what kinds of people those factions are made up of, and whom they represent and can mobilize on short order - are less keen on poisoning one side if that means giving unearned/undeserved aid to the other by default. ",
"Pox on both their house, really. ",
"They, the Egyptians, just might get a fleeting glimpse at how democracy really functions in the real world; and then revert back to their old selves...\n\nEgypt remains The Anchor to which Israel is tethered in the Middle East. ",
"The Peace with Israel is one of the bed rocks of US Middle Eastern Policy. ",
"The President's Political Calculus is therefore all around the Regime's Guarantee of that Peace. ",
"Evidently The President has been disavowed of the Notion not that Morsi would not guarantee the Peace, but that he simply could not reach beyond his Base. ",
"The Value of the MB Guarantee was essentially eroding at speed.",
"\n\nOutside Egypt, The President has been remarkably hard-nosed. ",
"The Toppling of Gaddafi via R2P, Bashar's Advances might only be phyrric because the Economy has been pounded. ",
"Iran must be a Pressure-Cooker. ",
"In the Context of the Geopolitical Aim of softening up Iran and its Allies, President Obama has been landing some punches.",
"\n\nThe President is a mercurial Man who could seriously ignite the Arab World but then he would have to bet on the People.",
"\n\nAmerican aid to Egypt is (or at least was initially) tied to equal amounts of aid to Israel. ",
"Does anyone know what the impact of suspending that aid to Egypt would have on the authroization of that aid to Israel?",
"\n.",
"\nAlso, since American aid to Egypt is so comparitively minor, as the article notes, and since Israel hardly needs American aid, why are we spending this money any more anyway?",
"\n\nFirst, they are separate. ",
"They are related in amount. ",
"There is a law about what happens when a coup occurs.",
"\n\nSecond, to be clear, the reasons for military aid to Egypt - separate from other aid - are complex and don't all relate to Israel. ",
"Egypt is not only a US ally but has a military production industry. ",
"The US essentially competes for a role in that business, with competition coming from France and Britain. ",
"The absolute size of the business may be considered for policy purposes less important than the fact it exists. ",
"The Egyptians have also bluntly said they expect the money as payment for keeping peace in the region. ",
"How much that is true today is unknown. (",
"I say that given the close connections between Israel and Egypt on military matters even under the Muslim Brotherhood - and despite mutual hatreds. ",
"They recently agreed, for example, that Egypt would increase troop numbers in Sinai and Egypt has been waging a rather harsh compaign on Gaza tunnels, if only because of attacks from Gaza on Egyptian soldiers.)",
"\n\nThird, it should be noted that Israel doesn't receive economic aid anymore. ",
"Egypt does. ",
"The amount has dropped and varies. ",
"It's been nearly $1B a year and has been as low as $250M a year.",
"\n\nFourth, any discussion of military aid needs to be put in the context of how it works. ",
"Almost all aid is actually \"given\" through the Foreign Military Financing program, which is paying US defense contractors to make stuff that is then handed to other countries. ",
"In other words, it's nearly all a subsidy to US companies. ",
"In absolute terms, it doesn't amount to much compared to other subsidies for corporations. ",
"The argument for FMF is that it keeps the US defense industry strong and reduces cost for the US military by increasing overall production. ",
"There is some truth in this but it's complicated. ",
"As a note, Israel gets perhaps the only actual military grant money as part of its military aid. ",
"The idea behind that is the US benefits from Israeli technology, that the money is essentially an investment. ",
"There is some truth to that but it's also complicated.",
"\n\nI must inquire about your 4th point... is what you say true? ",
"This disturbs me on many levels. ",
"Why would a responsible journalist lump the amount of military \"aid\" in to a cash figure and say this is \"given\" to Egypt when it is so much more complex than that, it leads the reader (or at least me) to believe that the US is making a deposit into the bank account of their Army.",
"\nThis causes me to have so much more anger towards our beloved \"job-creating\" defense contractors. ",
"Oh how we detest those spendmore socialists that want to give money and food stamps to poor Americans, yet to suggest that we stop giving money to U.S. companies to make weapons to give to our \"friends\" in the middle east-I could imagine the outrage. ",
"Couldn't we just revive the CCC?",
"\nWhy would we pay to arm any other country, let alone the ones in the Middles East? ",
"We are the shining light best example of free-market, rule-of-law, capitalism, but we do not actually believe in these things or trust them to work. ",
"Our agriculture is central-planned Soviet style, government mandates and protections create artificial markets and too-big-to-fail.",
"\nThe world would benefit from our rigorous pursuit of open markets, free trade, and general economic liberalization, and we do advocate these things a little. ",
"But unfortunately, we know what really makes the world go round is aggression, domination by force, military might, and of course outspending your ideological enemies on materiel.",
"\n\nU.S. military aid to Egypt, while comparatively minor in purely quantitative terms, is still superior to lump sum cash transfers or guaranteed loans from Middle Eastern countries because the funding also comes with access to U.S. defense technology and training (though that access is not unlimited). ",
"No amount of cash from the Middle East can give you access to some of the best avionics in the world unless you have a defense relationship with the United States, which Foreign Military Financing implies. ",
"This keeps the Egyptian military's equipment partly dependant on the U.S. for supplies, maintenance, and training. ",
"To fly F-16s, for instance, Egyptian pilots would go to the U.S. for training and therefore be exposed to U.S. military doctrine, the culture of logistics, etc. ",
"That doesn't stop the senior-most Egyptian Armed Forces officers from pursuing what's in their own interests, but it keeps open a line of dialogue that the U.S. lacks when it comes to countries with which we don't have a robust defense relationship. ",
"In a way, it also helps the U.S. keep an eye on Egyptian military capabilities, perhaps so they aren't too geared towards conflict with Israel and more towards policing the Sinai or taking out terrorist cells and smuggling routes.",
"\n\nAs for Israel, I don't think it's quite accurate to say they don't need our assistance. ",
"As a small country, Israel's defense industry relies on two sources to fund expensive R&D to maintain its qualitative military edge: 1) exports and 2) multi-year military aid commitments from the U.S., including joint development of missile defense technology like Iron Dome, David's Sling (what an awesome name!), ",
"and Arrow. ",
"If the Israeli defense industry could only rely on the national defense budget and export revenue, then it wouldn't be able to invest in high-tech military equipment for which the country is deservedly renowned.",
"\n\nI agree with you on points 1 and 2 and mostly on 3 (Israel does receive some non-military aid such as resettlement assistance and guaranteed loans, though I don't know when Israel last drew down on the loans).",
"\n\nOn the fourth point, I think it's a bit simplistic to say it's merely a subsidy for U.S. defense contractors. ",
"I agree that it does subsidize purchases for U.S. defense equipment and services, but with it comes access to U.S. technology, training, and maintenance. ",
"It's setting up a defense relationship between the U.S. and Egypt; while this doesn't mean control of Egyptian affairs, it does mean the U.S. has more of a direct line of communication with various levels of the Egyptian military than other countries. ",
"Sometimes it works in the Americans' favor, sometimes not.",
"\n\nNote also that Egypt receives Excess Defense Articles worth hundreds of millions of dollars, something which the defense industry probably doesn't like as much since the equipment transferred from old DoD stocks means less new procurements (though usually the old stuff needs contractor support for refurbishment, etc.). ",
"The U.S. also spends about a million or two each year just on military education.",
"\n\nThat said, the Egyptian military will always protect its own economic interests before bowing to U.S. influence. ",
"They're more interested in making sure their comfortable way of life isn't upended. ",
"In this case, they probably saw siding with the Tamarod movement as the only way to defend their future livelihood.",
"\n\n\"Why would a responsible journalist lump the amount of military \"aid\" in to a cash figure and say this is \"given\" to Egypt when it is so much more complex than that, it leads the reader (or at least me) to believe that the US is making a deposit into the bank account of their Army.\"",
"\n\nI'm not trying to insult you even though it may come off that way - reporters and journalists don't have time or the print space to type out history lessons and explain country-specific nuances in every article when they are justified and correct in simply saying it equates to direct military aid.",
"\n\n...military \"aid\" (which is paid to U.S. contractors to make military gear that is then given to Egypt).",
"\nThat would be enough to clarify I think.",
"\nThe way it is written, I could take the author's statement and in retelling the story say... \"Did you know we give the Egyptian military 1.5 billion dollars?\" ",
"My friend would then say... \"Why are my tax dollars being given to some country in the middle east?\" ",
"I believe it implies something totally different.",
"\nAnd then we come to learn that we actually pay U.S. companies to make missiles, guns, or whatever and then give these wonderful presents to our \"allies\" and we think... Oh, well that is good for American jobs, and it should bring peace to the region, and secure our influence and friendship with the countries in the region...\n\nAccording to the Congressional Research Service, Israel hasn't drawn on any loans guaranteed by the US since 2005. ",
"They still have authorization outstanding but have said there are no plans to do anything with that.",
"\n\nAs far as I can tell, the only non-military aid is $15M designated generally for \"refugees\", meaning originally from the USSR (!) ",
"but now taken to mean immigrants. ",
"I have no idea why that survives. ",
"It's not exactly a large number and can't be important. ",
"It isn't important as an amount to Israel either.",
"\n\nThe US also funds a bunch of other things that give money all over the place, including to institutions in Israel (and all over the Middle East). ",
"I don't include those; these institutions apply for grants and, it seems, are treated like applicants from anywhere.",
"\n\nHere are 2 examples. ",
"First, the US will \"sell\" Israel advanced fighters. ",
"That means they'll pay for them and send them. ",
"The same will be done in whole or in part with other countries. ",
"This extends US power & influence and also extends on a practical basis the production runs for the advanced fighter. ",
"To be blunt, if you don't make a lot, the cost per unit is even higher. ",
"So the actual net cost is somewhat less than the face value - which is an untrustworthy figure anyway.",
"\n\nSecond, the US has a joint missile development program which I believe gets about $100M. This ties into but is also separate from development of the anti-missile systems and ties into but is also separate from the Israeli military drone industry. ",
"It's hard to say how much of this stuff is cooperative and how much is aid and how much is spending to benefit yourself. ",
"That is why I say it's complicated.",
"\n\nBut on the whole, I repeat the most important thing to remember is that military aid is credits in the US Treasury paid out to US companies. ",
"BTW, I think a main purpose in sending Egypt older stuff is that they have a niche in making and marketing older US stuff to other nations. ",
"We want that business to thrive."
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"Tuesday, September 8, 2009\n\nFamiliar Road\n\nI woke up in my tent around one am, and since I've learned that I just don't sleep very soundly in the tent, I just listened to the sounds, and stared up at the stars for a while, hoping to see a meteorite or two. ",
"At one point, I could feel something, a weird sensation around the back of my neck, and then a crawling sensation, as if something were crawling in my neck. ",
"I reached my hand around and felt something insect-like, pulled it away, but then started to get that weird feeling you get when a bunch of bees sting you, sort of a head rush. ",
"Hard to describe. ",
"Then, a definite, sharp pain right at the nape of my neck. ",
"I feel the spot again, and already there's a big knot of swelling forming. ",
"I'm thinking \"what? ",
"Are there tarantulas in Wyoming?\" ",
"I had my screen door totally zipped shut too. ",
"Then I rolled over and-- woke up. ",
"The whole insect biting thing had been a dream, and confirmation that I did in fact fall asleep sometime between two-thirty and three-thirty. ",
"Relief.",
"\n\nI dozed until my alarms started going off (two each in my watch and cell phone). ",
"Quietly got up and walked to the showers. ",
"I hadn't noticed this before, but the men's showers at the Chuckwagon RV Campground have quite an art collection. ",
"There are five versions of the famous \"Dogs Playing Poker\" series, shellacked to pieces of tree bark. ",
"I have always liked those pictures. ",
"Maybe because I was very young when I first noticed them. ",
"I wonder what art the women get?",
"\n\nI still had about an hour of darkness, so I boiled water for coffee as I took down the tent and started packing the trailer. ",
"I knew this was the last time I'd be cooking, so I just let the stove go for a long time, burning off those pounds of white gas.",
"\n\nAnd now for a short commercial break: Did I mention that I have been drinking coffee from my very own Revolution Mugs coffee cup? ",
"If you haven't ordered yours, better get going. ",
"This mug has traveled hundreds of miles with me now. ",
"A much more satisfying weight to drag over mountain passes than the one-pound box of tasteless linguini, or the 6 heavy packs of albacore tuna and chicken breast.",
"\n\nNow back to our story.",
"\n\nToday was super humid, and this morning would be my first IFR departure of the trip. ",
"A heavy but shallow ground fog had formed, just at sunrise, and as I headed east on highway 26, the sun was a huge white disk you could stare at just as easily as the moon. ",
"The road shoulders were smooth, the highway was flat, and the miles just melted by. ",
"As always, rural drivers almost always move over half a lane to pass, even if I'm six feet away in an eight foot shoulder. ",
"I've even seen oncoming cars drive on the far side rumble strip, as if I'm radioactive or something. ",
"This would be amazing behavior in the Puget Sound basin.",
"\n\nHighway 26 is arrow-straight as it leads past these last few towns into Scottsbluff. ",
"Fortunately, the towns are spaced six to eight miles apart, so the entertainment value is a little higher. ",
"I had planned to have a real breakfast in Lingle, but the diner there apparently closed its doors (I was told this by a woman working at the gas station, where I picked up a maple-bar-ish twisty something pastry to tide me over) last summer. ",
"Next town: Torrington, Wyoming, where I found the 77 Grill at a big truck stop. ",
"Apparently the only place open, because it was hopping. ",
"I got my fill and continued down the road. ",
"Torrington is also the town closest to the farm of my Grandparents on my Mom's side. ",
"It's all dirt roads to get out there, and no one left who would know me, so I don't think I'll be riding out that way this time.",
"\n\nThe sequence of little towns down this road is a very familiar and nostalgic path for me, as I have counted my way down these last miles many times from the back seat of my parent's car as we traveled each summer to Scottsbluff. ",
"I remember sometimes we competed to see who could be the first across the state line. ",
"The best way to do this was to be crafty and pretend you weren't really thinking about it, and just happen to be in the front seat when that border was approaching. ",
"That way you could just make sure one of your feet was farther forward than the driver's right foot. ",
"But, you had to be careful, there might be a last-second lunge over the seat backs, and a pair of hands attempting to reach up under the dashboard.",
"\n\nI reached the Nebraska border at ten am, and almost immediately, the town of Henry, with the fading, hand painted sign \"Welcome to Henry, Scottsbluff County, Home to a Undergound Environmental Hazard.\" ",
"Then Morrill, Mitchell, and now I was finally in visual contact with the bluff itself, the national monument which shares its name with the town and county.",
"\n\nSo humid today, the air feels thick and hard to breathe, plus in the last few miles, of course I pick up a little headwind. ",
"Still it's exciting to arrive in this fashion, and I can hardly believe I'm finally here. ",
"One more Twilight Zone (Outer Limits?) ",
"moment just after Mitchell, where the grasshoppers which heretofore had been hopping out of my way as I ride, are suddenly jumping right at me, and onto me from all sides. ",
"Reminded me of that episode where the couple is marooned in the desert overnight, and have to deal with attacks from sage brush, and then frogs.",
"\n\nAs I ride along highway 26, I pass by Sunset Memorial, the cemetery where all my grandparents, and an uncle are buried. ",
"I stop and think about taking a break to go look at the markers, but after watching the traffic (65 speed limit, divided highway), I decide that a visit isn't worth the risk of joining them prematurely and permanently. ",
"I imagine my grandad Dale understanding my decision as I ride away. ",
"A couple of passing cars give a toot and hold out a peace sign, not sure what that's about. ",
"Finally, a right turn off the highway onto fifth avenue, past the Appleby's where a few fun family evenings were spent after my grandmother's funeral a few years back. ",
"A left turn, and... oops, streets are counting opposite the way I anticipated, I'm on sixth, u-turn, back the other way, there's the old Terry mansion, and another landmark, old Ford pickup, and I'm here!",
"\n\nThinking: Shower. ",
"Glass of wine. ",
"Pizza. ",
"Sitting. (",
"Made me think of Borat. \"",
"Look at me, I am sitting on a chair.\") ",
"Send a text to Theo. ",
"Oh, and finally I can make a quick run to the grocery store for shaving implements. ",
"I look like Gabby Whiskers. ",
"No wonder people are afraid of me when I roll into town.",
"\n\n8 comments:\n\nCongratulations, Kevin. ",
"What a feat!It never occurred to me before why they called it Scotts Bluff. ",
"It looks beautiful, in a Nebraska sort of way. ",
"When will you be home? ",
"Chris has been bugging me to have beer. ",
"We wait to drink you.",
"\n\nLu and I have enjoyed the hell out of reading your adventures - especially a big catch up the other day after getting back from our own (infinitely cushier) trip to Alaska. ",
"Lu has enjoyed listening to them - she got the audiobook version from me. ",
"So thanks for the privilege of tagging along on your strange Western odyssey. ",
"You had us on the edges of our tenterhooks a few times there, pal - but I have to say that this latest entry tops them all, with is recurrent David Lynch vibe. ",
"Nice mugs, yours & yours - gabby whiskers! ",
"On to the final post.",
"\n\nAbout Me\n\nI'll be using this blog to update my progress as I ride from Seattle, Washington to Scottsbluff, Nebraska. ",
"I'll be using a netbook to post new information, and I'll be hoping I can find islands of wifi often enough along the way to make this a worthwhile way for others to see where I am."
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"Removals New York\n\nArriving in a new country as part of an international relocation can be very emotional and overwhelming, particularly if your removal is as far afield as New York. ",
"With so much to think about and so many new experiences in a short space of time, it could be a good idea to engage the services of a New York removals expert to help you on arrival, and not just in the run-up to the move.",
"\n\nYour New York removals expert might be able to help you with things like:\n\nCollection from the airport\n\nThis would remove the need for you and your family to worry about how you’ll find your new home or temporary accommodation on your first day, especially when you’re tired from a long flight.",
"\n\nA welcome pack\n\nA good New York removals expert will put together a welcome pack at your accommodation, aimed at helping you to acclimatise. ",
"For example, it could include information like phone numbers for local services, customs information, maps of the local area and more.",
"\n\nFurnishing your home\n\nIf your belongings are still in transit from the UK, your New York removals consultant may be able to arrange for your new home or temporary accommodation to be furnished according to your needs. ",
"They could also arrange for utilities like water and power to be connected, and set up a mobile phone for you.",
"\n\nWant to find a New York removals company?",
"\n\nAt Premier Relocation, we’re expert New York movers and relocation professionals. ",
"We can carry out all of these services and more, so contact us today to see how we can help you.",
"\n\nFind Out More\n\nSearch\n\nBy continuing to use the site, you agree to the use of cookies. ",
"View our GDPR / Privacy Policy more information\n\nThe cookie settings on this website are set to \"allow cookies\" to give you the best browsing experience possible. ",
"If you continue to use this website without changing your cookie settings or you click \"Accept\" below then you are consenting to this."
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"\n\nThey were in the garage when Babin said the Eagles cut him. ",
"Had to be a joke. ",
"Not even a good one.",
"\n\n\"I thought he was playing around,\" Cole said. \"",
"I really thought he was just playing around. ",
"But then he said it's for real.",
"\n\n\"I didn't know what to say to him. ",
"I didn't have no words. ",
"It's a life-changing thing.\"",
"\n\nThe release of Babin, who led the Eagles with 18 sacks last season and again with 5½ this campaign, took all of his linemates by surprise.",
"\n\nThey texted it out in the group chat room they set up for the defensive linemen. ",
"They liked various statuses on Facebook.",
"\n\nBabin's teammates correctly predicted he would quickly get a job. ",
"The Jacksonville Jaguars scooped the pass rusher off waivers Wednesday. ",
"Who would have thought Babin would end up with a worse team than the Eagles (3-8)?",
"\n\n\"Babin was one of those guys I went and talked to besides Cullen Jenkins and Trent,\" rookie tackle Fletcher Cox said. \"",
"If I had a question about something on the field I asked him. ",
"I was sure he'd know.\"",
"\n\nCox paused.",
"\n\n\"Nobody had any idea this was going to happen,\" he said.",
"\n\nEagles head coach Andy Reid said Babin was released so that younger defensive ends could be evaluated.",
"\n\nBrandon Graham moved right into Babin's starting slot at left end. ",
"Phillip Hunt and rookie Vinny Curry also figure to get chunks of game time when the Eagles take on the Dallas Cowboys Sunday night at Cowboys Stadium.",
"\n\nThen again, it's not like the Eagles needed the roster space. ",
"They practiced Wednesday with only 51 players on their roster. ",
"Several were limited or unable to practice, the latter of which included quarterback Michael Vick and running back LeSean McCoy, both coming off concussions.",
"\n\nThe reality is the Eagles aren't going to the playoffs for the second straight season. ",
"They've lost seven straight games. ",
"And they apparently don't want to pay any more than they have to if this is the way it's going to be. ",
"They saved $1.69 million by getting rid of Babin, now being paid by the Jaguars.",
"\n\n\"When you lose, stuff like this happens,\" veteran wide receiver Jason Avant said. \"",
"It's one of those things. ",
"So the best thing that we can take from this is just win. ",
"Try to get some wins.",
"\n\n\"There is a message that everybody has to play better.\"",
"\n\nFor Graham, it's another chance to play like a first-round draft choice. ",
"And probably his last chance with the Eagles.",
"\n\n\"It lets me know I better make sure I stay on my game all through my career,\" Graham said. \"",
"Don't never take nothing for granted. ",
"I could be gone tomorrow, even though I think now is my shot. ",
"They must just say 'Graham is gone, too.' ",
"You never know. ",
"I'm just going to take advantage and go hard in practice today and take it one day at a time.\""
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"Telomeres protect the natural ends of linear chromosomes from recognition by the DNA damage response machinery. ",
"Not surprisingly, critically short or improperly 'capped' telomeres are major sources of genomic instability and have been linked to premature aging, hematological malignancies, and solid tumor formation. ",
"Thus, investigating how telomeres are maintained and how aberrant telomeres signal a DNA damage response, is essential to our understanding of cellular transformation. ",
"The objective of this proposal is to further define the mechanisms regulating the DNA damage response at telomeres and define how defects in this process promote genomic instability and ultimately, cancer progression. ",
"The studies described here will undoubtedly further our knowledge of the mechanisms involved in cancer progression and will lay the foundation for advances in disease diagnosis and/or treatment. ",
"The specific aims are outlined below. ",
"Specific Aim 1 will use a combination of biochemistry and cell biology to understand how the human POT1 protein protects telomere through two distinct functions. ",
"In mice, POT1 diverged into two genes encoding mPOT1a and mPOT1b proteins each possessing a unique function in telomere end protection. ",
"In this aim, I will use mouse mPOT1a and mPOT1b to determine how these two proteins differ in their ability to specifically repress ATR and how the single POT1 protein in humans accomplishes this function. ",
"Teasing out the functional domains in human POT1 will allow us to better understand how POT1 functions at telomeres and how mutations in POT1 may impair telomere end protection and contribute to tumorigenesis. ",
"Specific Aim 2 will use a combination of biochemistry and cell biology to determine how TRF2 represses ATM at telomeres. ",
"The goal of this aim is to understand mechanistically how TRF2 functions to inhibit ATM activation and ultimately, preserve genomic stability. ",
"Using new in vivo and in vitro assays, I will test the hypothesis TRF2 inhibits ATM activation by antagonizing binding of the DNA damage sensors at telomeric DNA. ",
"TRF2 is a key factor in telomere length maintenance and signaling, thus, dissecting the role of TRF2 in ATM inhibition will not only advance our current knowledge of how normal telomeres are maintained, but also how dysfunctional telomeres evoke a DNA damage response. ",
"The studies I have proposed here may shed light on how the telomere checkpoint is evoked and subsequently bypassed in cancers. ",
"Specific Aim 3 will use cell biology to investigate the role of the non-coding RNA TERRA in regulation of the DNA damage response at telomeres. ",
"TERRA is critical for maintaining genomic stability and is downregulated in a subset of human cancers suggesting that defects in TERRA contribute to telomere dysfunction and eventually, cellular transformation. ",
"The goal of this aim is to identify factors responsible for regulating the transcription, degradation, and/or localization of TERRA and to dissect the function of TERRA in maintaining genome stability. ",
"Despite my recent training in biochemistry, I will need 1-2 additional years of training to establish myself specifically as a telomere biochemist. ",
"This is a niche that is underrepresented in the field of telomere biology, and with additional training I feel I can make substantial contributions to the field. ",
"As an independent investigator, I will adapt my research from global DNA damage and genome maintenance, to telomere homeostasis and genome maintenance. ",
"In addition, I will continue to pursue research in the field of cancer biology and will begin to address the questions outlined in this proposal. ",
"Further defining the mechanism(s) regulating telomere stability will inevitably lead to a better understanding of cellular transformation and ma ultimately provide much needed therapeutic insight. ",
"I am eager to dissect the mechanisms regulating telomere homeostasis and would greatly appreciate the opportunity to conduct this research with the support of a K99 award. ",
"Receipt of this award would not only allow me to expand my research plan, but also establish myself as a primary investigator in the field of cancer biology. ",
"PUBLIC HEALTH RELEVANCE: Telomeres protect the natural ends of linear chromosomes from being recognized as sites of DNA damage. ",
"Given that DNA damage is a major source of genomic instability, the objective of this proposal is to further dissect the mechanisms regulating telomere maintenance and how defects in this process promote the genomic instability associated with cancer development."
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"/*\n * start.",
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"valette@crf.canon.fr\n *\n * The license and distribution terms for this file may be\n * found in the file LICENSE in this distribution or at\n * http://www.rtems.org/license/LICENSE.",
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"\n\t * Note: enabling HID1[RFXE] at this point makes no sense since\n\t * exceptions are not configured yet. ",
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"\n\t */\n /* pass result of 'save_boot_params' to 'boot_card' in R3 */\n\tbl\tboot_card\n\t/* point of no return: reset board here ? */",
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"\n\n\n\nHi bubs.",
"\n\nPlump Tokens\n\nPLUMP TOKENS\n\n\n\n$1 = 10 plump tokens (PT)\n\n\n\nYour name at the end of comics for one month (default at $3+ tiers) - 30 PT\n\n\n\nRustled Jimmies wallpaper pack (one desktop, one mobile) - 30 PT\n\n\n\nPatreon supporter instagram story shoutout - 50 PT\n\n\n\nRustled Jimmies vinyl sticker pack (4) - 140 PT\n\n\n\nQuick digital portrait - 150 PT black and white, 200 PT color\n\n\n\nOriginal physical sketch - 250 PT\n\n\n\nYou drawn as a RJ character in a comic - 500 PT\n\n\n\nMore rewards coming soon.",
"\n\n\n\nFAQ\n\n\n\nQ: How do I know how many plump tokens I have?",
"\n\nA: You can calculate your token amount by multiplying the current tier you’re subscribed to x the number of months you’ve been subscribed. ",
"Remember that $1 = 10 plump tokens each month. ",
"Or, you can always just ask me in the Discord server’s #shop section.",
"\n\n\n\nQ: When will my plump tokens be added to my account?",
"\n\nA: Plump tokens are added on the 1st of every month.",
"\n\n\n\nQ: What if I want more plump tokens than the current tiers allow?",
"\n\nA: You can always make a custom pledge. ",
"$1 = 10 monthly plump tokens.",
"\n\n\n\nQ: I don’t use Discord?",
"\n\nA: That’s fine! ",
"You can still redeem your plump tokens by sending me a message here on Patreon too.",
"\n\n\n\nQ: Does \"All previous rewards\" mean I get my tier's tokens + the other tier's?",
"\n\nA: No. ",
"Plump Tokens are not included under \"All previous rewards\".",
"\n\n\n\n**NOTE: All token redemptions must happen before the 7th of each month**\n\nYour name at the end of comics for one month (default at $3+ tiers) -Rustled Jimmies wallpaper pack (one desktop, one mobile) -Patreon supporter instagram story shoutout -Rustled Jimmies vinyl sticker pack (4) -Quick digital portrait -black and white,colorOriginal physical sketch -You drawn as a RJ character in a comic -More rewards coming soon.",
"A: You can calculate your token amount by multiplying the current tier you’re subscribed to x the number of months you’ve been subscribed. ",
"Remember that $1 = 10 plump tokens each month. ",
"Or, you can always just ask me in the Discord server’s #shop section.",
"A: Plump tokens are added on the 1st of every month.",
"A: You can always make a custom pledge. ",
"$1 = 10 monthly plump tokens.",
"A: That’s fine! ",
"You can still redeem your plump tokens by sending me a message here on Patreon too.",
"A: No. ",
"Plump Tokens are not included under \"All previous rewards\".",
"\n\nThanks for stopping by my Patreon page! ",
"My name's Sam and I make Rustled Jimmies - comics and cartoons about anything from sharks buying groceries, to flesh people sticking their hands in possums. ",
"I also enjoy long walks on the beach in full platemail armor, reading poetry under the moonlight, and sprinting through fields of dandelions screaming profanities. ",
"But that's enough about me.",
"Right now I'm making Rustled Jimmies in my free time, but hopefully someday I can focus all of my time towards it. ",
"Patreon is the #1 way to support my cartoons, and by becoming a patron you automatically become one of my favorite people. ",
"Plus, there's a lot of really cool rewards that you can unlock with yourGOOD QUESTION - Plump tokens are given to you at the beginning of every month depending on what tier you select. ",
"You can then use your plump tokens to redeem rewards in the shop down below! ",
"Take a look around and let me know in the Discord server'schannel if there's anything you'd want to buy. ",
"Remember, no shoes, no shirt, extra service.<3 Sam"
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"[terrain_type]\n\tident = flamefood_tree_snow\n\tname = \"Flamefood Tree (Snow)\"\n\tcharacter = a\n\tcolor = 128.128.32\n\tbase_terrain_type = snow\n\toverlay = true\n\tflag = land\n\tflag = forest\n\tflag = unpassable\n\tgraphics = \"terrain/flamefood_tree_snow.png\"\n\tresource = lumber\n\tsolid_tile = 18\n\tdestroyed_tile = 16\n\t[transition_tile]\n\t\ttransition_type = north\n\t\ttile = 2\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = south\n\t\ttile = 34\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = west\n\t\ttile = 17\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = east\n\t\ttile = 19\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = northwest_outer\n\t\ttile = 1\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = northeast_outer\n\t\ttile = 3\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = southwest_outer\n\t\ttile = 33\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = southeast_outer\n\t\ttile = 35\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = northwest_inner\n\t\ttile = 10\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = northeast_inner\n\t\ttile = 11\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = southwest_inner\n\t\ttile = 26\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = southeast_inner\n\t\ttile = 27\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = northwest_southeast_inner\n\t\ttile = 30\n\t[/transition_tile]\n\t[transition_tile]\n\t\ttransition_type = northeast_southwest_inner\n\t\ttile = 14\n\t[/transition_tile]\n[/terrain_type]\n"
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"\n\nReferences\n\nExternal links\nThe Ohio Ladies' Gallery: Shirley Smith\nProject Vote Smart - Senator Shirley Smith (OH) profile\nFollow the Money - Shirley Smith\n20102006 2004 2002 2000 1998 campaign contributions\n\nCategory:1950 births\nCategory:Living people\nCategory:African-American women in politics\nCategory:Cleveland State University alumni\nCategory:John F. Kennedy School of Government alumni\nCategory:Members of the Ohio House of Representatives\nCategory:Ohio state senators\nCategory:Politicians from Cleveland\nCategory:Women state legislators in Ohio\nCategory:African-American state legislators in Ohio\nCategory:21st-century American politicians\nCategory:21st-century American women politicians"
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"Wednesday, 16 November 2016\n\nNairobi Rumba Singles Splash\n\nOne comfort in an otherwise unsettling year\nhas been my own luck in finding some great second-hand records – 78rpm, 45rpm\nand LPs. ",
"I had the good fortune of visiting Nairobi three times this year, and\nmeeting a long-time Congo-Kenya Rhumba collector who no longer wanted to keep his\nvinyl collection, he wanted this music in digital format. ",
"So, come the summer\nholiday break I will be spending some time keeping my promise to him. ",
"Before I share with you my start on that project, I must mention two great compilations\nof Kenyan music that have this year been released world-wide. ",
"The sound reproduction on both\ncompilations is really excellent. ",
"My personal favourite is “Urgent Jumping: East\nAfrican Musiki Wa Dansi Classics” which features 27 tracks spanning close on\ntwo and a half hours of music from Kenyan, Tanzanian and Congolese bands\nthat plied their trade across the region over the period 1972 - 1982.",
"\n\n“Kenya Special Volume 2” follows the\nsuccessful release by Soundway Records of volume one in 2013. ",
"This compilation\noffers 20 tracks (90 minutes) sampling a wider variety of Kenyan music in the\n1970s and 1980s.",
"\n\nSoundway reckon that the resurgence of\ninterest in African music from the 1960s to the 1980s is boosted by a curious\nnew generation of music lovers: “The approach to musical rediscovery that is\nbehind Kenya Special has its origins in a youthful movement of\nvinyl collecting (and to some extent club culture), which has, in the past\ndecade and a half, carved out its own niche alongside the established music\nindustry.”",
"\n\n“Despite the renewed interest in music from Kenya’s past, finding these tracks\nand their rights holders hasn’t become any easier. ",
"Only a handful of music\narchives around the world harbour collections of Kenyan music, and just a few\nprivate collectors in Kenya and abroad have been sharing catalogue info online\nor privately. ",
"One of the problems with East African music of this era is that\nmuch of it was originally released only on 45 rpm, seven-inch vinyl singles,\nmany of which were only ever produced in tiny runs of a few hundred. ",
"45s with\ntheir thin, paper sleeves do not age as well as LPs and are often far more\nsusceptible to the elements. ",
"The compilers of Kenya Special 2 have\ngone to great lengths to disclose a small part of what is slowly being accepted\nas an essential element of East Africa’s cultural heritage: the history of\nrecorded popular music.”",
"\n\nSo, herewith a rather random sampling from\nthe pile of 45rpms that Henry sold me. ",
"Do leave a comment if you would like\nmore of the same.",
"\n\n1. ",
"Orch Bana Likasi: Lena Pts 1 & 2 (Kanema)\n\nOutstanding vocalist Lovy Mokolo Longomba\nwas the son of Vicky Longomba, a founding member of OK Jazz. ",
"The Muzikifan\nwebsite (well worth a visit) tells us that Longomba moved to Nairobi in August\n1978 and went through a string of bands, from Les Kinois, through Boma\nLiwanza, to Orch Shika Shika. ",
"He\nthen formed the band Super Lovy in\nMay 1981. ",
"Bana Likasi was the same\nband as Super Lovy, but was so named\nto avoid a contractual conflict. ",
"Longomba died in 1996 in a car crash in\nTanzania.",
"\n\n2. ",
"Orch. ",
"Les Jaca: Sikia Pts 1 & 2 (Ligbutu)\n\nThis from the Muzikifan website: “Les Jaca was created by Lovy (Longomba) when\nhe decided to leave Super Mazembe in 1981. ",
"He went into the studio with Siama,\nTabu Frantal, Mandefu, Roy Mosanda and other friends, but the attempt bore no\nfruit, so Lovy remained with Super Mazembe.”",
"\n\n3. ",
"L'Orch Baba National: Vituko Vya Mama\nMkwe Pts 1 & 2 (Baba Gaston)\n\nOne of the first Congolese musicians to\nsettle in East Africa (in 1971), first in Dar Es Salam, and then in 1976 in\nNairobi. ",
"Some reports indicate that Baba Gaston was not the easiest band leader\nto work with, and in July 1976 most his band walked out on him to form their\nown band, the highly successful Les\nMangalepa. ",
"Baba Gaston remained a super star in Nairobi until his\nretirement in 1989.",
"\n\n4. ",
"Orch Les Wanyika: Nisaidie Baba Pts 1\n& 2 (D.J. Ngereza) 1980.",
"\n\nAn offshoot of Simba Wanyika formed by a group of Kenyan and Tanzanian musicians\nleft the band to form Les Wanyika.",
"\nFamous for classic Swahili rumba hit hits like Sina Makossa, Paulina and Pamela (these singles were even sold in\nSouth Africa at the time). ",
"Band leader and guitarist John Ngereza composed this\nsong. ",
"In 2010, ten years after Ngereza died, four surviving members put aside\ntheir differences and re-grouped and started playing again. ",
"You can read about\nthe reunion of Rashid Juma, Alfani Tommy Malanga, Sijali Zuwa\n‘Usikajali’ and Joseph Justy ‘Yellow Man’ here.",
"\n\nAnother off-shoot band though the details are not as clear. – ",
"this time from Super Volcano. ",
"Some sources say that Les Volcano were co-founded by Charles Ray\nKasembe and Mohamed Mazingazinga in 1976. ",
"Info on the Kentanza site suggests that Les Volcano came about following the death of Mbaraka Mwinishe (leader of Super Volcano) in 1979. ",
"Doug Paterson sees it the same way: \"I don't think Les Volcano\nwas a group until after the death of Mbaraka Mwinshehe. ",
"I think Mbaraka\nstarted Super Volcano after leaving Morogoro Jazz (in about 1974?). ",
"After\nMbaraka died, Ray Charles Kasembe tried to keep a subset of Super Volcano\nmembers together under the name Les Volcanos. ",
"I don't think this group\nexisted prior to Mbaraka's death (I could be mistaken). ",
"I never had the\nopportunity to see Super Volcano, though I went to the Kenya coast try to find\nthem in 1975 (just missed them).\" ",
"Thanks Doug.",
"\n\n7. ",
"Orch. ",
"Super Bwambe: Atikapo Pts 1 &\n2\n\nMuzikifan tells us that John Negereza of\nLes Wanyika was a member of this Congolese band that also included: George\nKalombo Mwanza, sax; John Ngereza, guitar; Chou chou, vocals; Kayembe Nyonga,\nvocals; Luboya wa Tshiteyai; Matabu Kunyanga. ",
"With Thomy Lomboto, bass, and\nKabeya Ilombo from Viva Makale.",
"\n\n8. ",
"Orchestre Matonge: Pesa Moselebende Pts\n1 & 2 (Jean-Claude K)\n\nNamed after the musical heart of Congo\nKinshasa, this was another of the bands that sold really well in Nairobi. ",
"I\ncannot find much info on this band, though Tim Clifford on the ever useful\nKentanza Vinyl site says that the Kamanyola label is named “after an area in\neastern DRC Congo close to the Rwandan border. ",
"The name featured heavily in\nMobutu’s Zaire as in 1964 it was where the young army officer led troops in the\ncapture of a rebel-held bridge. ",
"When he rose to power, he named the\npresidential yacht, an army division and Kinshasa’s sports stadium (now the\nStade des Martyrs) after this victory. ",
"There is still a Boulevard Kamanyola in\nLubumbashi. ",
"Appropriately enough, one of the roads leading off it is the Avenue\ndes Chutes - Falls or Collapses Avenue.”",
"\n\nThanks Chris, I am very excited to hear about this. ",
"I am going to download this immediately and woza Friday can start a bit earlier this week. ",
"Didn't manage to comment the last months(spamfilter?) , ",
"but be sure I have been following you guys from close. ",
"Please continue preserving the heritage. ",
"Rik"
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"New Year's Day homicide was a 'hit' set up by Port St. Lucie woman over drugs, police say\n\nFORT PIERCE — The New Year’s Day shooting that fatally injured an 18-year-old man was a “hit” set up by a Port St. Lucie woman stemming from an earlier drug rip-off, according to statements made to police.",
"\n\nFort Pierce police late on New Year’s Day found Jamal Booker face down with gunshot wounds to his head at an apartment complex in the 1200 block of North 29th Street.",
"\n\nMore: Four arrests in New Year's Day homicide in Fort Pierce\n\nAfter a complex investigation with links to Palm Beach, Broward and Miami-Dade counties, four people were charged in the killing.",
"\n\nAffidavits for two of those arrested, Amir Tafari Muhammad, 27, of Wellington; and Megan Nichole Craig, 23, of Port St. Lucie, were sealed, but records relating to the arrest of Dwayne Anthony Fisher, 31, of Coconut Creek, recently were released. ",
"Anastasia Benincasa, 17, of Fort Lauderdale, also was arrested in the case.",
"\n\nThe Fisher affidavit showed police used surveillance video and determined a Toyota 4Runner backed into a parking space of the Pinecreek Village apartment complex shortly before the shooting.",
"\n\nOne person got out and stood in front of a hallway, talking on a phone and smoking. ",
"Booker, on his cell phone, walked by as the person who got out of the Toyota pulled a pistol. ",
"A second person on the video watched the encounter.",
"\n\nMore: Records sealed in New Year's Day fatal shooting of 18-year-old Jamal Booker\n\nPhone call helps police\n\nThe first person fired once at Booker from point-blank range, firing two more shots after Booker fell. ",
"Both of the people got in the Toyota and left the scene.",
"\n\nPolice researched Booker’s phone and found a call seconds before he was shot. ",
"Investigators were led to Muhammad, who was released from state prison in March 2018 after an attempted murder conviction, according to court records.",
"\n\nCraig is Muhammad’s girlfriend, who told police she met Booker through the dating site Plenty of Fish “to offer her services as a marijuana dealer,” records said.",
"\n\nMore: Port St. Lucie woman charged with murder in shooting death of 18-year-old Fort Pierce man\n\nCraig told investigators she and Booker were supposed to meet Dec. 30 so Booker could buy $900 worth of marijuana, a half pound. ",
"She said Booker robbed her of the drugs and Booker said he had a gun.",
"\n\nCraig said she and Muhammad were very upset about the stolen marijuana. ",
"Craig said Muhammad told her he was going to kill Booker and she drove him to the apartments to scope out the area.",
"\n\nPlan in place\n\n“She said she knew the plan was for Amir to kill Jamal and then evade police detection,” records said. “",
"Megan confessed to being the person who ultimately set up a hit on Jamal Booker.”",
"\n\nMore: Man drives car into house, stabs another man, PSL police say\n\nFort Pierce police Detective Jeffrey Wachendorfer said on the night of the shooting, investigators believe Fisher and his friend, Muhammad, traveled to the apartment complex in the Toyota 4Runner.",
"\n\nWachendorfer said Benincasa is Fisher’s girlfriend. ",
"Benincasa told investigators that on the day of the killing, she was at a “tattoo party” when Fisher and Muhammad said they wanted to “take care of something.”",
"\n\nBenincasa said she knew they planned on killing someone, and drove them to Fort Pierce.",
"\n\nShe said she was in a three-way call with Booker and Muhammad. ",
"She said after the shooting, she drove them to Muhammad’s apartment so he could change. ",
"They then returned to the “tattoo party.”",
"\n\nBenincasa said they threw the firearm, described as a pink gun, into a canal in Fort Lauderdale.",
"\n\n“She was also upset about how sloppy Amir was, knowing that there was cameras and using his cell phone,” the affidavit said.",
"\n\nWhile customarily not naming juveniles accused of crimes, TCPalm.com and Treasure Coast Newspapers does name some juveniles depending on the seriousness of the crime, their age, aggravating circumstances and other factors.",
"\n\nWant more Treasure Coast breaking news coverage? ",
"If you're already a subscriber to TCPalm, thank you! ",
"If not, please subscribe to keep coverage of Treasure Coast breaking news coming."
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"If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nghostfacekillahrabbit:If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nI mean about Sam, and Ted loves to pick every sort of tweener DE-OLB with a hard-nosed pedigree if he can get a bargain on 'em... That's what i want to hear from serious fans of other teams who pay attention to the inside scoop... what do you think your team's current GM, who you may love or hate or not give a shiat about would think about Sam?",
"\n\nWith a high school coach that looks like this, I'm surprised he wasn't scared straight\n\nTHAT PIC and its caption:\n\nIn junior high school, Craig Smith, the football coach, saw that Sam was athletically blessed and, even better, hungry for guidance and camaraderie. ",
"The coaches drafted him to carry equipment and hang around the squad.",
"\n\nMy mind immediately played out THE nightmare scenario for how this young man discovered his sexuality.",
"\n\nghostfacekillahrabbit:ghostfacekillahrabbit: If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nI mean about Sam, and Ted loves to pick every sort of tweener DE-OLB with a hard-nosed pedigree if he can get a bargain on 'em... That's what i want to hear from serious fans of other teams who pay attention to the inside scoop... what do you think your team's current GM, who you may love or hate or not give a shiat about would think about Sam?",
"\n\nIf he is as good a player as they say he is, I could see Seattle taking a flyer on him. ",
"We have a large gay community here, Pete Carroll would be like, \"hey that's great\", and he would have a chance to play if he can really hit. ",
"I could see Seattle as a possible destination.",
"\n\nghostfacekillahrabbit:ghostfacekillahrabbit: If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nI mean about Sam, and Ted loves to pick every sort of tweener DE-OLB with a hard-nosed pedigree if he can get a bargain on 'em... That's what i want to hear from serious fans of other teams who pay attention to the inside scoop... what do you think your team's current GM, who you may love or hate or not give a shiat about would think about Sam?",
"\n\nBy the draft, I may be Browns Gm. ",
"I'd take him in the 5th due to size if you convert him. ",
"Now if my dog ends up as Browns GM, he's drafting 3 QBs, 2 RBs, and the rest skill players until he finds someone who is slightly below average.",
"\n\nghostfacekillahrabbit:If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nHe's very much a 3-4 DE, so teams like the Panthers that are pretty well entrenched in the 4-3 aren't going to have as much need for him.",
"\n\nghostfacekillahrabbit:If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nThe Lions GM is a big fan of taking guys who are really athletic but have something that causes other teams to downgrade them. ",
"I could see him overlooking the tweener aspect. ",
"No clue, though, about the sexual orientation thing.",
"\n\nmeanmutton:ghostfacekillahrabbit: If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nThe Lions GM is a big fan of taking guys who are really athletic but have something that causes other teams to downgrade them. ",
"I could see him overlooking the tweener aspect. ",
"No clue, though, about the sexual orientation thing.",
"\n\nIf he had an injury and/or concussion history, he'd be a sure thing to go to the Lions in the second round.",
"\n\nThe party stopped cold. \"",
"I couldn't eat no more, so I went to Applebee's to have drinks,\" Sam Sr. ",
"said. \"",
"I don't want my grandkids raised in that kind of environment.",
"\"I'm old school,\" he added. \"",
"I'm a man-and-a-woman type of guy.\" ",
"As evidence, he pointed out that he had taken an older son to Mexico to lose his virginity.",
"\n\nghostfacekillahrabbit:ghostfacekillahrabbit: If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nI mean about Sam, and Ted loves to pick every sort of tweener DE-OLB with a hard-nosed pedigree if he can get a bargain on 'em... That's what i want to hear from serious fans of other teams who pay attention to the inside scoop... what do you think your team's current GM, who you may love or hate or not give a shiat about would think about Sam?",
"\n\nIn a vacuum he's the exact type of player the Bears under Emery would take a good look at. ",
"In reality, they've already got one tweener they don't know what to do with (McClellen) so despite an awful defense in need of upgrades, there's no way they'll even consider him.",
"\n\nJusticeandIndependence:The party stopped cold. \"",
"I couldn't eat no more, so I went to Applebee's to have drinks,\" Sam Sr. ",
"said. \"",
"I don't want my grandkids raised in that kind of environment.",
"\"I'm old school,\" he added. \"",
"I'm a man-and-a-woman type of guy.\" ",
"As evidence, he pointed out that he had taken an older son to Mexico to lose his virginity.",
"\n\nFTA \"A sister drowned when she was 2, before Michael was born, when another child accidentally knocked her off a fishing pier. ",
"Another brother, Russell, was 15 when he was shot and killed trying to break into a home, in what his father said was part of a gang initiation. ",
"Another brother, Julian, has not been heard from since he left for work one day in 1998; his family believes he is dead. ",
"Two others are in jail. \"",
"HIs mother \" JoAnn is a member of the Jehovah's Witnesses, who do not celebrate holidays or believe in most birth control and generally shun participation in organized sports. ",
"Michael Sr. ",
"comes from a large Baptist family, and his father was a long-serving deacon at a church in nearby Texas City, Tex. \"",
"\n\nI bet coming out is one of the easier things this guy has gone through.",
"\n\nghostfacekillahrabbit:If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nI think he'd fit in well with the Steelers defense, they typically like those hybrids to play their rush positions in the 3-4. ",
"But they just drafted one (Jarvis Jones) last year so I doubt they're thinking about him anytime before round 3 or 4. ",
"I don't think they'd have a problem drafting him because of his penis preference but who really knows? ",
"Scouts, GMs, and owners especially tend to be old white guys, otherwise known as the most likely demographic to have a problem with gay people.",
"\n\nI like to think the NFL is a meritocracy (Tebow phenom excluded) and that if he's good enough my team would take him. ",
"What I'm not excited about is if he goes to the combine and stinks it up, runs a 5.2 or something, and he drops way down in the draft. ",
"That shiatstorm won't be fun to hear about. ",
"I hope he dominates the combine and leaves no question he can play in the NFL.",
"\n\nSemper IvXx:ghostfacekillahrabbit: If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nI think he'd fit in well with the Steelers defense, they typically like those hybrids to play their rush positions in the 3-4. ",
"But they just drafted one (Jarvis Jones) last year so I doubt they're thinking about him anytime before round 3 or 4. ",
"I don't think they'd have a problem drafting him because of his penis preference but who really knows? ",
"Scouts, GMs, and owners especially tend to be old white guys, otherwise known as the most likely demographic to have a problem with gay people.",
"\n\nI like to think the NFL is a meritocracy (Tebow phenom excluded) and that if he's good enough my team would take him. ",
"What I'm not excited about is if he goes to the combine and stinks it up, runs a 5.2 or something, and he drops way down in the draft. ",
"That shiatstorm won't be fun to hear about. ",
"I hope he dominates the combine and leaves no question he can play in the NFL.",
"\n\nIf you think that old white guys are the ones most likely to have a problem with gay men, you haven't met many old black guys.",
"\n\nmeanmutton:Semper IvXx: ghostfacekillahrabbit: If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nI think he'd fit in well with the Steelers defense, they typically like those hybrids to play their rush positions in the 3-4. ",
"But they just drafted one (Jarvis Jones) last year so I doubt they're thinking about him anytime before round 3 or 4. ",
"I don't think they'd have a problem drafting him because of his penis preference but who really knows? ",
"Scouts, GMs, and owners especially tend to be old white guys, otherwise known as the most likely demographic to have a problem with gay people.",
"\n\nI like to think the NFL is a meritocracy (Tebow phenom excluded) and that if he's good enough my team would take him. ",
"What I'm not excited about is if he goes to the combine and stinks it up, runs a 5.2 or something, and he drops way down in the draft. ",
"That shiatstorm won't be fun to hear about. ",
"I hope he dominates the combine and leaves no question he can play in the NFL.",
"\n\nIf you think that old white guys are the ones most likely to have a problem with gay men, you haven't met many old black guys.",
"\n\nI think there aren't old black owners. ",
"Not sure about GMs, though there probably are scouts.",
"\n\nHey NYT (and the rest of the media), thanks for helping to make the life of this guy, his family, his coach, and everyone who knew him a living hell because he had the guts to come out as gay. ",
"You really deserve a lot of credit. ",
"My two local sports stations haven't STFU about him for three days now. ",
"No idea what's going on in the Olympics or the NBA.",
"\n\nghostfacekillahrabbit:If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nWell, Elway came out and said the right things, but I highly doubt after Denver's experience with Tebowmania he would want to deal with the media and the distractions Sam would bring. ",
"Very doubtful the Broncos would draft him, regardless of 'fit'.",
"\n\nnot sure what \"being straight\" has to do with buying prostitution to end your son's virginity... ever thought of ya know, dating a girl?",
"\n\nOw! ",
"That was my feelings!:ghostfacekillahrabbit: If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nWell, Elway came out and said the right things, but I highly doubt after Denver's experience with Tebowmania he would want to deal with the media and the distractions Sam would bring. ",
"Very doubtful the Broncos would draft him, regardless of 'fit'.",
"\n\nthe best decision Elway has made so far is bringing in Peyton, after he retires that team will be a hot mess\n\nironically i think the Bears could draft him, their non-existent pass-rush needs all the help it can get, the Ravens are another candidate with Suggs possibly being a Cap casaulty\n\nskrame:meanmutton: I like to think the NFL is a meritocracy (Tebow phenom excluded)\n\nC'mon, man...\n\nPlease explain how, based only on his college career, Tebow didn't merit a chance in the NFL.",
"\n\n/obviously he didn't prove that he had what it takes to stay in the NFL, but it looks like you're talking about the draft and a shot at the NFL.",
"\n\nI wasn't talking about the draft in relation to Tebow. ",
"I didn't say he didn't deserve a chance, and regardless if I did or not, not only did he get a chance he got drafted about 5 rounds early. ",
"He got a chance and sucked, yet kept his job (for a bit anyway) and was treated as if he didn't suck, at least by the media and some weirdly over zealous Tebowners who felt the need to white knight him online.",
"\n\nMy point was he was an outlier in a bottom line business. ",
"But you certainly protected his honor from imaginary insults.",
"\n\nSo a black man chooses his farked up values over his unconditional love for his son. ",
"The same values some dicks trumpeted to put his people in chains 150 years ago. ",
"This kind of shiat makes my brain hurt. ",
"Also interesting to note that most anti-gay quotes from the bible are from Leviticus. ",
"Which also say its cool to rape a biatch but if you knock her up you have to marry her or buy her dad a donkey. ",
"Truly values to base your life on.",
"\n\nBowtoMogul:FTA \"A sister drowned when she was 2, before Michael was born, when another child accidentally knocked her off a fishing pier. ",
"Another brother, Russell, was 15 when he was shot and killed trying to break into a home, in what his father said was part of a gang initiation. ",
"Another brother, Julian, has not been heard from since he left for work one day in 1998; his family believes he is dead. ",
"Two others are in jail. \"",
"HIs mother \" JoAnn is a member of the Jehovah's Witnesses, who do not celebrate holidays or believe in most birth control and generally shun participation in organized sports. ",
"Michael Sr. ",
"comes from a large Baptist family, and his father was a long-serving deacon at a church in nearby Texas City, Tex. \"",
"\n\nI bet coming out is one of the easier things this guy has gone through.",
"\n\nHoly hell. ",
"You'd think the parents would have realized their values led them to being some of the worst parents in history and chilled out a bit.",
"\n\nOw! ",
"That was my feelings!:ghostfacekillahrabbit: If this kid is a tweener DE who could be converted to OLB, and put up player of the year numbers in a tough conference, and is projected to go in third through fifth rounds, then Ted Thompson will agonize over picking him or passing on him starting with the 4th or 5th, and then if he's still there trade up in the 6th to take him. ",
"Fans of other teams, how do you think your team's GM thinks about this?",
"\n\nWell, Elway came out and said the right things, but I highly doubt after Denver's experience with Tebowmania he would want to deal with the media and the distractions Sam would bring. ",
"Very doubtful the Broncos would draft him, regardless of 'fit'.",
"\n\nDude, Elwsy's best defensive player is a knucklehead. ",
"He was suspended for violating the drug policy and has done some seriously immature shiat off the field.",
"\n\nElway got rid of Tebow because his stock was never going to be higher and he wasn't that good. ",
"Plus Peyton.",
"\n\nI see the Broncos, Steelers, Pats, Hawks, Niners, and Giants amongst others that would be good places for Sam to land just because they are stable teams and can tell folks to STFU."
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"Q:\n\nr - add multiple markers to rMaps\n\nQuestion\nHow do you add multiple markers using rMaps?",
"\nData\ncoords <- structure(list(stop_id = 19841:19843, stop_name = c(\"Flagstaff Railway Station (Melbourne City)\", \n\"Melbourne Central Railway Station (Melbourne City)\", \"Parliament Railway Station (Melbourne City)\"\n), stop_lat = c(-37.8119813073807, -37.8099387667386, -37.8110540555305\n), stop_lon = c(144.955653760429, 144.962593535096, 144.972910916416\n)), .Names = c(\"stop_id\", \"stop_name\", \"stop_lat\", \"stop_lon\"\n), sorted = \"stop_id\", row.names = 17:19, class = c(\"data.table\", \n\"data.frame\"))\n\nExamples\nUsing library(rMaps) I can create a map and add a single marker, as per the examples on Ramnath's github page:\nlibrary(rMaps)\n\nl <- Leaflet$new()\nl$setView(c(-37.8602828, 145.079616), zoom=11)\nl$tileLayer(provider = \"Acetate.terrain\")\n\n## add one marker:\nl$marker(LatLng = c(-37.81198,144.9557))\n\nBut I can't figure out how to add multiple markers from the coords data frame, without writing a l$marker line for each one.",
"\nI've tried to use GeoJSON but I'm new to this so haven't got my head around it yet and must be doing something wrong.",
"\n# library(rgdal)\n# coords.sp <- SpatialPointsDataFrame(coords[,.(stop_lon, stop_lat)], coords[,.(stop_id, stop_name)])\n# writeOGR(obj=coords.sp, dsn='coords.geojson', layer='OGRGeoJSON', driver='GeoJSON')\n\n# gj <- readOGR(\"./coords.geojson\", layer=\"OGRGeoJSON\")\n# l$geoJson(gj)\n# l$geoJson(\"./coords.geojson\")\n\nI'm happy to use JSON/GeoJSON if it's the way to go.",
"\nDesired output\nI would like the result to be the same as if I used library(leaflet) to show multiple markers\nlibrary(leaflet)\nleaflet() %>%\n addProviderTiles(\"Acetate.terrain\") %>%\n setView(lat = -37.8602828, lng = 145.079616, zoom=11) %>%\n addMarkers(data=coords, lat=~stop_lat, lng=~stop_lon)\n\nGeoJSON\nHere is the GeoJSON code that's produced from the writeOGR command, and I've validated it at GeoJSONLint\n{\n\"type\": \"FeatureCollection\",\n\n\"features\": [\n{ \"type\": \"Feature\", \"id\": 1, \"properties\": { \"stop_id\": 19841, \"stop_name\": \"Flagstaff Railway Station (Melbourne City)\" }, \"geometry\": { \"type\": \"Point\", \"coordinates\": [ 144.955653760428987, -37.811981307380698 ] } },\n{ \"type\": \"Feature\", \"id\": 2, \"properties\": { \"stop_id\": 19842, \"stop_name\": \"Melbourne Central Railway Station (Melbourne City)\" }, \"geometry\": { \"type\": \"Point\", \"coordinates\": [ 144.962593535096005, -37.809938766738597 ] } },\n{ \"type\": \"Feature\", \"id\": 3, \"properties\": { \"stop_id\": 19843, \"stop_name\": \"Parliament Railway Station (Melbourne City)\" }, \"geometry\": { \"type\": \"Point\", \"coordinates\": [ 144.972910916415998, -37.811054055530498 ] } }\n]}\n\nA:\n\nI can make use of library(geojsonio) to correctly read the GeoJSON file, and then use l$geoJson to load the markers onto the map\n## create spatial object and save as GeoJSON\n# library(rgdal)\ncoords.sp <- SpatialPointsDataFrame(coords[,.(stop_lon, stop_lat)], coords[,.(stop_id, stop_name)])\nwriteOGR(obj=coords.sp, dsn='coords.geojson', layer='OGRGeoJSON', driver='GeoJSON')\n\nl <- Leaflet$new()\nl$setView(c(-37.8602828, 145.079616), zoom=11)\nl$tileLayer(provider = \"Acetate.terrain\")\n\n## Read the GeoJSON data\nlibrary(geojsonio)\ngj <- geojson_read(\"./coords.geojson\")\nl$geoJson(gj)\n\n"
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"41 F.3d 35\n74 A.F.T.R.2d 94-7036\nUNITED STATES of America, Appellee,v.Lawrence MACCHIA, Defendant-Appellant.",
"\nNo. ",
"1037, Docket 94-1497.",
"\nUnited States Court of Appeals,Second Circuit.",
"\nSubmitted Oct. 17, 1994.Decided Oct. 19, 1994.Filed Nov. 21, 1994.",
"\n\nGerald B. Lefcourt, Joshua L. Dratel, Lefcourt & Dratel, P.C., New York City, for defendant-appellant.",
"\nRobert E. Lindsay, Dept. ",
"of Justice, Washington, DC, for appellee.",
"\nBefore: NEWMAN, Chief Judge, ALTIMARI and LEVAL, Circuit Judges.",
"\nJON O. NEWMAN, Chief Judge:\n\n\n1\nThis interlocutory appeal in a criminal case endeavors to secure pretrial review of a ruling denying a motion to dismiss an indictment alleged to have been returned in violation of an immunity agreement. ",
" The appeal is brought by Lawrence Macchia from the August 24, 1994, ruling of the District Court for the Eastern District of New York (Leonard D. Wexler, Judge), 861 F.Supp. ",
"182, construing the immunity agreement to have conferred only derivative use immunity and not transactional immunity. ",
" We conclude that recent Supreme Court pronouncements have altered the law of this Circuit, which formerly permitted such interlocutory appeals, and we therefore dismiss the appeal for lack of appellate jurisdiction.",
"\n\nBackground\n\n2\nMacchia and seven co-defendants were indicted for conspiracy to defraud the United States, 18 U.S.C. Sec. ",
"371, and attempting to evade excise taxes, 26 U.S.C. Sec. ",
"7201, in connection with an alleged scheme to avoid paying federal gasoline taxes. ",
" The offenses are detailed in United States v. Macchia, 35 F.3d 662 (2d Cir.1994), an unsuccessful pretrial appeal by two of Macchia's co-defendants claiming insulation from trial on double jeopardy grounds. ",
" Macchia sought to prevent his trial on the basis of an immunity agreement containing the following language:\n\n\n3\nAny truthful statements made by Lawrence Macchia in response to questions asked of him by government attorneys and agents during this interview or any information arising from or relating thereto will not be used against Lawrence Macchia in any criminal prosecution by the United States government, or by the State of New York, or its political subdivisions (emphasis added).",
"\n\n\n4\nMacchia contended in the District Court that the emphasized words, especially the words \"or relating thereto,\" added at the insistence of his counsel, broadened the agreement beyond use immunity and, in effect, conferred transactional immunity since any evidence relevant to the charge against him would necessarily \"relat[e] to\" his statements. ",
" Judge Wexler disagreed, denied the motion to dismiss, and also ruled that a so-called Kastigar hearing, see Kastigar v. United States, 406 U.S. 441, 92 S.Ct. ",
"1653, 32 L.Ed.2d 212 (1972), to determine whether the protection of derivative use immunity was observed, would be deferred until after the trial. ",
" This appeal followed.",
"\n\n\n5\nOn September 23, 1994, the Government moved to dismiss for lack of appellate jurisdiction. ",
" After considering opposing papers and with the scheduled trial date approaching, we issued an order on October 19, 1994, granting the motion to dismiss and indicating that an opinion would follow. ",
" We now issue that opinion.",
"\n\nDiscussion\n\n6\nThe Government contends that a pretrial interlocutory appeal is not available to consider a defendant's claim that an immunity agreement has conferred insulation from trial. ",
" This jurisdictional issue has evoked varying responses from courts. ",
" Compare United States v. Abbamonte, 759 F.2d 1065 (2d Cir.1985) (order denying dismissal of indictment on ground of prior plea agreement appealable), with United States v. Bird, 709 F.2d 388 (5th Cir.1983) (order denying dismissal of indictment on ground of prior plea agreement not appealable), and United States v. Eggert, 624 F.2d 973 (10th Cir.1980) (same). ",
" The disagreement has stemmed from uncertainty as to the continued force of the Supreme Court's 1910 decision in Heike v. United States, 217 U.S. 423, 30 S.Ct. ",
"539, 54 L.Ed. ",
"821 (1910). ",
" In Heike, the Supreme Court unequivocally rejected an interlocutory appeal in the context of a claim of statutory immunity flowing from a grand jury appearance, 217 U.S. at 433, 54 S.Ct. ",
"at 542-43. ",
" Sixty-seven years later, however, the continued force of that decision was put in doubt by Abney v. United States, 431 U.S. 651, 97 S.Ct. ",
"2034, 52 L.Ed.2d 651 (1977), which allowed an interlocutory appeal of an adverse double jeopardy ruling. ",
" Abney rendered Heike uncertain because Heike had explicitly relied on the unavailability of pretrial appeals to challenge adverse double jeopardy rulings. ",
" Heike, 217 U.S. at 432, 54 S.Ct. ",
"at 542.",
"\n\n\n7\nThe tension between Abney and Heike began to be resolved the year after Abney was decided. ",
" In United States v. MacDonald, 435 U.S. 850, 98 S.Ct. ",
"1547, 56 L.Ed.2d 18 (1978), the Court ruled that an interlocutory appeal was not available from a ruling denying a motion to dismiss for alleged violation of the Sixth Amendment right to a speedy trial. ",
" Though the speedy trial claim, if upheld in the trial court, would have insulated the defendant from having to defend against the criminal charges, the Court ruled that the Sixth Amendment right would be adequately vindicated by a reversal of the conviction on appeal from a final judgment. ",
" In a footnote, the Court pointed out that not every claim that, if successful, would require dismissal of an indictment may be the basis for interlocutory review, and cited Heike, presumably as an example of claims that must await post-conviction review. ",
" Id. at 860 n. 7, 98 S.Ct. ",
"at 1552-53 n. 7. ",
" However, the MacDonald footnote did not comment on the fact that Abney had allowed pre-conviction appeal of a double jeopardy claim, the analogy on which Heike had relied.",
"\n\n\n8\nMore recently, in Midland Asphalt Corp. v. United States, 489 U.S. 794, 109 S.Ct. ",
"1494, 103 L.Ed.2d 879 (1989), the Court elevated the language of the MacDonald footnote to the text, id. at 801, 109 S.Ct. ",
"at 1499, and gave a very strong signal that a rejected immunity claim may not be raised on an interlocutory appeal. ",
" Midland Asphalt ruled that an interlocutory appeal was not available to challenge the denial of a motion to dismiss an indictment on the ground of an alleged violation by the Government of Rule 6(e) of the Federal Rules of Criminal Procedure. ",
" The Court explained that interlocutory appeal may occur not merely when the right at issue, if vindicated, will result in dismissal of the indictment, but only when the right at issue is itself a constitutional or statutory right not to be tried, such as the protections of the Double Jeopardy Clause, see Abney v. United States, supra, or the Speech or Debate Clause, see Helstoski v. Meanor, 442 U.S. 500, 99 S.Ct. ",
"2445, 61 L.Ed.2d 30 (1979).",
"\n\n\n9\nIt is arguable that an immunity claim satisfies even the limited right of interlocutory appeal recognized in Midland Asphalt. ",
" The argument would be at its strongest if the immunity claim were grounded on statutory insulation, such as that formerly contained in 18 U.S.C. Sec. ",
"3486, repealed by Pub.",
"L. 91-452, tit. ",
" II, Sec. ",
"228(a), 84 Stat. ",
"930 (1970). ",
" See Ullmann v. United States, 350 U.S. 422, 76 S.Ct. ",
"497, 100 L.Ed. ",
"511 (1956). ",
" Statutes such as former section 3486, which conferred transactional immunity, typically provided that \"no such [immunized] witness shall be prosecuted.\" ",
" 18 U.S.C. Sec. ",
"3486(c). ",
" Such a prohibition seems to be an \"explicit statutory ... guarantee that trial shall not occur,\" Midland Asphalt, 489 U.S. at 801, 109 S.Ct. ",
"at 1499. ",
" Nevertheless, we cannot ignore the fact that Midland Asphalt quoted with approval the MacDonald footnote's reference to the unavailability of an interlocutory appeal to challenge immunity claims:\n\n\n10\n\"Dismissal of the indictment is the proper sanction when a defendant has been granted immunity from prosecution, when his indictment is defective, or, usually, when the only evidence against him was seized in violation of the Fourth Amendment. ",
" Obviously, however, this has not led the Court to conclude that such defendants can pursue interlocutory appeals.\" ",
" MacDonald, supra, 435 U.S. at 860 n. 7, 98 S.Ct. ",
"at 1552-53 n. 7.",
"\n\n\n11\nMidland Asphalt, 489 U.S. at 801, 109 S.Ct. ",
"at 1499. ",
" Apparently, the insulation from prosecution, and thus from trial, provided by a transactional immunity statute (like the one in Heike, 217 U.S. at 426, 30 S.Ct. ",
"at 540), is not the \"explicit statutory ... guarantee that trial will not occur\" that Justice Scalia had in mind.",
"\n\n\n12\nThe pending case presents a weaker claim for interlocutory appeal than would arise from a claim grounded on a statute explicitly insulating from prosecution a witness given transactional immunity. ",
" Macchia relies not on a statute conferring transactional immunity, but rather on an alleged agreement with the United States Attorney, the breach of which, he contends, violates the Due Process Clause of the Fifth Amendment. ",
" There can be virtually no doubt that a violation of the general prohibition of the Due Process Clause is not a violation of an \"explicit statutory or constitutional guarantee that trial will not occur,\" as that phrase is used in Midland Asphalt.",
"\n\n\n13\nThe appellant seeks to draw comfort from the fact that his argument is grounded on a constitutional protection--the Due Process Clause, but that argument misses Justice Scalia's point. ",
" The test is not whether the right alleged to be impaired is grounded in the Constitution; MacDonald's claim, for which interlocutory appeal was denied, was based on the Sixth Amendment. ",
" The test is whether the pertinent protection, constitutional or statutory, explicitly guarantees a right not to be tried.",
"\n\n\n14\nIn this Circuit, we have heretofore permitted interlocutory appeals to assert breach of plea agreements alleged to confer immunity. ",
" See United States v. Abbamonte, 759 F.2d at 1070-71, United States v. Alessi, 544 F.2d 1139 (2d Cir.) ",
" (Alessi III ), cert. ",
"denied, 429 U.S. 960, 97 S.Ct. ",
"384, 50 L.Ed.2d 327 (1976); United States v. Alessi, 536 F.2d 978 (2d Cir.1976) (Alessi I ). ",
" Alessi I, which predated Abney, allowed the interlocutory appeal, analogizing, for appeal purposes, claims based on immunity provisions of plea agreements to double jeopardy claims. ",
" 536 F.2d at 980 (relying on United States v. Beckerman, 516 F.2d 905 (2d Cir.1975)). ",
" In Alessi III, which reached us after certiorari had been granted in Abney, Abney v. United States, 426 U.S. 934, 96 S.Ct. ",
"2646, 49 L.Ed.2d 385 (1976) (granting certiorari), the appealability issue evoked a strong opinion by Judge Friendly, arguing against pretrial review of double jeopardy and immunity claims. ",
" This view was evidently not shared by Judge Feinberg, who concurred in the result on the appealability issue. ",
" Alessi III, 544 F.2d at 1155. ",
" Though Judge Friendly, and Judge Van Graafeiland, who concurred in Judge Friendly's opinion, would have disallowed the interlocutory appeal and followed Heike \"[i]f the point were open in this circuit,\" id. at 1152, they felt bound by Alessi I.\n\n\n15\nWhen the issue arose again after Abney, we said in Abbamonte that \"[w]e are obliged to follow the holding on appealability in Alessi [I and III ] unless we have some basis for concluding that that ruling has been eroded by subsequent decisions of the Supreme Court.\" ",
" 759 F.2d at 1071. ",
" We did not find such erosion in Abney, and, though the MacDonald footnote had cited Heike, we evidently concluded that there was an insufficient basis for considering the law of this Circuit to have been altered. ",
" Now, however, Midland Asphalt has sent a clear message that at least an immunity claim grounded on the Due Process Clause, as we have in the pending case, and very likely an immunity claim grounded on a statute conferring transactional immunity, as in Heike, can no longer be raised on an interlocutory appeal. ",
" In Midland Asphalt, the Supreme Court has not only promoted to text the MacDonald footnote, which had cited Heike, but has also provided the explicit rationale that an interlocutory appeal will lie in the criminal context only where the constitutional or statutory protection relied upon confers a right not to be tried, as distinguished from a right to be free of some adverse action for which the remedy is dismissal of the indictment. ",
" Moreover, we need not consider the merits of appellant's claim for transactional immunity in order to determine our appellate jurisdiction: even if he is right that his immunity agreement should be construed to confer transactional immunity, his claim for dismissal of the indictment would still rest on the general prohibition of the Due Process Clause, rather than an immunity statute explicitly protecting the right not to be tried.",
"\n\n\n16\nWe therefore conclude that the appealability rulings in Alessi I, Alessi III, and Abbamonte have been eroded, and we overrule those rulings and dismiss this appeal for lack of appellate jurisdiction.1\n\n\n\n1\n This opinion has been circulated to the active judges of the Court\n\n\n"
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"A Life Table for Liriomyza trifolii (Diptera: Agromyzidae) in a Temperate Zone of Northeast Egypt With Key Factor Analysis.",
"\nThe American serpentine leaf miner, Liriomyza trifolii (Burgess) (Diptera: Agromyzidae), is a perennial pest of leguminous crops in the Mediterranean region. ",
"A life table was constructed for L. trifolii infesting broad bean, Vicia faba L., in northern Egypt. ",
"Key factor analysis was used to rank sources of immature mortality over two seasons. ",
"Leaf miners had three successive generations, and a partial fourth, in each year, with peak abundance in March-April. ",
"Less than 15 and 22% of L. trifolii survived to adult in seasons one and two, respectively. ",
"The largest contributor of immature leaf miner mortality in both seasons was unknown (41.2 and 39.1% of total mortality, respectively), and likely comprised a combination of abiotic factors, parasitoid-inflicted mortality (host-feeding), and predation. ",
"Parasitism was second, contributing 36.2 and 35.6% of total mortality in the two seasons, respectively, primarily due to larval parasitism by Diglyphus isaea (Walker) (Hymenoptera: Eulophidae), and low levels of larval-pupal parasitism by Opius pallipes Wesmael (Hymenoptera: Braconidae) and Halticoptera circulus (Walker) (Hymenoptera: Pteromalidae). ",
"Residual mortality resulted from malformed pupae or failed adult emergence. ",
"Key factor analysis revealed malformation to be the major cause of pupal mortality. ",
"Sequential regression confirmed that unknown mortality and D. isaea were the top stage-specific factors, both acting on larvae. ",
"Parasitoid abundance tracked host abundance across generations, but density dependence was not observed for any mortality factor, and the magnitudes of regression slopes were small. ",
"The results indicate the potential importance of conservation biological control in management of L. trifolii, given that naturally occurring parasitoids and other biotic/abiotic factors exert significant mortality on immature leaf miners."
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"SF examines impacts of its many budget set-asides\n\nSupervisors Katy Tang and Aaron Peskin at a Board of Supervisors meeting last year. ",
"Supervisors Katy Tang and Aaron Peskin at a Board of Supervisors meeting last year. ",
"Photo: Gabrielle Lurie, Special To The Chronicle Photo: Gabrielle Lurie, Special To The Chronicle Image 1 of / 1 Caption Close SF examines impacts of its many budget set-asides 1 / 1 Back to Gallery\n\nSan Francisco lawmakers are beginning to take a hard look at the long-term consequences of the city’s abundant use of budget set-asides — voter-approved ballot measures that mandate how the government spends money on specific projects.",
"\n\nSince the mid-1990s, the amount of money San Francisco has allocated based on voter-adopted initiatives has ballooned dramatically, according to a report compiled by the city controller’s office. ",
"That has alarmed some city supervisors.",
"\n\nIn the next five years, the report said, San Francisco will spend $1.6 billion on set-asides— 30 percent of the city’s $5 billion general fund, its primary source of discretionary funding. ",
"Self-supporting departments such as the airport comprise the remainder of the city’s $10.1 billion budget.",
"\n\nThat’s a sharp increase from just over a decade ago. ",
"By comparison, in the 1994-95 fiscal year city spending on set-aside initiatives was $200 million, according to the report.",
"\n\nAt a hearing before the Budget and Finance Committee Tuesday, Supervisors Katy Tang and Aaron Peskin expressed concern about constraints the set-asides place on policymakers as budget priorities shift year to year.",
"\n\nWhile set-asides provide a fixed amount of funding for essential city services, like transportation and libraries, once funding is locked up by the voters, city officials are essentially powerless to make adjustments, Tang said.",
"\n\n“Future boards of supervisors and mayors essentially have their hands tied for many of these measures,” she said. “",
"We don’t have the ability to be flexible in these funding decisions.”",
"\n\nSan Francisco currently has 19 set-aside mandates, far more than any other large city in the state, according to the controller’s office. ",
"In fact, local governments in the rest of California have a total of 10 similar funding requirements.",
"\n\n“For comparison to other cities, Los Angeles has two adopted requirements, San Diego has one and San Jose has none,” the controller’s report said.",
"\n\nMost of San Francisco’s set-aside requirements — nearly $418 million — are allocated for the Municipal Transportation Agency.",
"\n\n“All of the set-asides are for remarkably worthy causes, but it really constrains the city, the Board of Supervisors and the mayor from being able to make tough decisions on an annual basis,” Peskin said. “",
"It’s the easy way out — it doesn’t require us to look for new sources of revenues” and it “constrains the ability to govern and allocate money in the best and most mature way that’s most responsive to the needs of the day.”",
"\n\nCity officials on Tuesday briefly discussed placing expiration dates on all existing set-asides, which would allow legislators to reconsider spending mandates after a period of time. ",
"Such a move would have to be approved by voters. ",
"Currently, only 10 of the city’s 19 set-asides have fixed expiration dates.",
"\n\n“My goal is to work on a set of policies to address our spending” set-asides, Tang said.",
"\n\nDominic Fracassa is a San Francisco Chronicle staff writer. ",
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"---\nabstract: 'In this paper, we establish optimal rates of adaptive estimation of a vector in the multi-reference alignment model, a problem with important applications in fields such as signal processing, image processing, and computer vision, among others. ",
"We describe how this model can be viewed as a multivariate Gaussian mixture model under the constraint that the centers belong to the orbit of a group. ",
"This enables us to derive matching upper and lower bounds that feature an interesting dependence on the signal-to-noise ratio of the model. ",
"Both upper and lower bounds are articulated around a tight local control of Kullback-Leibler divergences that showcases the central role of moment tensors in this problem.'",
"\naddress:\n- |\n [Afonso S. Bandeira]{}\\\n [Department of Mathematics]{}\\\n [Courant Institute of Mathematical Sciences]{}\\\n [Center for Data Science]{}\\\n [New York Univeristy,]{}\\\n [New York, NY 10012, USA]{}\\\n- |\n [Philippe Rigollet]{}\\\n [Department of Mathematics]{}\\\n [Massachusetts Institute of Technology]{}\\\n [77 Massachusetts Avenue,]{}\\\n [Cambridge, MA 02139-4307, USA]{}\\\n- |\n [Jonathan Weed]{}\\\n [Department of Mathematics]{}\\\n [Massachusetts Institute of Technology]{}\\\n [77 Massachusetts Avenue,]{}\\\n [Cambridge, MA 02139-4307, USA]{}\\\nauthor:\n- \n- \n- \nbibliography:\n- 'mra\\_moment\\_matching.bib'\ntitle: 'Optimal rates of estimation for multi-reference alignment'\n---\n\n,\n\n,\n\nMulti-reference alignment, Orbit retrieval, Mixtures of Gaussians\n\nIntroduction\n============\n\nA fundamental problem arising in various scientific and engineering domains is the presence of heterogenous data. ",
"In many applications, each observation of an object of interest is corrupted not only by noise but also by a latent transformation, which can often be modeled as the action of an unknown element of a known group. ",
"The presence of these latent transformations raises serious challenges, both in theory and in practice.",
"\n\nOur goal in this work is to inaugurate the statistical study of such models and establish optimal rates of estimation for a particular version known in the computer science literature as *multi-reference alignment*, a simple problem arising in fields such as structural biology [@SchValNun05; @TheSte12; @Sad89], image recognition [@Bro92], and signal processing [@ZwaHeiGel03]. ",
"The tools we develop to prove these bounds provide a unified theoretical framework for statistical estimation in the presence of algebraic structure.",
"\n\nAlgebraically structured models and cryo-EM\n-------------------------------------------\n\nA primary motivation to study models with algebraic structure is cryo-electron microscopy (cryo-EM), an important technique to determine three-dimensional structures of biological macromolecules. ",
"The citation for the 2017 Nobel prize in Chemistry, awarded to its inventors, reads:\n\n> The Nobel Prize in Chemistry 2017 was awarded to Jacques Dubochet, Joachim Frank and Richard Henderson “for developing cryo-electron microscopy for the high-resolution structure determination of biomolecules in solution\".",
"\n\nIn this imaging technique, each measurement consists of a noisy tomographic projection of a rotated—by an unknown rotation in $\\mathrm{SO}(3)$—copy of an unknown molecule. ",
"The task is then to reconstruct the molecule density from many such measurements. ",
"This reconstruction problem has received significant attention, primarily from computational perspectives, but its statistical properties remain largely unexplored. ",
"This problem features three singular characteristics: (i) The latent group action in each observation—here a rotation—(ii) the tomographic projection and (iii) the presence of high noise as illustrated by Figure \\[FIG:noisy\\_image\\].",
"\n\n[r]{}[0.4]{}\n\nAs a first step toward the statistical analysis of this class of algebraically structured models, we focus on a simpler model that features two of the aforementioned characteristics, namely (i) the presence of a group action and (iii) the presence of high noise. ",
"This model is simpler to analyze and already presents fundamentally novel statistical features that manifest themselves in nonclassical rates of estimation.",
"\n\nDenote by ${\\mathcal{G}}$ a known compact subgroup of the group $\\mathrm{O}(L)$ of orthogonal transformations of ${{\\rm I}\\kern-0.18em{\\rm R}}^L$. Throughout this paper, we identify the action of a group element $G \\in {\\mathcal{G}}\\subset \\mathrm{O}(L)$ on ${{\\rm I}\\kern-0.18em{\\rm R}}^L$ by left-multiplication with an orthogonal matrix $G$. We slightly abuse terminology by referring to $G$ as a group element. ",
"Our goal is to recover a parameter $\\theta \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$, which we often refer to as a *signal*, on the basis of very noisy observations corrupted by unknown elements of ${\\mathcal{G}}$. Concretely, we observe $$\\label{eqn:basic_model}\nY_i = G_i \\theta + \\sigma \\xi_i\\,,$$ where $G_i \\in {\\mathcal{G}}$ is unknown and $\\xi_i$ is standard Gaussian noise independent of $G_i$. The parameter $\\theta$ is only identifiable up to the action of ${\\mathcal{G}}$, so we focus on obtaining an estimator $\\tilde \\theta$ whose distance to the orbit of $\\theta$ as defined by $$\\rho(\\tilde \\theta, \\theta) := \\min_{G \\in {\\mathcal{G}}} \\|\\tilde \\theta - G \\theta\\|$$ is small in expectation. ",
"We call an *algebraically structured model*. ",
"For normalization purposes, we assume that $c^{-1} \\leq \\|\\theta\\| \\leq c$ for some universal positive constant $c$. Fixing the scaling of $\\theta$ in such a way allows us to control the signal-to-noise ratio of the problem only via the parameter $\\sigma$, which plays a central role in the sequel.",
"\n\nPrior work: The synchronization approach {#sec:synchro}\n----------------------------------------\n\nThe difficulty of algebraically structured models resides in the fact that both the signal $\\theta$ and the transformations $G_1,\\ldots,G_n \\in {\\mathcal{G}}$ are unknown and the latter are therefore latent variables. ",
"If the group elements were known, one could easily estimate the vector $\\theta$ by taking the average of $G_i^{-1} Y_i, i =1, \\ldots, n$. In fact, this simple observation is the basis of the leading current approach to this problem, called the “synchronization approach\" [@BanChaSin14; @BanCheSin15]. ",
"Specifically, synchronization aims at recovering the latent variables $G_{i}$ by solving a problem of the form $$\\label{eqn:latent_rotations}\n\\min_{H_{1}, \\dots, H_{n} \\in {\\mathcal{G}}} \\sum_{1 \\leq i, j \\leq n} \\big\\|{H}_{i}^{-1} Y_i - {H}_{j}^{-1} Y_j\\big\\|^2\\,.$$ Denoting by $\\tilde H_{i}$ the solutions of , one can then estimate $\\theta$ by the average of ${\\tilde H}^{-1}_i Y_i, i =1, \\ldots, n$.\n\nDespite synchronization problems being computationally hard in general [@BanChaSin14], certain theoretical guarantees have been derived under specific noise models that are unfortunately not realistic for the problems of interest in this paper. ",
"For example, it is often assumed that each *pair* of observations is corrupted by independent noise, so that the terms in the sum in are independent. ",
"Instead, our model adopts the more relevant assumption of independent noise on each observation. ",
"Among the most prominent methods to date are spectral methods [@Sin11; @BanSinSpi11], semidefinite relaxations [@BanChaSin14; @BanCheSin15; @AbbBanBra14b; @BanBouSin16; @JavMonRic16; @BanBouVor16], methods based on Approximate Message Passing [@PerWeiBan16a] and other modified power methods [@Boumal_ProjPowerMethod; @Chen_Candes_ProjPowerMethod]. ",
"Synchronization also enjoys many interesting connections with geometry (see, e.g., [@Gao_GeometrySynchronization]).",
"\n\nAnother fundamental drawback of the synchronization approach is its intolerance to large noise levels $\\sigma$. When $\\sigma$ is significantly smaller than $\\|\\theta\\|$, the prior work referenced above has demonstrated empirically and theoretically that the synchronization approach yields excellent results. ",
"Intuitively, the success of this approach relies on the fact that when the noise is small, macroscopic features of the underlying signal are still visible. ",
"However, as noted in our discussion of cryo-EM, the noise level in applications is often significantly larger than the signal [@Sig16], which renders the synchronization approach unusable. ",
"An illustration of the difference between these regimes appears in Figure \\[fig:noise\\_low\\_high\\].",
"\n\n![",
"Instances of the multi-reference alignment problem, defined in Section \\[sec:mra\\], at low (left column) and high (right column) noise levels. ",
"The true underlying signal appears in gray, and the noised version appears in red. ",
"When the noise level is low, large features of the signal are still visible despite the noise; in the presence of large noise, however, the signals cannot reliably be synchronized.[]{data-label=\"fig:noise_low_high\"}](noise_low_high.pdf){width=\"\\textwidth\"}\n\nFrom a theoretical standpoint, this fact implies that the low- and high-noise regimes are very different: when the noise is sufficiently large, prior work has shown that the transformations are impossible to reliably estimate, regardless of the number of samples [@Weinstein_limitssignalalignment; @Sapiro_limitsimagealignment]. ",
"Thus, for the high-noise regime, new techniques are required. ",
"We therefore focus in this work on the case where the variance of the noise is bounded below by a constant.",
"\n\nThe Gaussian mixture approach {#SEC:setup}\n-----------------------------\n\nWe propose an alternative to the synchronization approach that completely bypasses the estimation of the transformations $G_1,\\dots,G_n$ in favor of estimating $\\theta$ directly. ",
"To do so, we first show how to recast our model as a continuous mixture of Gaussians whose centers are algebraically constrained.",
"\n\nTo reinterpret as a Gaussian mixture model, we replace the latent group elements $G_1, \\dots, G_n$ by group elements drawn independently and uniformly at random (according to the Haar measure) from ${\\mathcal{G}}$. This is a worst-case assumption, which is appropriate since we prove minimax rates.[^1] Indeed, we can always reduce to this case: since the Gaussian distribution is invariant under the action of the orthogonal group, we can transform each observation $Y_i$ into $H_i Y_i$, where $H_i$ is uniformly distributed over ${\\mathcal{G}}$ and independent of all other random variables. ",
"Since $H_i G_i$ is also uniformly distributed over ${\\mathcal{G}}$, these new observations are drawn from a mixture of Gaussians whose centers are given by $G\\theta$, $G \\in {\\mathcal{G}}$, with uniform mixing weights. ",
"In particular, these centers are linked together by a rigid algebraic structure: they are the orbit of $\\theta \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$ under the action of ${\\mathcal{G}}$.\n\nWe therefore specify the following Gaussian mixture model. ",
"Given a noise level $\\sigma$, group ${\\mathcal{G}}\\subseteq \\mathrm{O}(L)$, and parameter of interest $\\theta \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$, denote by ${\\mathrm}{P}_\\theta$ the distribution of a random variable $Y$ satisfying $$\\label{eq:MRA:model}\nY = G\\theta + \\sigma \\xi\\,,$$ where $G$ is drawn uniformly from ${\\mathcal{G}}$ and $\\xi \\sim {\\mathcal{N}}(0, I_L)$ is independent Gaussian noise.",
"\n\nWe assume throughout that the noise variance $\\sigma^2$ is known. ",
"This assumption is realistic in many applications such as imaging or signal processing, where it is inexpensive to collect pure-noise samples from ${\\mathrm}{P}_0$ and thereby estimate $\\sigma^2$ to arbitrary accuracy.",
"\n\nIn this work, we analyze the maximum likelihood estimator (MLE) $\\tilde \\theta_n$ for : $$\\label{eq:ML:mixtureGaussian}\n\\tilde \\theta_n=\\operatorname*{argmax}_{\\phi \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L} \\sum_{i=1}^n \\log {{\\rm I}\\kern-0.18em{\\rm E}}\\big[ \\exp\\big(- \\frac{1}{2\\sigma^2} \\|Y_i - G \\phi\\|^2\\big)\\big|Y_i\\big]\\,.$$ We focus on obtaining the optimal scaling of the quantity $\\rho(\\tilde \\theta_n, \\theta)$ with the signal-to-noise ratio of the problem.[^2] This question is central to signal processing problems where $\\sigma$ is quite large, since it determines the order of magnitude of the sample size $n$ required to achieve a certain accuracy. ",
"Moreover, in many applications, technological improvements to the measurement apparatus can directly improve the effective value of $\\sigma$—in cryo-EM, for instance, this is a focus of active research [@Sig16]. ",
"For these reasons, understanding the scaling of $\\rho(\\tilde \\theta_n, \\theta)$ with $\\sigma$ is a core question both in theory and in practice. ",
"Our main upper bound result gives a *uniform* analysis of this maximum likelihood estimator, valid for any algebraically constrained model. ",
"We complement this analysis with lower bounds which are equally universal. ",
"In both cases, we proceed by controlling the Fisher information of the model.",
"\n\nGaussian mixture models have been extensively studied in the statistical literature since their introduction by [@Pea94] in the nineteenth century (see, e.g., [@McLPee00] for an overview). ",
"As illustrated by the extant literature, mixture models are quite rich and broadly applicable to a variety of statistical problems ranging from clustering to density estimation. ",
"It is known that the rate of estimation of the parameters of a Gaussian mixture with $k$ components can scale like $n^{-O(1/k)}$ (see for example [@Che95; @MoiVal10] and more recently [@HeiKah15] for an interesting explanation from the point of view of model misspecification). ",
"In this work, our analysis of the multi-reference alignment problem focuses on a setting where the convergence of $\\tilde \\theta_n$ to $\\theta$ occurs at the parametric $n^{-1/2}$ rate; nevertheless, our results show that even in this benign setting, the optimal dependence of this rate on $\\sigma$ can still be extremely poor.",
"\n\nMulti-reference alignment {#sec:mra}\n-------------------------\n\nAs an application of our techniques, we analyze and establish optimal rates for a model known as multi-reference alignment, a simple algebraically structured model. ",
"Multi-reference alignment is a special case of cryo-EM, where instead of three-dimensional rotations we consider phase shifts of a periodic signal. ",
"This represents a special case of cryo-EM because it corresponds to the situation when the axis of rotation of the molecule is known, but not its angle.",
"\n\nIn addition to being a toy model for cryo-EM, this simpler model is also of independent interest in several applications including in structural biology [@TheSte12] and radar classification [@ZwaHeiGel03]. ",
"A discrete version of this problem where the group is the cyclic group ${\\mathbb{Z}}/L$ acting on the coordinates of $\\theta$ was introduced in [@BanChaSin14] to permit approaches based on semi-definite programming; however, our results indicate that this simplification is not actually benign, in the sense that the discretized model admits significantly worse rates of estimation than the model we describe below. ",
"We compare our more general model with theirs in Appendix \\[supp\\].",
"\n\nLet $f: [0, 1] \\to {{\\rm I}\\kern-0.18em{\\rm R}}$ be an unknown function, and let $g_{s}$ be the shift operator which acts on $f$ by $g_{s}\\hspace{-.5ex}\\circ\\hspace{-.5ex} f(x) = f(x + s)$, where $s \\in [0, 1)$ and the addition is performed modulo $1$. These operators clearly form a group, denoted ${\\mathcal{S}}$, which is isomorphic to ${{\\rm I}\\kern-0.18em{\\rm R}}/{\\mathbb{Z}}$. We observe independent copies of $$\\label{eq:continuous_model}\nY = g_{s} \\hspace{-.5ex}\\circ\\hspace{-.5ex} f(x) + \\sigma \\xi\\,,$$ where where $g_{s} \\hspace{-.5ex}\\circ\\hspace{-.5ex} f(x) \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$ denotes the vector $(g_s\\hspace{-.5ex}\\circ\\hspace{-.5ex} f(x_j))_{j=1}^L$ for the fixed design $x_j = j/L$, $s$ is drawn uniformly at random from $[0, 1]$, and $\\xi \\sim {\\mathcal{N}}(0, I_L)$ is independent of $s$. To put into the same form as , assume that the function $f$ is *band limited*—i.e., ",
"the Fourier transform of $f$ vanishes outside the interval $[-B, B]$ for some positive integer $B$—and that the measurements are performed above the *Nyquist frequency*—i.e., ",
"$L > 2B$. This assumption ensures that the function $f$ is identifiable, in that the discrete measurements $f(x_1), \\dots, f(x_L)$ suffice to recover $f$.\n\nThe action of ${\\mathcal{S}}$ on $f(x)$ can be identified with a subgroup of the orthogonal group $\\mathrm{O}(L)$ by passing to the Fourier domain. ",
"Indeed, since $f$ is band limited, it can be identified with the vector of its $2B + 1$ Fourier coefficients $(\\hat f(-B), \\dots, \\hat f(B))$, which we denote by $\\hat f$. Writing $$f(x) = \\sum_{k=-B}^{B} \\hat f(k) e^{-2\\pi \\mathrm{i} k x}\\,,$$ yields the relation $$\\widehat{g_{s} \\hspace{-.5ex}\\circ\\hspace{-.5ex} f}(k) = \\hat f(k) e^{-2 \\pi \\mathrm{i} s k} =: \\hat f(k) z^k\\,,$$ where $z = e^{-2 \\pi \\mathrm{i} s}$ is a complex number of unit norm. ",
"This identifies ${\\mathcal{S}}$ with the circle group $U(1)$. Writing $\\theta$ for the vector $f(x)$, we obtain an example of : we observe independent copies of $$\\label{eq:phase_shift}\nY = G_{z} \\theta + \\sigma \\xi\\,,$$ where $\\theta \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$ is the parameter to be estimated, $z$ is drawn uniformly at random from $U(1)$, $\\xi$ is a standard Gaussian random variable independent of $z$, and $G_{z} \\theta$ is defined by its Fourier coefficients: $$\\label{eq:discrete_action}\n\\widehat{G_z \\theta}_k = z^k \\hat \\theta_k \\text{ for $k = -\\lfloor L/2 \\rfloor, \\dots, \\lfloor L/2 \\rfloor$,}$$ where we use the notation $\\hat \\theta$ to represent the discrete Fourier transform of $\\theta$. If we restrict $z$ to be of the form $\\omega^L$, where $\\omega$ is a primitive $L$th root of unity, then we recover the discrete model of [@BanChaSin14]. ",
"We call model , the *phase shift* model.",
"\n\nOrganization of the paper\n-------------------------\n\nIn Section \\[section\\_mainresults\\] we present our main results, Theorems \\[thm:main\\_lower\\] and \\[thm:main\\_upper\\], providing minimax rates for the multi-reference alignment problem under the phase shift model . ",
"The proofs of these theorems rely on developing general tools for analyzing algebraically structured models and controlling the Kullback-Leibler divergence between distributions corresponding to two different signals.",
"\n\nIn Section \\[sec:uniform\\_upper\\], we give guarantees on the maximum likelihood estimator (MLE) under a condition on the curvature of the KL divergence. ",
"We then specialize to the phase shift model in Section \\[SEC:modMLE\\] and develop a modified MLE for the phase shift model which achieves the optimal rates in Theorem \\[thm:main\\_upper\\]. ",
"Section \\[sec:multi\\] concludes by establishing the lower bound in Theorem \\[thm:main\\_lower\\]; the proof involves finding pairs of different signals with several matching invariant moment tensors. ",
"Both lower and upper bounds depend on an analysis of the KL divergence for algebraically structure models, which appears in Appendix \\[sec:tight\\_bounds\\].",
"\n\nNotation\n--------\n\nWe define the Fourier transform $\\hat \\theta$ of $\\theta \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$ by $$\\hat \\theta_j = \\frac{1}{\\sqrt L} \\sum_{k=1}^L e^{2 \\pi \\mathrm{i} j k /L} \\theta_k\\,, \\quad \\quad - \\lfloor L/2 \\rfloor \\leq j \\leq \\lfloor L/2 \\rfloor\\,.$$ We assume for convenience throughout that $L$ is odd.",
"\n\nThe symbol $\\|\\cdot \\|$ denotes the $\\ell_2$ norm on ${{\\rm I}\\kern-0.18em{\\rm R}}^L$. For any positive integer $k$, we write $[k]=\\{1, \\ldots, k\\}$. We use $z^*$ to denote the complex conjugate of $z \\in \\mathbb{C}$\n\nGiven a vector $t$, let $t^{\\otimes k}$ denote the order-$k$ tensor formed by taking the $k$-fold tensor product of $t$ with itself. ",
"Denote by $\\|A\\|$ the Hilbert-Schmidt norm of a tensor $A$, defined by $\\|A\\|^2 = \\langle A, A\\rangle$, where $\\langle \\cdot, \\cdot \\rangle$ denotes the entrywise inner product. ",
"It is easy to check that, for any two column vectors $t, u$ of the same size, the identity $\\langle t^{\\otimes k}, u^{\\otimes k}\\rangle=(t^\\top u)^k$ holds.",
"\n\nA tensor $A$ is *symmetric* if $A_{i_1\\dots i_k} = A_{i_{\\pi(1)} \\dots i_{\\pi(k)}}$ for any permutation $\\pi$ of $[k]$. For such tensors, the value $A_{i_1\\dots i_k}$ depends only on the multiset $\\{i_1,\\dots,i_k\\}$, or equivalently on the multi-index $\\alpha$ defined by $\\alpha_\\ell = |\\{j \\in [k]: i_j = \\ell\\}|$.\n\nThe Kullback-Leibler (KL) divergence between two distributions $P$ and $Q$ such that $P \\ll Q$ is given by $$D(P {\\,\\|\\,}Q) = \\int\\log \\big(\\frac{dP}{dQ}\\big) \\,dP\\,.$$ It is well known that $D(P {\\,\\|\\,}Q) \\geq 0$, with equality holding iff $P = Q$.\n\nMain results {#section_mainresults}\n============\n\nAs mentioned above, the rescaled loss $\\sqrt{n}\\rho(\\tilde \\theta_n, \\theta)$ of the maximum depends asymptotically on the Fisher information of the model, which can be related to the curvature of the Kullback-Leibler divergence around its minimum. ",
"Conversely, (lack of) curvature of the Kullback-Leibler divergence around its minimum is what controls minimax lower bounds that are valid for *any* esitmator. ",
"We provide a unified framework for proving upper and lower bounds based on the curvature of the divergence function, following an idea originally introduced in [@LepNemSpo99] in the context of functional estimation and further developed by [@JudNem02; @CaiLow11; @WuYan16; @CarVer17; @BlaCarGut17]. ",
"In the multi-reference alignment model, this approach allows us to relate Kullback-Leibler divergence to moment tensors, which can in turn be controlled using Fourier-theoretic arguments.",
"\n\nOur analysis establishes that the difficulty of estimating a particular signal $\\theta$ depends on the support of the Fourier transform $\\hat \\theta$ of $\\theta$. Define the *positive support* $\\operatorname{psupp}(\\hat \\theta)$ of $\\hat \\theta$ by $$\\operatorname{psupp}(\\hat \\theta) = \\{j \\mid j \\in \\{1, \\dots, L/2 \\}, \\hat \\theta_j \\neq 0\\}\\,.$$ We focus only on the positive indices because the signal $\\theta$ is real, so the Fourier transform is conjugate symmetric: $\\hat \\theta_j^* = \\hat \\theta_{-j}$. We make the following assumptions.",
"\n\n\\[assumption:norm\\] There exists an absolute constant $c > 1$ such that $c^{-1} \\leq \\|\\theta\\| \\leq c$.\n\n\\[assumption\\] Moreover, there exists an absolute constant $c_0$, not depending on $n$, such that $c_0 \\leq |\\hat \\theta_j|$ for all $j \\in \\operatorname{psupp}(\\hat \\theta)$.\n\nWe denote by ${\\mathcal{T}}$ the set of vectors satisfying Assumptions \\[assumption:norm\\] and \\[assumption\\]. ",
"Assumption \\[assumption:norm\\] is benign and is adopted for normalization purposes, so that $\\sigma$ captures entirely the signal-to-noise ratio of the problem. ",
"Regarding Assumption \\[assumption\\], we emphasize that this is the situation of most interest to practitioners: the existence of very small, but non-zero, coordinates whose values approach $0$ with $n$ should rightly be considered pathological. ",
"Assumption \\[assumption\\] rules out certain artificial situations analogous to classical difficulties arising in estimating mixtures of Gaussians, such as distinguishing the mixture $.5{\\mathcal{N}}(+{\\varepsilon}, 1) + .5{\\mathcal{N}}(-{\\varepsilon}, 1)$ from the single Gaussian ${\\mathcal{N}}(0, 1)$ for very small ${\\varepsilon}$. Determining minimax rates of estimation without Assumption \\[assumption\\] is certainly of theoretical interest, and we leave this question for future work.",
"\n\nAs noted above, our results focus on understanding how minimax rates of estimation for the multi-reference alignment problem scale with $\\sigma$. This is the question of primary interest in algebraically structured problems like cryo-EM, since in these applications $\\sigma$ is the only part of the model that can be improved by the development of new imaging technologies and techniques. ",
"We note that our results do not address the dependence on the dimension $L$, and obtaining sharp dependence on $L$ is an attractive open problem.",
"\n\nThe following theorem reveals a surprising phenomenon: even under Assumption \\[assumption\\], the multi-reference alignment problem suffers from the curse of dimensionality. ",
"We prove the following lower bound for the phase shift model.",
"\n\n\\[thm:main\\_lower\\] Let $0 \\leq s \\leq \\lfloor L/2 \\rfloor$. Let ${\\mathcal{T}}_s$ be the set of vectors $\\theta \\in {\\mathcal{T}}$ satisfying $\\operatorname{psupp}(\\hat \\theta) \\subset [s]$. For any $\\sigma \\geq \\max_{\\theta \\in {\\mathcal{T}}_s} \\|\\theta\\|$, the phase shift model satisfies $$\\label{eqn:minimax_lower}\n\\inf_{T_n} \\sup_{\\theta \\in {\\mathcal{T}}_s} {{\\rm I}\\kern-0.18em{\\rm E}}_\\theta[{\\rho}(T_n, \\theta)] \\geq C\\Big( \\frac{\\sigma^{(2s-1) \\vee (s+1)}}{\\sqrt n} \\wedge 1\\Big)\\,,$$ where the infimum is taken over all estimators $T_n$ of $\\theta$ and where $C$ is a universal constant.",
"\n\nThe $\\sigma^{s+1}/\\sqrt n$ rate in Theorem \\[thm:main\\_upper\\] holds only in the edge case when $s \\in \\{0, 1\\}$; for $2 \\leq s \\leq \\lfloor L/2 \\rfloor$ the rate scales as $\\sigma^{2s-1}/\\sqrt n$.\n\nIn Section \\[sec:upper\\_proof\\], we show that a modified version of the MLE achieves the optimal rate asymptotically for $2 \\leq s \\leq \\lfloor L/2 \\rfloor$. This estimator is also adaptive to the class ${\\mathcal{T}}_s$.\n\n\\[thm:main\\_upper\\] For any $\\sigma \\geq 1$ and $2 \\leq s \\leq \\lfloor L/2 \\rfloor$, the modified MLE $\\check \\theta_n$ for the phase shift model satisfies $$\\label{eqn:minimax_upper}\n\\sup_{\\theta \\in {\\mathcal{T}}_s} {{\\rm I}\\kern-0.18em{\\rm E}}_\\theta[{\\rho}(\\check \\theta_n, \\theta)] \\leq C' \\frac{\\sigma^{2s -1}}{\\sqrt n} + C_\\sigma \\frac{\\log n}{n}\\,,$$ where $C_\\sigma \\leq C'' \\sigma^{12 s - 11}$ and $C'$ and $C''$ are constants depending on $L$ and $c_0$ but on no other parameter.",
"\n\nTheorem \\[thm:main\\_upper\\] excludes the cases where $s = 0$ or $s = 1$. The behavior of these cases is different, and is significantly easier to analyze.",
"\n\n\\[thm:small\\_s\\] If $s \\in \n\\{0,1\\}$ and $\\sigma \\geq 1$, then the phase shift model satisfies $$\\inf_{T_n} \\sup_{\\theta \\in {\\mathcal{T}}_s} {{\\rm I}\\kern-0.18em{\\rm E}}_\\theta[{\\rho}(T_n, \\theta)] \\leq C''' \\frac{\\sigma^{s+1}}{\\sqrt n}\\,,$$ where $C'''$ is a constant depending on $c_0$ but on no other parameter.",
"\n\nTheorem \\[thm:small\\_s\\] is proved in Appendix \\[supp\\], where we exhibit a computationally efficient estimator achieving the upper bounds for $s \\in \\{0, 1\\}$.\n\nA few remarks are in order. ",
"We have given rates over the classes ${\\mathcal{T}}_s$ because, in the context of cryo-EM, it is generally assumed that band-limited signals, that is, signals lying in ${\\mathcal{T}}_s$ for $s$ small, are easier to estimate. ",
"Our work offers partial validation for this view. ",
"However, we stress that the dependence on $s$ present in Theorems \\[thm:main\\_lower\\] and \\[thm:main\\_upper\\] is a consequence of the minimax paradigm. ",
"Indeed, our proof of the lower bound involves a class of signals with very specific support in the Fourier domain. ",
"Such signals drive the worst case bound of order $\\sigma^{2s-1}/\\sqrt{n}$. This is in striking contrast to the behavior for signals which are likely to arise in practice—in a companion paper [@PerWeeBan17], we show that signals whose Fourier transform has full support can be estimated at the rate $\\sigma^3/\\sqrt n$.\n\nSecond, our proof techniques do not allow us to remove the $\\sigma$ dependence of the term $C_\\sigma \\log n/n$ in the upper bound. ",
"In particular, for small values of $n$, this term may actually dominate. ",
"We conjecture that this issue is an artifact of our proof technique and note that preliminary numerical results in [@PerWeeBan17] support this claim.",
"\n\nThird, though we focus on the “high-noise regime,” we note that Theorems \\[thm:main\\_lower\\] and \\[thm:main\\_upper\\] do not require that $\\sigma \\to \\infty$; we merely require that $\\sigma$ be bounded below by a (small) constant.",
"\n\nThe rest of this paper is devoted to the proof of the main results in Theorems \\[thm:main\\_lower\\] and \\[thm:main\\_upper\\].",
"\n\nMaximum likelihood estimation {#sec:uniform_upper}\n=============================\n\nLet $Y_1, \\dots, Y_n$ be i.i.d observations from the phase shift model and consider the MLE $\\tilde \\theta_n$ that was defined in . ",
"In this section, we prove our main statistical result, that is a *uniform* upper bound on the rate of convergence of the MLE in terms of the curvature of the divergence $D(\\theta {\\,\\|\\,}\\phi)$ near its minimum. ",
"Note that this analysis departs from the classical *pointwise* rate of convergence for MLE that guarantees a rate of convergence $n^{-1/2}$ *for each fixed choice of parameter* as $n \\to \\infty$. Our tools strengthen this result considerably. ",
"Indeed, we show that for reasonable choices of $\\theta$, the MLE achieves a rate of $n^{1/2}$ *uniformly* over all choices of $\\theta$. We refer the reader to [@HeiKah15] for examples of Gaussian mixture problems where the pointwise and uniform rates of estimation differ.",
"\n\nThe following theorem establishes an upper bound for the MLE under a general lower bound for the KL divergence for any algebraically structured model. ",
"Our proof technique applies to any subgroup ${\\mathcal{G}}$ of $\\mathrm{O}(L)$ and can be broadly applied to derive uniform rates of convergence for the MLE from the tight bounds on the KL divergence given in Theorem \\[thm:tight\\_KL\\_bound\\]. ",
"In the following section, we specialize this result to obtain the minimax upper bounds for the phase shift model over ${\\mathcal{T}}_s$ that are presented in Theorem \\[thm:main\\_upper\\]. ",
"From here on, positive constants may depend on $L$ unless noted otherwise.",
"\n\n\\[thm:non\\_asymptotic\\] Let ${\\mathcal{T}}$ be any subspace of ${{\\rm I}\\kern-0.18em{\\rm R}}^L$. Assume that there exist $k \\geq 1$ and positive constants $c$ and $C$ such that for all $\\theta, \\phi \\in {\\mathcal{T}}$ satisfying $c^{-1} \\leq \\|\\theta\\| \\leq c$ and $\\sigma \\in {{\\rm I}\\kern-0.18em{\\rm R}}$ satisfying $\\sigma \\geq \\|\\theta\\|$, $$D(\\theta {\\,\\|\\,}\\phi)\\geq C \\sigma^{-2k}\\rho^2(\\theta, \\phi)\\,.\\label{EQ:universal_curvature}$$ Then there exists positive constants $C'$ and $C''$ such that the MLE $\\tilde \\theta_n$ constrained to lie in ${\\mathcal{T}}$ satisfies $$\\label{eq:theorem:non_asymptotic}\n{{\\rm I}\\kern-0.18em{\\rm E}}_\\theta[{\\rho}(\\tilde \\theta_n, \\theta)] \\leq C'\\frac{\\sigma^{k}}{\\sqrt{n}} + C_\\sigma\\frac{\\log n}{n}\\,,$$ uniformly over $\\theta \\in {\\mathcal{T}}$ satisfying $c^{-1} \\leq \\|\\theta\\| \\leq c$ and $\\sigma \\in {{\\rm I}\\kern-0.18em{\\rm R}}$ satisfying $\\sigma \\geq \\|\\theta\\|$, where $C_\\sigma\\le C''\\sigma^{6k-5}$.\n\nThe symbols $c$ and $C$ denote constants whose value may change from line to line. ",
"In the rest of this proof, we write $\\tilde \\theta=\\tilde \\theta_n$ to denote the constrained MLE. ",
"Since $\\theta$ and $\\tilde \\theta$ are both constrained to lie in ${\\mathcal{T}}$, we restrict all functions of this proof to this subspace without loss of generality. ",
"By rescaling by a constant, we can assume $\\|\\theta\\| = 1$ and $\\sigma \\geq 1$.\n\nThe proof strategy is to combine control of the curvature of the function $D$ with control of the deviations of the log-likelihood function.",
"\n\nDefine the event ${\\mathcal{E}}=\\{{\\rho}(\\tilde \\theta, \\theta) \\le {\\varepsilon}\\}$ where ${\\varepsilon}$ is to be specified. ",
"Since $D$ is invariant under the action of ${\\mathcal{G}}$, we can assume without loss of generality that ${\\rho}(\\tilde \\theta, \\theta) = \\|\\tilde \\theta - \\theta\\|$. We first establish that on this event, $\\|\\tilde \\theta - \\theta\\|$ can be controlled in terms of the metric induced by the Hessian of $D$ at $\\theta$. Fix $\\theta \\in {\\mathcal{T}}$ and denote by $H$ the Hessian of the function $\\phi \\mapsto D(\\theta {\\,\\|\\,}\\phi)$ evaluated at $\\phi = \\theta$. For any $u \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$, define $\\|u\\|_H = \\sqrt{u^\\top H u}$.\n\nIt follows from a Taylor expansion (Lemma \\[lem:d\\_taylor\\] in Appendix \\[supp\\]) that on ${\\mathcal{E}}$, $$\\Big|D(\\theta {\\,\\|\\,}\\tilde \\theta) - \\frac 1 2 \\|\\tilde \\theta -\\theta\\|_H^2\\Big| \\leq C\\frac{\\|\\tilde \\theta -\\theta\\|^3}{\\sigma^3}<\\frac12D(\\theta {\\,\\|\\,}\\tilde \\theta)\\,.$$ as long as ${\\varepsilon}\\leq c \\sigma^{3 - 2k}$ for some sufficiently small constant $c$. This yields $$\\frac 1 3 \\|\\tilde \\theta - \\theta\\|_H^2 \\leq D(\\theta \\, \\| \\, \\tilde \\theta) \\leq \\|\\tilde \\theta - \\theta\\|_H^2\\,, \\label{eqn:d_to_h_bound}$$ and, by , $$\\|\\tilde \\theta - \\theta\\|_H^2 \\geq c \\sigma^{-2k} \\|\\tilde \\theta - \\theta\\|^2\\,. ",
"\\label{eqn:rho_to_h_bound}$$ for some constant $c$.\n\nWe now control the geometry of the log-likelihood function near $\\theta$. Define $$D_n(\\theta {\\,\\|\\,}\\phi) = \\frac 1 n \\sum_{i=1}^n \\log \\frac{f_\\theta}{f_\\phi}(Y_i)\\,,$$ where $Y_i$ are i.i.d from ${\\mathrm}{P}_\\theta$ and $f_\\zeta$ is the density of ${\\mathrm}{P}_\\zeta, \\zeta \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$. Note that $D_n(\\theta {\\,\\|\\,}\\theta) = 0$ and recall that $\\tilde \\theta$ minimizes $\\phi \\mapsto D_n(\\theta {\\,\\|\\,}\\phi)\n$ so that $D_n(\\theta {\\,\\|\\,}\\tilde \\theta) \\leq 0$.\n\nSince $\\theta$ is held fixed throughout the proof, we abbreviate $D(\\theta {\\,\\|\\,}\\phi)$ and $D_n(\\theta {\\,\\|\\,}\\phi)$ as $D(\\phi)$ and $D_n(\\phi)$, respectively.",
"\n\nUsing Taylor expansion and $D(\\theta)=D_n(\\theta)=\\nabla D(\\theta)=0$, we get $$D(\\tilde \\theta) - D_n(\\tilde \\theta) = -\\nabla D_n(\\theta)^\\top h + \\frac 1 2 h^\\top \\nabla^2 (D - D_n)(\\bar \\theta)h\\,,$$ where $h=\\tilde \\theta -\\theta$ and $\\bar \\theta$ lies on a segment between $\\tilde \\theta$ and $\\theta$. For all $\\zeta \\in {\\mathcal{T}}$, write $H_n(\\zeta)$ for the Hessian of $D_n(\\phi)$ evaluated at $\\phi = \\zeta$, and similarly let $H(\\zeta)$ be the Hessian of $D(\\phi)$ evaluated at $\\phi = \\zeta$.\n\nCombining the above equation with and the fact that $D_n(\\tilde \\theta) \\leq 0$ yields $$\\label{eqn:second_order_expansion}\n\\frac 1 3 \\|h\\|_H^2 \\leq D(\\tilde \\theta) - D_n(\\tilde \\theta) \\leq - \\nabla D_n(\\theta)^\\top h + \\frac 1 2 h^\\top (H(\\bar \\theta) - H_n(\\bar \\theta)) h\\,.$$\n\nFor the first term, we employ the bound $|\\nabla D_n(\\theta)^\\top h| \\leq \\|\\nabla D_n(\\theta)\\|_H^* \\|h\\|_H$, where $\\|\\cdot\\|_H^*$ denotes the dual norm to $\\|\\cdot\\|_H$.\n\nTo control the second, note first that by , it holds $\\|h\\| \\le C\\sigma^k \\|h\\|_H$. Therefore, $$h^\\top (H(\\bar \\theta) - H_n(\\bar \\theta)) h \\leq C\\sigma^k\\|h\\|_H\\|h\\| \\sup_{\\phi \\in {\\mathcal{B}}_{\\varepsilon}} \\|H(\\phi) - H_n(\\phi)\\|_\\text{op} \\,,$$ where ${\\mathcal{B}}_{\\varepsilon}:=\\{\\phi \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L\\,:\\, \\rho(\\phi, \\theta) \\le {\\varepsilon}\\}$.\n\nCombining the above bounds and dividing by $\\|h\\|_H$, we get that on ${{\\mathcal{E}}}$, $$\\begin{aligned}\n\\sigma^{-k}\\|h\\| \\leq C \\|h\\|_H & \\leq C\\|\\nabla D_n(\\theta)\\|_{H}^* + C\\sigma^k\\|h\\| \\sup_{\\phi \\in {\\mathcal{B}}_{\\varepsilon}} \\|H(\\phi) - H_n(\\phi)\\|_\\text{op}\\\\\n& \\leq C\\|\\nabla D_n(\\theta)\\|_{H}^* + C\\sigma^{k+3} \\sup_{\\phi \\in {\\mathcal{B}}_{\\varepsilon}} \\|H(\\phi) - H_n(\\phi)\\|_\\text{op}^2+ \\sigma^{k-3}\\|h\\|^2\\,,\\end{aligned}$$ where we applied Young’s inequality.",
"\n\nSince ${\\varepsilon}^{-1}{\\rho}(\\tilde \\theta, \\theta) \\geq 1$ on ${\\mathcal{E}}^c$, we get $$\\begin{aligned}\n{{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\tilde \\theta, \\theta)] &= {{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\tilde \\theta, \\theta){{\\rm 1}\\kern-0.24em{\\rm I}}_{{\\mathcal{E}}}] + {{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\tilde \\theta, \\theta){{\\rm 1}\\kern-0.24em{\\rm I}}_{{{\\mathcal{E}}}^c}]\\\\\n&\\le C\\sigma^{k}{{\\rm I}\\kern-0.18em{\\rm E}}[\\|h\\|_H{{\\rm 1}\\kern-0.24em{\\rm I}}_{{\\mathcal{E}}}] +{\\varepsilon}^{-1}{{{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}^2(\\tilde \\theta, \\theta)]}\\,.\\end{aligned}$$\n\nChoose ${\\varepsilon}= c \\sigma^{3-2k}$ for some small constant $c$. We obtain $$\\label{EQ:pr_mainbd}\n{{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\tilde \\theta, \\theta)] \\le C\\Big(\\sigma^k{{\\rm I}\\kern-0.18em{\\rm E}}\\|\\nabla D_n(\\theta)\\|_{H}^* + \\sigma^{2k+3} {{\\rm I}\\kern-0.18em{\\rm E}}\\sup_{\\phi \\in {\\mathcal{B}}_{\\varepsilon}} \\|H(\\phi) - H_n(\\phi)\\|_\\text{op}^2+ \\sigma^{2k-3}{{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}^2(\\tilde \\theta, \\theta)]\\Big)$$\n\nIt suffices to control the right side of the above inequality. ",
"The main term is the first one. ",
"We note that if $H$ were invertible, and hence $\\|\\cdot\\|_H$ a genuine metric, then it is well known (see, e.g., [@HirLem01short]) that $\n\\|\\nabla D_n(\\theta)\\|_H^*=\\|\\nabla D_n(\\theta)\\|_{H^{-1}}\\,.",
"\n$ In general, $H$ is not invertible, but we still have $$\\|u\\|_H^* = \\left\\{\\begin{array}{ll}\n\\sqrt{u^\\top H^\\dagger u} & \\text{ if $u$ lies in the row space of $H$,} \\\\\n\\infty & \\text{ otherwise,}\n\\end{array}\\right.$$ where $H^\\dagger$ denotes the Moore-Penrose pseudo-inverse of the matrix $H$. The Bartlett identities state that $$\\begin{aligned}\n{{\\rm I}\\kern-0.18em{\\rm E}}[\\nabla_\\phi \\log f_\\phi(Y_i)|_{\\phi = \\theta}] & = 0\\,,\\\\\n{{\\rm I}\\kern-0.18em{\\rm E}}[(\\nabla_\\phi \\log f_\\phi(Y_i)|_{\\phi = \\theta}) (\\nabla_\\phi \\log f_\\phi(Y_i)|_{\\phi = \\theta})^\\top] & = H\\,,\\end{aligned}$$ and since $\\nabla D_n(\\theta) = \\frac 1 n \\sum_{i=1}^n \\nabla_\\phi \\log f_\\phi(Y_i)|_{\\phi = \\theta}$ and $Y_1, \\dots, Y_n$ are independent, we obtain $${{\\rm I}\\kern-0.18em{\\rm E}}[\\nabla D_n(\\theta) \\nabla D_n(\\theta)^\\top] = \\frac 1 n H\\,.$$ In particular, $\\nabla D_n(\\theta)$ lies in the row space of $H$ almost surely. ",
"Jensen’s inequality implies $$\\label{EQ:bdGradient}\n{{\\rm I}\\kern-0.18em{\\rm E}}\\|\\nabla D_n(\\theta)\\|_{H}^* \\le \\Big(\\operatorname{tr}( H^{\\dagger} {{\\rm I}\\kern-0.18em{\\rm E}}[\\nabla D_n(\\theta) \\nabla D_n(\\theta)^\\top])\\Big)^{1/2} = \\Big(\\frac 1 n \\operatorname{tr}(H^{\\dagger} H)\\Big)^{1/2} \\leq \\sqrt{\\frac L n}\\,.$$ For the second term, standard matrix concentration bounds can be applied to show $$\\label{EQ:bdHessian}\n{{\\rm I}\\kern-0.18em{\\rm E}}\\sup_{\\phi \\in {\\mathcal{B}}_{\\varepsilon}} \\|H(\\phi) - H_n(\\phi)\\|_\\text{op}^2 \\leq C \\frac{\\log n}{n\\sigma^4}\\,.$$ A proof of appears as Lemma \\[lem:bdHessian\\] in Appendix \\[supp\\].",
"\n\nLikewise, a standard slicing argument, Lemma \\[eventA\\] in Appenddix \\[supp\\], implies $$\\label{EQ:slicing_ub}{{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}^2(\\tilde \\theta, \\theta)] \\le C\\frac{\\sigma^{4k-2}}{n}\\,.$$\n\nPlugging , , and into , we get $${{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\tilde \\theta, \\theta)] \\le C\\Big(\\frac{\\sigma^k}{\\sqrt{n}} + \\frac{\\sigma^{2k-1}\\log n}{n}+ \\frac{\\sigma^{6k-5}}{n}\\Big)\\,,$$ as desired.",
"\n\nMinimax upper bounds {#SEC:modMLE}\n====================\n\nIn this section, we apply the results of Section \\[sec:uniform\\_upper\\] to the phase shift model . ",
"Note that rather than the MLE, we study a constrained MLE because the lower bound may only hold for a proper subset ${\\mathcal{T}}\\subset {{\\rm I}\\kern-0.18em{\\rm R}}^L$ in Theorem \\[thm:non\\_asymptotic\\]. ",
"This phenomenon does occur in the specific case of phase shifts: the divergence $D(\\phi)$ is not curved enough in directions that perturb a null Fourier coefficient of $\\theta$. To overcome this limitation, we split the sample $Y_1, \\ldots, Y_n$ into two parts: with the first part we estimate the support of $\\hat \\theta$ under Assumption \\[assumption\\] and with the second part, we compute a maximum likelihood estimator constrained to have the estimated support.",
"\n\nSpecifically, assume for simplicity that we have a sample $Y_1, \\ldots, Y_{2n}$ of size $2n$ and split it into two samples ${\\mathcal{Y}}_1=\\{Y_1, \\ldots, Y_n\\}$ and ${\\mathcal{Y}}_2=\\{Y_{n+1}, \\ldots, Y_{2n}\\}$ of equal size.",
"\n\nFourier support estimation\n--------------------------\n\nWe use the first subsample ${\\mathcal{Y}}_1$ to construct a set $\\tilde S$ that coincides with $\\operatorname{psupp}(\\hat \\theta)$ with high probability. ",
"For any $j =1, \\ldots, \\lfloor L/2\\rfloor $, define, $$M_j = \\frac 1 n \\sum_{i=1}^n |\\widehat{(Y_i)}_j|^2 - \\sigma^2\\,.$$ Recall that, by Assumption \\[assumption\\], there exists a positive constant $c_0$ such that $|\\hat \\theta_j| \\geq c_0$ for all $j \\in \\operatorname{psupp}(\\hat \\theta)$. Define the set $\\tilde S$ by $$\\tilde S=\\Big\\{ j \\in \\{1, \\ldots, \\lfloor L/2\\rfloor\\}\\,:\\, M_j \\geq \\frac 12 c_0^2| \\Big\\}$$\n\nThe following proposition shows that $\\tilde S=\\operatorname{psupp}(\\hat \\theta)$ with high probability.",
"\n\n\\[prop:projection\\_construction\\] There exists a positive constant $c$ depending on $c_0$ such that $${{\\rm I}\\kern-0.18em{\\rm P}}[\\tilde S \\neq \\operatorname{psupp}(\\hat \\theta)] \\le 2L \\exp(-c n \\sigma^{-4})\\,.$$\n\nThis follows from standard concentration arguments. ",
"A full proof appears in Appendix \\[supp\\].",
"\n\nConstrained MLE\n---------------\n\nWe use the second sample to construct a constrained MLE. ",
"To that end, for any $S \\subset \\{1, \\ldots, \\lfloor L/2\\rfloor\\}$, define the projection $P_S$ by $$\\widehat{P_S(\\phi)}_j = \\left\\{\\begin{array}{ll}\n\\hat \\phi_j & \\text{ if $j \\in S \\cup -S$} \\\\\n\\hat \\phi_0 & \\text{ if $j = 0$} \\\\\n0 & \\text{ otherwise.}",
"\n\\end{array}\n\\right.$$ The image $\\operatorname{Im}(P_{\\tilde S})$ of $P_S$ is a $(2|S|+1)$-dimensional real vector space. ",
"For convenience, write $\\phi_S = P_S \\phi$ for any vector $\\phi \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$.\n\nHaving constructed the set $\\tilde S$, we use the samples in ${\\mathcal{Y}}_2$ to calculate a modified MLE $\\check \\theta_n$ constrained to lie in the subspace $\\operatorname{Im}(P_{\\tilde S})$. To analyze the performance of this constrained MLE, we check that holds on this subspace.",
"\n\n\\[prop:lambda\\_P\\_bound\\] Fix $2 \\leq s \\leq \\lfloor L/2 \\rfloor$ and $\\theta \\in {\\mathcal{T}}_s$. Let $S = \\operatorname{psupp}(\\hat \\theta)$. If ${\\mathcal{T}}= \\operatorname{Im}(P_S)$, then there exists $C>0$ such that for all $\\sigma\\ge \\|\\theta\\|, \\phi \\in {\\mathcal{T}}$, it holds $$D(\\theta {\\,\\|\\,}\\phi)\\geq C \\sigma^{-4s+2}\\rho^2(\\theta, \\phi) \\,.\\label{EQ:shift_universal_curvature}$$\n\nFor the sake of exposition, we only prove for $\\phi$ such that ${\\rho}(\\theta, \\phi) \\leq {\\varepsilon}_0$ for some small ${\\varepsilon}_0$ to be specified. ",
"The complete proof is deferred to Appendix \\[supp\\]. ",
"In what follows, the symbols $c$ and $C$ will refer to unspecified positive constants whose value may change from line to line. ",
"By rescaling by a constant, we can assume $\\|\\theta\\| = 1$ and $\\sigma \\geq 1$.\n\nBy Lemma \\[lem:first\\_moment\\], $$\\label{eqn:k1_boundmt}\nD(\\phi) = \\frac{1}{2\\sigma^2}\\| {{\\rm I}\\kern-0.18em{\\rm E}}[G \\theta - G \\phi]\\|^2 + D(\\vartheta {\\,\\|\\,}\\varphi)\\,,$$ where $\\vartheta = \\theta - {{\\rm I}\\kern-0.18em{\\rm E}}G\\theta$ and $\\varphi = \\phi - {{\\rm I}\\kern-0.18em{\\rm E}}G \\phi$. If $|\\hat \\theta_0 - \\hat \\phi_0| \\geq \\frac 1 2 {\\rho}(\\theta, \\phi)$, then implies $$D(\\phi) \\geq \\frac{1}{2\\sigma^2}\\| {{\\rm I}\\kern-0.18em{\\rm E}}[G \\theta - G \\phi]\\|^2 = \\frac{1}{2\\sigma^2}(\\hat \\theta_0 - \\hat \\phi_0)^2 \\ge \\frac{{\\rho}^2(\\theta, \\phi)}{8\\sigma^2} \\geq \\frac 1 8 \\sigma^{-4s+2} {\\rho}^2(\\theta, \\phi)\\,.$$ On the other hand, if $|\\hat \\theta_0 - \\hat \\phi_0| < \\frac 1 2 {\\rho}(\\theta, \\phi)$, then $${\\rho}(\\vartheta, \\varphi)^2 = {\\rho}(\\theta_S, \\phi)^2 - |\\hat \\theta_0 - \\hat \\phi_0|^2 \\ge 3{\\rho}^2(\\theta, \\phi)/4\\,.$$ Thus, by , it suffices to show that $$D(\\vartheta {\\,\\|\\,}\\varphi) \\ge C \\sigma^{-4s+2} {\\rho}(\\vartheta, \\varphi)^2\\,,$$ for vectors $\\vartheta$ and $\\varphi$ satisfying ${{\\rm I}\\kern-0.18em{\\rm E}}G \\vartheta = {{\\rm I}\\kern-0.18em{\\rm E}}G \\varphi = 0$. In what follows, write ${\\rho}(\\vartheta, \\varphi) = {\\varepsilon}$. Since $D(\\varphi)=D(G\\varphi)$ for all $G \\in {\\mathcal{S}}$, we may assume that $\\|\\vartheta - \\varphi\\| = {\\varepsilon}$. We will show that there exists a small positive constant $c$ such that for some $m \\leq 2 s - 1$, $$\\|\\Delta_m\\| := \\|{{\\rm I}\\kern-0.18em{\\rm E}}[(G\\vartheta)^{\\otimes m} - (G\\varphi)^{\\otimes m}]\\| \\geq c {\\varepsilon}\\,,$$ and the claim will follow from Theorem \\[thm:tight\\_KL\\_bound\\]. ",
"We denote by $\\kappa$ a small constant whose value will be specified. ",
"There are two cases: either $\\vartheta$ and $\\varphi$ have essentially the same power spectrum (i.e., $|\\hat \\vartheta_k| \\approx |\\hat \\varphi_k|$ for all $k$) or their power spectra are very different. ",
"We will treat these two cases separately.",
"\n\nRecall that for each $j \\in S$, by Assumptions \\[assumption:norm\\] and \\[assumption\\], the bounds $c_0^{-1}\\le| \\hat \\vartheta_j| \\le c$ hold. ",
"Consider the polar form $\\hat \\varphi_j/\\hat \\vartheta_j = r_j e^{\\mathrm{i} \\delta_j}$, where $r_j\\ge 0$.\n\n#### Case a: There exists $j \\in S$ such that $|1 - r_j| \\ge \\kappa {\\varepsilon}$\n\nThe fact that $|\\hat \\vartheta_j| \\geq c_0^{-1}$ implies $$\\begin{aligned}\n\\|\\Delta_2\\|^2 & = \\|{{\\rm I}\\kern-0.18em{\\rm E}}[(G \\vartheta)^{\\otimes 2} - (G \\varphi)^{\\otimes 2}]\\|^2 \\\\\n& = \\sum_{k = -\\lfloor L/2 \\rfloor}^{\\lfloor L/2 \\rfloor} (|\\hat \\vartheta_k|^2 - |\\hat \\varphi_k|^2)^2 \\\\\n& \\geq (|\\hat \\vartheta_j|^2 - |\\varphi_j|^2)^2 \\\\\n& = |\\hat \\vartheta_j|^4 (1 - r_j^2)^2 \\\\\n& \\geq c_0^{-4} (1+r_j)^2(1-r_j)^2 \\\\\n& \\geq c_0^{-4} \\kappa^2 {\\varepsilon}^2\\,,\\end{aligned}$$ so that $\\|\\Delta_2\\| \\geq c {\\varepsilon}$.\n\n#### Case b: $|1 - r_j| < \\kappa {\\varepsilon}$ for all $j \\in S$\n\nDenote by $p$ the smallest integer in $S$ and observe that $${\\varepsilon}^2= {\\rho}(\\vartheta, \\varphi)^2 = \\min_{z: |z| = 1} 2 \\sum_{j \\in S} |1-r_jz^je^{\\mathrm{i}\\delta_j}|^2|\\hat \\vartheta_j|^2 \\le C \\sum_{j \\in S} |1 - r_j e^{\\mathrm{i}(p\\delta_j-j \\delta_p)/p}|^2\\,,$$ where the inequality follows from choosing $z = e^{-\\mathrm{i} \\delta_p/p}$. Therefore, there exists a coordinate $\\ell \\in S$ such that $$\\label{EQ:deltaphases}\n|1 - e^{\\mathrm{i} (p \\delta_\\ell - \\ell \\delta_p)/p}| \\geq |1 - r_\\ell e^{\\mathrm{i}(p \\delta_\\ell - \\ell \\delta_p)/p}| - \\kappa {\\varepsilon}\\geq c {\\varepsilon}\\,,$$ as long as $\\kappa $ is chosen sufficiently small. ",
"In particular, $|1 - e^{\\mathrm{i} (p \\delta_\\ell - \\ell \\delta_p)/p}| > 0$, so $\\ell \\neq p$. Note that this fact implies that, if $|1 - r_j| < \\kappa {\\varepsilon}$ for all $j \\in S$, then $|S| \\geq 2$.\n\nChoose $m = \\ell + p$. Since $\\ell, p \\in S \\subseteq [s]$ and $\\ell \\neq p$, the bound $m \\leq 2s - 1$ holds. ",
"As in the proof of Proposition \\[prop:generalized\\_matching\\], we have that $$\\begin{aligned}\n\\|\\Delta_m\\|^2 & = \\sum_{j_1 + \\dots + j_{m} = 0} \\left|\\prod_{n = 1}^{m} \\hat \\vartheta_{j_n}- \\prod_{n = 1}^{m} \\hat \\varphi_{j_n}\\right|^2 \\\\\n& = \\sum_{j_1 + \\dots + j_{m} = 0} \\left|1 - \\prod_{n = 1}^{m} r_{j_n} e^{\\mathrm{i} \\delta_{j_n}}\\right|^2 \\prod_{n = 1}^{m} |\\hat \\vartheta_{j_n}|^2\\,.\\end{aligned}$$ Each term in the above sum is positive. ",
"One valid solution to the equation $j_1 + \\dots + j_m = 0$ is $j_1 = \\dots = j_\\ell = -p$ and $j_{\\ell + 1} = \\dots = j_m = \\ell$. We obtain $$\\begin{aligned}\n\\|\\Delta_m\\|^2 &\\ge C \\left|1 - e^{i (p\\delta_\\ell - \\ell \\delta_p)} \\prod_{n = 1}^{m} r_{j_n} \\right|^2 \\\\\n& \\geq C |1 - e^{i (p\\delta_\\ell - \\ell \\delta_p)}|^2 - C\\left|1 - \\prod_{n = 1}^{m} r_{j_n}\\right|^2\\,.\\end{aligned}$$ As long as $\\kappa {\\varepsilon}$ is small enough, $\\left|1 - \\prod_{n = 1}^{m} r_{j_n}\\right| \\leq 2 m \\kappa {\\varepsilon}$. Moreover, as long as ${\\varepsilon}_0 $ is chosen sufficiently small, $\\delta_\\ell$ and $\\delta_p$ can both be chosen small enough that $|p \\delta_\\ell - \\ell \\delta_p| \\leq 1$, in which case it holds $$|1 - e^{i (p\\delta_\\ell - \\ell \\delta_p)}|^2 \\ge |1 - e^{i (p \\delta_\\ell - \\ell \\delta_p)/p}|^2 \\ge c^2{\\varepsilon}^2\\,,$$ where the last inequality follows from . ",
"So for ${\\varepsilon}_0$ and $\\kappa$ chosen sufficiently small, this proves the existence of an $m \\leq 2s -1$ for which $\\|\\Delta_m\\| \\geq c {\\varepsilon}$.\n\nProof of Theorem \\[thm:main\\_upper\\] {#sec:upper_proof}\n------------------------------------\n\nDefine ${\\mathcal{R}}=\\{\\tilde S=\\operatorname{psupp}(\\hat \\theta)\\}$ and observe that $${{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta)] = {{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta){{\\rm 1}\\kern-0.24em{\\rm I}}_{{\\mathcal{R}}}] + {{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta){{\\rm 1}\\kern-0.24em{\\rm I}}_{{\\mathcal{R}}^c}]\\,.$$ The first term is controlled by combining Proposition \\[prop:lambda\\_P\\_bound\\] and Theorem \\[thm:non\\_asymptotic\\] to get $${{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta){{\\rm 1}\\kern-0.24em{\\rm I}}_{{\\mathcal{R}}}]\\le C\\frac{\\sigma^{2s-1}}{\\sqrt{n}} + C_\\sigma\\frac{\\log n}{n}\\,,$$ where $C_\\sigma \\le C\\sigma^{12s-11}$.\n\nTo bound the second term, we use the Cauchy-Schwarz inequality and Proposition \\[prop:projection\\_construction\\] to get $${{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta){{\\rm 1}\\kern-0.24em{\\rm I}}_{{\\mathcal{R}}^c}]\\le 2L \\exp(-c n \\sigma^{-4}) \\sqrt{{{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta)^2]} \\leq C \\frac{\\sigma^4}{n} \\sqrt{{{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta)^2]}\\,.$$ We now show that ${{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta)^2]$ is bounded uniformly over all choices of $\\tilde S$ by a constant multiple of $\\sigma^2$ using a similar slicing argument as the one employed in the proof of Lemma \\[eventA\\] in Appendix \\[supp\\].",
"\n\nBy the triangle inequality, $${\\rho}(\\check \\theta_{n}, \\theta) \\leq {\\rho}(\\check \\theta_{n}, \\theta_{\\tilde S}) + {\\rho}(\\theta_{\\tilde S}, \\theta) \\leq {\\rho}(\\check \\theta_{n}, \\theta_{\\tilde S}) + 1\\,.$$ By Lemma \\[lem:far\\_quadratic\\] in Appendix \\[supp\\], when ${\\rho}(\\check \\theta_{n}, \\theta_{\\tilde S}) \\geq 3 \\sqrt 2 \\sigma$, the divergence satisfies $D(\\theta_{\\tilde S} {\\,\\|\\,}\\check \\theta_{n}) \\geq c \\sigma^{-2} {\\rho}(\\check \\theta_{n}, \\theta_{\\tilde S})$. We therefore have $${\\rho}(\\check \\theta_{n}, \\theta_{\\tilde S})^2\\le 18 \\sigma^2 + c^{-1} \\sigma^2 D(\\theta_{\\tilde S} {\\,\\|\\,}\\check \\theta_{n}) \\le (C^\\circ \\sigma)^2(1+ \\mathfrak{G}_n(\\theta_{\\tilde S} {\\,\\|\\,}\\check \\theta_{n}))\\,,$$ for some constant $C^\\circ$, where $$\\mathfrak{G}_n(\\theta_{\\tilde S} {\\,\\|\\,}\\check \\theta_{n}) = D(\\theta_{\\tilde S} {\\,\\|\\,}\\check \\theta_{n}) - D_n(\\theta_{\\tilde S} {\\,\\|\\,}\\check \\theta_{n})$$ and where we have used the fact that $D_n(\\theta_{\\tilde S} {\\,\\|\\,}\\check \\theta_{n}) \\leq D_n(\\theta_{\\tilde S} {\\,\\|\\,}\\theta_{\\tilde S}) = 0$.\n\nFor $j \\ge 0$, define $T_j=\\{\\phi \\in {{\\rm I}\\kern-0.18em{\\rm R}}^d\\,:\\, 2^j\\sigma \\le \\rho(\\phi, \\theta_{\\tilde S})\\le 2^{j+1}\\sigma\\}$ and let $J$ be such that $C^\\circ \\le 2^J \\le 2C^\\circ$. Observe that $$\\begin{aligned}\n{{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta_{\\tilde S})^2]&\\le 4 (C^\\circ \\sigma)^2 + \\sum_{j\\ge J} {{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta_{\\tilde S})^2{{\\rm 1}\\kern-0.24em{\\rm I}}(\\check \\theta_{n} \\in T_j)]\\\\\n&\\le 4 (C^\\circ \\sigma)^2 + \\sigma^2\\sum_{j>J} 2^{2j+2}{{\\rm I}\\kern-0.18em{\\rm P}}[\\sup_{\\phi \\in T_j}\\mathfrak{G}_n(\\theta_{\\tilde S} {\\,\\|\\,}\\phi)>2^{2j}]\\\\\n&\\le 4 (C^\\circ \\sigma)^2 + C \\sigma^2 \\sum_{j\\ge 0} 2^{2j}\\exp(-C2^{2j})\\le C \\sigma^2\\,,\\end{aligned}$$ where we used from Appendix \\[supp\\] in the third inequality. ",
"We obtain $${{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta)^2] \\leq 2 {{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta_{\\tilde S})^2] + 2 \\leq C\\sigma^2\\,.$$\n\nWe have established that $${{\\rm I}\\kern-0.18em{\\rm E}}[{\\rho}(\\check \\theta_{n}, \\theta)] \\le C\\Big(\\frac{\\sigma^{2s-1}}{\\sqrt{n}}+\\sigma^{12s - 11}\\frac{\\log n}{n}+\\frac{\\sigma^5}{n} \\Big)\\,,$$ which, since $s \\geq 2$, completes the proof of Theorem \\[thm:main\\_upper\\].",
"\n\nMinimax lower bounds {#sec:multi}\n====================\n\nOur minimax lower bounds rely ultimately on Le Cam’s classical two-point testing method [@LeC73]. ",
"For this reason, our lower bounds do not capture the optimal dependence in $L$ but only in $\\sigma$ and $n$. In particular, the version that we use requires an upper bound on the KL divergence, which can be obtained using Theorem \\[thm:tight\\_KL\\_bound\\] and a moment matching argument.",
"\n\nMoment matching\n---------------\n\nTheorem \\[thm:tight\\_KL\\_bound\\] implies that ${\\mathrm}{P}_\\theta$ and ${\\mathrm}{P}_\\phi$ are hard to distinguish when the quantities $$\\|\\Delta_m\\| = \\|{{\\rm I}\\kern-0.18em{\\rm E}}[(G\\theta)^{\\otimes m}] - {{\\rm I}\\kern-0.18em{\\rm E}}[(G\\phi)^{\\otimes m}]\\|$$ vanish for $m \\in [k]$. In this section, we show that, in the phase shift model, $\\Delta_m, m \\in [k]$ can be made to vanish for large $k$ by appropriately choosing the support of the Fourier transforms $\\hat \\theta$ and $\\hat \\phi$.\n\nBefore stating our main results, we first give a brief sketch of the technique. ",
"As we show in the proof of Proposition \\[prop:generalized\\_matching\\] below, the tensor ${{\\rm I}\\kern-0.18em{\\rm E}}[(G \\theta)^{\\otimes m}]$ has a simple form in the Fourier basis: $${{\\rm I}\\kern-0.18em{\\rm E}}[(\\widehat{G \\theta})^{\\otimes m}]_{j_1 \\dots j_m} = \\left\\{\\begin{array}{ll}\n\\hat \\theta_{j_1} \\cdots \\hat \\theta_{j_m} & \\text{ if $j_1 + \\dots + j_m = 0$,} \\\\\n0 & \\text{ otherwise.}",
"\n\\end{array}\\right.$$\n\nFor example, the first moment tensor ${{\\rm I}\\kern-0.18em{\\rm E}}[G \\theta]$ contains only the term $\\hat \\theta_0$, and the second moment tensor ${{\\rm I}\\kern-0.18em{\\rm E}}[(G \\theta)^{\\otimes 2}]$ contains the term $\\hat \\theta_j \\hat \\theta_{-j}$ for each index $j$. Since $\\theta$ has real entries, $\\hat \\theta_j \\hat \\theta_{-j} = |\\hat \\theta_j|^2$, so that ${{\\rm I}\\kern-0.18em{\\rm E}}[(G \\theta)^{\\otimes 2}]$ contains enough information to reconstruct the magnitudes of the Fourier coefficients, but not their phases. ",
"This implies that if $\\theta$ and $\\phi$ satisfy $|\\hat \\theta_j| = |\\hat \\phi_j|$ for all $j$, then $\\|{{\\rm I}\\kern-0.18em{\\rm E}}[(G\\theta)^{\\otimes 2}] - {{\\rm I}\\kern-0.18em{\\rm E}}[(G\\phi)^{\\otimes 2}]\\| = 0$.\n\nTo exhibit two signals whose higher moments also match, we employ the following idea: if a tuple $(j_1, \\dots, j_m)$ is of the form $(j_1, -j_1, j_2, -j_2, \\dots)$, then ${{\\rm I}\\kern-0.18em{\\rm E}}[(\\widehat{G \\theta})^{\\otimes m}]_{j_1 \\dots j_m} = |\\hat \\theta_{j_1}|^2 \\cdots |\\hat \\theta_{j_m}|^2$. In other words, this entry of the $m$th moment tensor also only depends on the magnitudes of the Fourier coefficients of $\\theta$ and not on their phases. ",
"Therefore, if the only nonzero entries of ${{\\rm I}\\kern-0.18em{\\rm E}}[(\\widehat{G \\theta})^{\\otimes m}]$ and ${{\\rm I}\\kern-0.18em{\\rm E}}[(\\widehat{G \\phi})^{\\otimes m}]$ correspond to tuples of this form and if the magnitudes of the Fourier coefficients of $\\theta$ and $\\phi$ agree, then $\\|{{\\rm I}\\kern-0.18em{\\rm E}}[(G\\theta)^{\\otimes m}] - {{\\rm I}\\kern-0.18em{\\rm E}}[(G\\phi)^{\\otimes m}]\\| = 0$.\n\nThis argument is formalized in Proposition \\[prop:generalized\\_matching\\].",
"\n\n\\[prop:generalized\\_matching\\] Fix $2 \\leq s \\leq \\lfloor L/2 \\rfloor$ and let $\\theta, \\phi \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$ satisfy $$\\hat \\theta_j = \\hat \\phi_j = 0 \\quad \\text{for $j \\notin \\{\\pm(s-1), \\pm s \\}$}$$ and $$|\\hat \\theta_{j}| = |\\hat \\phi_{j}| \\quad \\text{for $j \\in \\{\\pm(s-1), \\pm s\\}$.}$$ If $G$ is drawn uniformly from ${\\mathcal{S}}$, then for any $m =1, \\dots, 2s-2$, $${{\\rm I}\\kern-0.18em{\\rm E}}[(G \\theta)^{\\otimes m}] = {{\\rm I}\\kern-0.18em{\\rm E}}[(G \\phi)^{\\otimes m}]$$\n\nFix $m \\leq 2s-2$. Since ${{\\rm I}\\kern-0.18em{\\rm E}}[(G \\theta)^{\\otimes m}]$ and ${{\\rm I}\\kern-0.18em{\\rm E}}[(G \\phi)^{\\otimes m}]$ are symmetric tensors, to show that they are equal it suffices to show that $$\\langle {{\\rm I}\\kern-0.18em{\\rm E}}[(G \\theta)^{\\otimes m}], u^{\\otimes m}\\rangle = \\langle {{\\rm I}\\kern-0.18em{\\rm E}}[(G \\phi)^{\\otimes m}], u^{\\otimes m}\\rangle \\qquad \\forall\\, u \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$$ or equivalently, that $$\\label{EQ:equaltensors}\n{{\\rm I}\\kern-0.18em{\\rm E}}[(u^\\top G \\theta)^m] = {{\\rm I}\\kern-0.18em{\\rm E}}[(u^\\top G \\phi)^m]\\qquad \\forall\\, u \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L\\,.$$\n\nConsider the set ${\\mathcal{P}}= \\{\\zeta \\,:\\, |\\hat \\zeta_j| = |\\hat \\theta_j| \\,, \\ \\forall\\, j\\}$ and note that $\\theta, \\phi \\in {\\mathcal{P}}$. We show that the function $\\zeta \\mapsto {{\\rm I}\\kern-0.18em{\\rm E}}[(u^\\top G \\zeta)^m]$ is constant on ${\\mathcal{P}}$, which readily yields . ",
"For a fixed shift $G_z \\in {\\mathcal{F}}$, we obtain $$u^\\top G_z \\zeta = \\langle \\hat u, \\widehat{G_z \\zeta}\\rangle = \\sum_{j = -\\lfloor L/2 \\rfloor}^{\\lfloor L/2 \\rfloor} \\hat u_{-j} \\hat \\zeta_j z^{j}\\,,$$ so $$(u^\\top G_z \\zeta)^m = \\sum_{j_1, \\dots, j_m = -\\lfloor L/2 \\rfloor}^{\\lfloor L/2 \\rfloor} z^{j_1 + \\dots + j_m} \\prod_{n = 1}^m \\hat u_{-j_n} \\hat \\zeta_{j_n}\\,.$$\n\nTaking expectations with respect to a uniform choice of $z$ yields $$\\label{EQ:sumofji}\n{{\\rm I}\\kern-0.18em{\\rm E}}[(u^\\top G_z \\zeta)^m] = \\sum_{j_1 + \\dots + j_m = 0} \\prod_{n = 1}^m \\hat u_{-j_n} \\hat \\zeta_{j_n}\\,,$$ where the sums are over all choices of coordinates $j_1, \\dots, j_m \\in \\{-\\lfloor L/2 \\rfloor, \\dots, \\lfloor L/2 \\rfloor\\}$ whose sum is $0$.\n\nThe Fourier transform of $\\zeta$ is supported only on coordinates $\\pm (s-1)$ and $\\pm s$, so we may restrict our attention to sums involving only those coordinates. ",
"Suppose $j_1 + \\dots + j_m = 0$. Define $$\\begin{aligned}\n\\alpha & = |\\{i : j_i = s-1\\}| &\n\\beta & = |\\{i : j_i = -(s-1)\\}| \\\\\n\\gamma & = |\\{i : j_i = s\\}| &\n\\delta & = |\\{i : j_i = -s\\}| \\end{aligned}$$\n\nBy assumption $j_1 + \\dots + j_m = 0$, so the tuple $(\\alpha, \\beta, \\gamma, \\delta)$ is a solution to $$\\alpha (s-1) + \\beta (-(s-1)) + \\gamma (s) + \\delta (-s) = 0$$ or, equivalently, $$\\begin{aligned}\n(\\alpha-\\beta) (s-1) + (\\gamma-\\delta) s & = 0 \\nonumber\\end{aligned}$$\n\nSince $s-1$ and $s$ are coprime, $(\\alpha - \\beta)$ and $(\\gamma - \\delta)$ must be multiples of $s$ and $s-1$, respectively. ",
"Since $|\\alpha - \\beta| + |\\gamma - \\delta| \\leq m < 2s - 1$, in fact $\\alpha - \\beta = \\gamma - \\delta = 0$.\n\nTherefore the only $m$-tuples $(j_1, \\ldots, j_m)$ that appear in the sum on the right-hand side of are those in which $+(s-1)$ and $-(s-1)$ occur an equal number of times and $+s$ and $-s$ occur an equal number of times. ",
"For such $m$-tuples, the product $\\prod_{n = 1}^m \\hat u_{-j_n} \\hat \\zeta_{j_n}$ can be reduced to a product of terms of the form $\\hat u_{-(s-1)}\\hat u_{s-1}\\hat \\zeta_{s-1} \\hat \\zeta_{-(s-1)}$ and $\\hat u_{-s}\\hat u_{s}\\hat \\zeta_s \\hat \\zeta_{-s}$. Since $u$ and $\\zeta$ are real vectors, $\\hat u_j \\hat u_{-j} = |u_j|^2$ and $\\hat \\zeta_j \\hat \\zeta_{-j} = |\\hat \\zeta_{j}|^2$ for all $j = -\\lfloor L/2 \\rfloor, \\dots, \\lfloor L/2 \\rfloor$, so $$\\prod_{n = 1}^m \\hat u_{-j_n} \\hat \\zeta_{j_n} = (|\\hat u_{s-1}|^2 |\\hat \\zeta_{s-1}|^2)^{\\alpha+\\beta} (|\\hat u_s|^2 |\\hat \\zeta_s|^2)^{\\gamma+\\delta}\\,.$$ This quantity depends only on the moduli $|\\hat \\zeta_s|$ and $|\\hat \\zeta_{s-1}|$, hence it is the same for all $\\zeta \\in {\\mathcal{P}}$. This completes the proof of and therefore the proof of the proposition.",
"\n\nProof of Theorem \\[thm:main\\_lower\\]\n------------------------------------\n\nFix $n \\geq 1$. We will select $\\phi, \\theta_n \\in {\\mathcal{T}}_s$ such that $\\rho(\\phi, \\theta_n) \\geq c \\big(\\frac{\\sigma^{(2s-1) \\vee (s+1)}}{\\sqrt n} \\wedge 1\\big)$ for some small universal constant $c > 0$ but $D({\\mathrm}{P}_{\\theta_n}^n {\\,\\|\\,}{\\mathrm}{P}_{\\phi}^n) \\leq \\frac 12$. The bound will then follow from standard techniques.",
"\n\nIf $s = 0$, then let $\\phi = 0$ and $\\theta_n = \\frac{\\sigma}{\\sqrt{nL\n}} {\\mathbf{1}}$, where ${\\mathbf{1}}$ denotes the all-ones vector of ${{\\rm I}\\kern-0.18em{\\rm R}}^L$. Note that ${\\rho}(\\theta_n, \\phi) = \\sigma/\\sqrt n$. Moving to the Fourier domain, we have $(\\widehat{\\theta_n})_0 = \\sigma/\\sqrt{n}$ and $(\\widehat{\\theta_n})_j = 0$ for $j \\neq 0$, so that $\\theta_n, \\phi \\in {\\mathcal{T}}_0$. By Lemma \\[lem:first\\_moment\\], $$D({\\mathrm}{P}_{\\theta_n}^n {\\,\\|\\,}{\\mathrm}{P}_{\\phi}^n) = \\frac{n\\sigma^2}{2 n\\sigma^2} = \\frac{1}{2}\\,.$$\n\nIf $s = 1$, let $\\phi$ be given by $$\\hat \\phi_j = \n\\left\\{ \\begin{array}{ll}\n1/\\sqrt{2} & \\text{if $j \\in \\{\\pm 1\\}$,} \\\\\n0 & \\text{otherwise.}\\end{array}\\right.$$ and let $\\theta_n$ satisfy $$(\\widehat{\\theta_n})_j =\n\\left\\{ \\begin{array}{ll}\n1/\\sqrt{2} + c_1 \\frac{\\sigma^2}{\\sqrt{2n}} & \\text{if $j \\in \\{\\pm 1\\}$,} \\\\\n0 & \\text{otherwise,}\\end{array}\\right.$$ for some constant $c_1 > 0$ to be specified. ",
"Clearly $\\theta_n, \\phi \\in {\\mathcal{T}}_1$, and ${\\rho}(\\theta_n, \\phi) = c_1 \\frac{\\sigma^2}{\\sqrt n}$.\n\nTheorem \\[thm:tight\\_KL\\_bound\\] implies that $$D({\\mathrm}{P}_{\\theta_n}^n {\\,\\|\\,}{\\mathrm}{P}_{\\phi}^n) \\leq \\overline C \\frac{n{\\rho}(\\theta_n, \\phi)^2}{\\sigma^4} \\leq \\frac{1}{2}\\,,$$ by choosing $c_1$ small enough.",
"\n\nFinally, suppose $s \\geq 2$. Fix $z = e^{\\mathrm{i} \\delta}$ for $\\delta = c_1 (\\sigma^{2s-1}/\\sqrt{n} \\wedge 1)$ for some positive constant $c_1 \\leq 1$ to be specified. ",
"Let $\\phi$ be given by $$\\hat \\phi_j =\n\\left\\{ \\begin{array}{ll}\n1/2 & \\text{if $j \\in \\{\\pm(s-1), \\pm s \\}$,} \\\\\n0 & \\text{otherwise.}",
"\n\\end{array}\\right.$$ Let $\\theta_n$ be given by $$\\begin{aligned}\n(\\widehat{\\theta_n})_j =\n\\left\\{ \\begin{array}{ll}\n1/2 & \\text{if $j \\in \\{\\pm(s-1)\\}$,} \\\\\nz/2 & \\text{ if $j = s$,} \\\\\nz^*/2 & \\text{ if $j = -s$,} \\\\\n0 & \\text{otherwise.}",
"\n\\end{array}\\right.\\end{aligned}$$ Note that $\\theta_n$ and $\\phi$ both lie in ${\\mathcal{T}}_s$.\n\nFor any unit complex number $w \\in U(1)$, we have $$\\|\\theta_n - G_w \\phi\\|^2 = \\frac 12 (\\|1 - w\\|^2 + \\|z - w\\|^2) \\ge \\frac14 \\|1 - z\\|^2\\,.$$ So $$\\frac 1 4 \\|1-z\\|^2 \\leq \\rho^2(\\theta_n, \\phi) \\leq \\|\\theta_n - \\phi\\|^2 = \\frac 12 \\|1 - z\\|^2\\,,$$ and under the assumption that $\\delta \\leq 1$, we have $\\frac 12 \\delta^2 \\leq \\|1 - z\\|^2 \\leq \\delta^2$.\n\nTherefore $$c \\delta \\leq \\rho(\\theta_n, \\phi) \\leq C \\delta$$ for absolute positive constants $c$ and $C$. ",
"Theorem \\[thm:tight\\_KL\\_bound\\] and Proposition \\[prop:generalized\\_matching\\] imply that$$D({\\mathrm}{P}_{\\theta_n}^n {\\,\\|\\,}{\\mathrm}{P}_{\\phi}^n) \\le \\overline{C} n \\sigma^{-4s+2}{\\rho}^2(\\phi_n, \\tau)\\le \\frac12\\,,$$ by taking $c_1$ small enough.",
"\n\nIn all three cases, we have a bound $D({\\mathrm}{P}_{\\theta_n}^n {\\,\\|\\,}{\\mathrm}{P}_{\\phi}^n) \\le 1/2$ for $\\theta_n, \\phi \\in {\\mathcal{T}}_s$ satisfying ${\\rho}(\\theta_n, \\phi) \\geq c (\\sigma^{(2s -1) \\wedge (s+1)}/\\sqrt n \\wedge 1)$ for some constant $c > 0$. Using standard minimax lower bound techniques [@Tsy09], we get the desired result.",
"\n\nInformation geometry for algebraically structured models {#sec:tight_bounds}\n========================================================\n\nOur proof techniques rely on understanding the curvature of the Kullback-Leibler divergence around its minimum, which is known to control the information geometry of the problem. ",
"In this section, we obtain precise bounds on the divergence $D({\\mathrm}{P}_\\theta {\\,\\|\\,}{\\mathrm}{P}_\\phi)$ for pairs of signals $\\theta$ and $\\phi$ for any choice of a subgroup ${\\mathcal{G}}$ of the orthogonal group $\\mathrm{O}(L)$.\n\nWe extend the approach of [@CaiLow11] to bound the divergence between $D({\\mathrm}{P}_\\theta {\\,\\|\\,}{\\mathrm}{P}_\\phi)$ in terms of the Hilbert-Schmidt distance between the moment tensors ${{\\rm I}\\kern-0.18em{\\rm E}}[(G \\theta)^{\\otimes m}]$ and ${{\\rm I}\\kern-0.18em{\\rm E}}[(G \\phi)^{\\otimes m}]$. Recall that $G$ is uniformly distributed over ${\\mathcal{G}}$. In what follows, we write $$\\Delta_m := {{\\rm I}\\kern-0.18em{\\rm E}}[(G\\theta)^{\\otimes m}] - {{\\rm I}\\kern-0.18em{\\rm E}}[(G \\phi)^{\\otimes m}]\\,.$$ Our results imply that when $\\sigma$ is bounded below by a constant, the divergence can be bounded above and below by an infinite series of the form $\\sum_m \\frac{c^m\\|\\Delta_m\\|^2}{\\sigma^{2m} m!}$. We note that the assumption that $\\sigma$ be bounded below is essential: when $\\sigma \\to 0$, it is not hard to show that $D(\\theta {\\,\\|\\,}\\phi) = \\frac{{\\rho}^2(\\phi, \\theta)}{2 \\sigma^2} + o(\\sigma^{-2})$.\n\nFor convenience, we write $D(\\theta {\\,\\|\\,}\\phi)$ for $D({\\mathrm}{P}_\\theta {\\,\\|\\,}{\\mathrm}{P}_\\phi)$. We begin by establishing the effect of the first moments ${{\\rm I}\\kern-0.18em{\\rm E}}G \\theta$ and ${{\\rm I}\\kern-0.18em{\\rm E}}G \\phi$ on $D(\\theta {\\,\\|\\,}\\phi)$.\n\n\\[lem:first\\_moment\\] If $\\vartheta = \\theta - {{\\rm I}\\kern-0.18em{\\rm E}}G \\theta$ and $\\varphi = \\phi - {{\\rm I}\\kern-0.18em{\\rm E}}G \\phi$, then $$D(\\theta {\\,\\|\\,}\\phi) =D(\\vartheta {\\,\\|\\,}\\varphi)+ \\frac{1}{2\\sigma^2} \\|\\Delta_1\\|^2 \\,.$$\n\nLemma \\[lem:first\\_moment\\] implies that it suffices to bound $D(\\theta {\\,\\|\\,}\\phi)$ for vectors $\\theta$ and $\\phi$ satisfying ${{\\rm I}\\kern-0.18em{\\rm E}}G \\theta = {{\\rm I}\\kern-0.18em{\\rm E}}G \\phi = 0$, which we accomplish in the following theorem.",
"\n\n\\[thm:tight\\_KL\\_bound\\] Let $\\theta, \\phi$ in ${{\\rm I}\\kern-0.18em{\\rm R}}^L$ satisfy $3 {\\rho}(\\theta, \\phi) \\leq \\|\\theta\\| \\le \\sigma$ and ${{\\rm I}\\kern-0.18em{\\rm E}}G \\theta = {{\\rm I}\\kern-0.18em{\\rm E}}G \\phi = 0$. For any $k \\geq 1$, there exist universal constants $\\underline C$ and $\\overline C$ such that $$\\underline C \\sum_{m=1}^{\\infty} \\frac{\\|\\Delta_m\\|^2}{(\\sqrt 3\\sigma)^{2m} m!} ",
"\\leq D(\\theta {\\,\\|\\,}\\phi) \\leq 2 \\sum_{m=1}^{k-1} \\frac{\\|\\Delta_m\\|^2}{\\sigma^{2m} m!} ",
"+ \\overline C\\frac{\\|\\theta\\|^{2k-2}{\\rho}(\\theta, \\phi)^2}{\\sigma^{2k}} \\,.$$\n\nIn particular, Theorem \\[thm:tight\\_KL\\_bound\\] implies that if $\\|\\Delta_m\\| = 0$ for $m = 1, \\dots, k-1$ and $\\|\\Delta_k\\| \\geq c {\\rho}(\\theta, \\phi)$ for some constant $c$, then $D(\\theta \\,\\|\\, \\phi)$ is of order $\\sigma^{-2k} {\\rho}^2(\\theta, \\phi)$.\n\nWe first prove the following simple expression: $$D({\\mathrm}{P}_\\theta {\\,\\|\\,}{\\mathrm}{P}_\\phi) = D(\\theta {\\,\\|\\,}\\phi) = \\frac{1}{2 \\sigma^2}(\\|\\phi\\|^2 - \\|\\theta\\|^2) + {{\\rm I}\\kern-0.18em{\\rm E}}\\log \\frac{{{\\rm I}\\kern-0.18em{\\rm E}}[e^{\\frac{1}{\\sigma^2} (\\theta + \\sigma \\xi)^\\top G \\theta} \\mid \\xi ]}{{{\\rm I}\\kern-0.18em{\\rm E}}[e^{\\frac{1}{\\sigma^2} (\\theta + \\sigma \\xi)^\\top G \\phi} \\mid \\xi ]}\\,,$$ where $\\xi\\sim {\\mathcal{N}}(0,I_L)$ and $G \\in {\\mathcal{G}}$ is uniform and independent of $\\xi$.\n\nThis claim follows directly from the definition of divergence. ",
"Let ${\\mathsf{g}}$ the density of a standard Gaussian random variable with respect to the Lebesgue measure on ${{\\rm I}\\kern-0.18em{\\rm R}}^L$. It holds $$\\begin{aligned}\n\\frac{d{\\mathrm}{P}_\\theta}{d{\\mathrm}{P}_\\phi}(y) & = \\frac{{{\\rm I}\\kern-0.18em{\\rm E}}[{\\mathsf{g}}((y - G\\theta)/\\sigma)]}{{{\\rm I}\\kern-0.18em{\\rm E}}[{\\mathsf{g}}((y - G\\phi)/\\sigma)]} \\\\\n& = \\frac{{{\\rm I}\\kern-0.18em{\\rm E}}\\left[\\exp\\left(-\\frac{1}{2\\sigma^2}(\\|y\\|^2 - 2 y^\\top G \\theta + \\|G \\theta\\|^2)\\right)\\right]}{{{\\rm I}\\kern-0.18em{\\rm E}}\\left[\\exp\\left(-\\frac{1}{2\\sigma^2}(\\|y\\|^2 - 2 y^\\top G \\phi + \\|G \\phi\\|^2)\\right)\\right]} \\\\\n& = \\exp\\left(\\frac{1}{2\\sigma^2}(\\|\\phi\\|^2 - \\|\\theta\\|^2)\\right)\\frac{{{\\rm I}\\kern-0.18em{\\rm E}}\\left[\\exp\\left(\\frac{1}{\\sigma^2}y^\\top G \\theta\\right)\\right]}{{{\\rm I}\\kern-0.18em{\\rm E}}\\left[\\exp\\left(\\frac{1}{\\sigma^2} y^\\top G \\phi\\right)\\right]}\\,,\\end{aligned}$$ since $G$ is orthogonal. ",
"Hence, if $Y \\sim {\\mathrm}{P}_\\theta$, we have $$\\begin{aligned}\nD(\\theta {\\,\\|\\,}\\phi) & = {{\\rm I}\\kern-0.18em{\\rm E}}\\log \\frac{d{\\mathrm}{P}_\\theta}{d{\\mathrm}{P}_\\phi}(Y) = \\frac{1}{2\\sigma^2}(\\|\\phi\\|^2 - \\|\\theta\\|^2) + {{\\rm I}\\kern-0.18em{\\rm E}}\\log \\frac{{{\\rm I}\\kern-0.18em{\\rm E}}[e^{\\frac{1}{\\sigma^2} Y^\\top G \\theta} \\mid Y ]}{{{\\rm I}\\kern-0.18em{\\rm E}}[e^{\\frac{1}{\\sigma^2} Y^\\top G \\phi} \\mid Y]}\\,.\\end{aligned}$$ Note that we can write $Y = G' \\theta + \\sigma \\xi$ for a standard Gaussian vector $\\xi$ and $G' \\in {\\mathcal{G}}$ an independent copy of $G$. Since $G'\\in O(L)$, $Y$ has the same distribution as $G'(\\theta + \\sigma \\xi)$. If $G$ and $G'$ are independent and uniform, then $(G')^\\top G$ has the same distribution as $G$, so $$Y^\\top G \\theta \\stackrel{d}{=} (\\theta + \\sigma \\xi)^\\top G \\theta\\,,$$ where the above equality holds in distribution. ",
"It yields $$\\label{EQ:KLexp}\nD(\\theta {\\,\\|\\,}\\phi) = \\frac{1}{2\\sigma^2}(\\|\\phi\\|^2 - \\|\\theta\\|^2) + {{\\rm I}\\kern-0.18em{\\rm E}}\\log \\frac{{{\\rm I}\\kern-0.18em{\\rm E}}[e^{\\frac{1}{\\sigma^2} (\\theta + \\sigma \\xi)^\\top G \\theta} \\mid \\xi ]}{{{\\rm I}\\kern-0.18em{\\rm E}}[e^{\\frac{1}{\\sigma^2} (\\theta + \\sigma \\xi)^\\top G \\phi} \\mid \\xi]}\\,.$$\n\nWe now turn to the proof of Lemma \\[lem:first\\_moment\\]. ",
"For convenience write $\\bar \\theta = {{\\rm I}\\kern-0.18em{\\rm E}}G \\theta$ and $\\bar \\phi = {{\\rm I}\\kern-0.18em{\\rm E}}G \\phi$. These vectors satisfy $G \\bar \\theta = \\bar \\theta$ and $G \\bar \\phi = \\bar \\phi$ almost surely and $\\vartheta^\\top\\bar \\theta=0$. Hence, almost surely, $$\\begin{aligned}\n(\\theta + \\sigma \\xi)^\\top G \\theta & = (\\bar \\theta + \\vartheta + \\sigma \\xi)^\\top G (\\bar \\theta + \\vartheta) \\\\\n& = (\\vartheta + \\sigma \\xi)^\\top G \\vartheta + (\\bar \\theta + \\sigma \\xi)^\\top \\bar \\theta\\,,\\end{aligned}$$ and similarly $$(\\theta + \\sigma \\xi)^\\top G \\phi = (\\vartheta + \\sigma \\xi)^\\top G \\varphi + (\\bar \\theta + \\sigma \\xi)^\\top \\bar \\phi\\,.$$\n\nPlugging these quantities into yields $$\\begin{aligned}\nD(\\theta &{\\,\\|\\,}\\phi) = \\frac{1}{2\\sigma^2}(\\|\\phi\\|^2 - \\|\\theta\\|^2) + {{\\rm I}\\kern-0.18em{\\rm E}}\\log \\frac{{{\\rm I}\\kern-0.18em{\\rm E}}[e^{\\frac{1}{\\sigma^2} (\\vartheta + \\sigma \\xi)^\\top G \\vartheta + (\\bar \\theta + \\sigma \\xi)^\\top \\bar \\theta} \\mid \\xi ]}{{{\\rm I}\\kern-0.18em{\\rm E}}[e^{\\frac{1}{\\sigma^2} (\\vartheta + \\sigma \\xi)^\\top G \\varphi + (\\bar \\theta + \\sigma \\xi)^\\top \\bar \\phi} \\mid \\xi]} \\\\\n& = \\frac{1}{2\\sigma^2}(\\|\\phi\\|^2 - \\|\\theta\\|^2) + \\frac{1}{\\sigma^2}{{\\rm I}\\kern-0.18em{\\rm E}}(\\bar \\theta + \\sigma \\xi)^\\top (\\bar \\theta - \\bar \\phi) + {{\\rm I}\\kern-0.18em{\\rm E}}\\log \\frac{{{\\rm I}\\kern-0.18em{\\rm E}}[e^{\\frac{1}{\\sigma^2} (\\vartheta + \\sigma \\xi)^\\top G \\vartheta} \\mid \\xi ]}{{{\\rm I}\\kern-0.18em{\\rm E}}[e^{\\frac{1}{\\sigma^2} (\\vartheta + \\sigma \\xi)^\\top G \\varphi} \\mid \\xi]} \\\\\n& = \\frac{1}{2\\sigma^2}(\\|\\bar \\phi\\|^2 - \\|\\bar \\theta\\|^2) + \\frac{1}{\\sigma^2} (\\|\\bar \\theta\\|^2 - \\bar \\theta^\\top \\bar \\phi) + D(\\vartheta {\\,\\|\\,}\\varphi) \\\\\n& = \\frac{1}{2 \\sigma^2}\\|{{\\rm I}\\kern-0.18em{\\rm E}}[G \\theta - G \\phi]\\|^2 + D(\\vartheta {\\,\\|\\,}\\varphi)\\,.\\end{aligned}$$\n\nIf $\\theta = 0$, then the conditions of the theorem imply that $\\phi = 0$, so the statement is vacuous. ",
"We therefore assume $\\theta \\neq 0$. The divergence $D(\\theta {\\,\\|\\,}\\phi)$ and the moment tensors ${{\\rm I}\\kern-0.18em{\\rm E}}[(G \\phi)^{\\otimes m}]$ are unaffected if we replace $\\phi$ by $G_0 \\phi$ for any $G_0 \\in {\\mathcal{G}}$. Hence without loss of generality, we can assume that $\\|\\theta - \\phi\\| = {\\rho}(\\theta, \\phi) =: {\\varepsilon}$. Moreover, the quantity $D(\\theta {\\,\\|\\,}\\phi)$ and the bounds in question are all unaffected upon replacing $\\theta$, $\\phi$, and $\\sigma$ by $\\theta/\\|\\theta\\|$, $\\phi/\\|\\theta\\|$, and $\\sigma/\\|\\theta\\|$, respectively, so we assume in what follows that $\\|\\theta\\| = 1$ and $\\sigma \\geq 1$.\n\nWe first prove the upper bound. ",
"Denote by ${\\mathsf{g}}$ the density of a standard $L$-dimensional Gaussian random variable. ",
"For all $\\zeta \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$, let $f_\\zeta$ denote the density of ${\\mathrm}{P}_\\zeta$ defined in . ",
"Recall that in this model, $G$ is drawn uniformly from the Haar measure on ${\\mathcal{G}}$. Then, for any $y \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$ we have $$f_\\zeta(y) = {{\\rm I}\\kern-0.18em{\\rm E}}\\frac{1}{\\sigma^d} {\\mathsf{g}}(\\sigma^{-1}(y - G \\zeta)) = \\frac{1}{\\sigma^d} {\\mathsf{g}}(\\sigma^{-1} y) e^{-\\frac{\\|\\zeta\\|^2}{2 \\sigma^2}}{{\\rm I}\\kern-0.18em{\\rm E}}e^{\\frac{y^\\top G \\zeta}{\\sigma^2}}\\,.$$ Let $\\chi^2(\\theta, \\phi)$ denote the $\\chi^2$-divergence between ${\\mathrm}{P}_\\theta$ and ${\\mathrm}{P}_\\phi$, defined by $$\\chi^2(\\theta, \\phi) = \\int \\frac{(f_\\theta(y) - f_\\phi(y))^2}{f_\\theta(y)} \\mathrm dy\\,.$$\n\nSince ${{\\rm I}\\kern-0.18em{\\rm E}}G \\theta = 0$ by assumption, Jensen’s inequality implies $$f_\\theta(y) \\geq \\frac{1}{\\sigma^d} {\\mathsf{g}}(\\sigma^{-1} y) e^{-\\frac{\\|\\theta\\|^2}{2\\sigma^2}}e^{{{\\rm I}\\kern-0.18em{\\rm E}}\\frac{y^\\top G \\theta}{\\sigma^2}} = \\frac{1}{\\sigma^d} {\\mathsf{g}}(\\sigma^{-1} y) e^{-\\frac{\\|\\theta\\|^2}{2\\sigma^2}} \\,.$$\n\nHence $$\\frac{(f_\\theta(y) - f_\\phi(y))^2}{f_\\theta(y)} \\leq e^{\\frac{\\|\\theta\\|^2}{2 \\sigma^2}} \\big(e^{-\\frac{\\|\\theta\\|^2}{2 \\sigma^2}}{{\\rm I}\\kern-0.18em{\\rm E}}e^{\\frac{y^\\top G \\theta}{\\sigma^2}} - e^{-\\frac{\\|\\phi\\|^2}{2 \\sigma^2}}{{\\rm I}\\kern-0.18em{\\rm E}}e^{\\frac{y^\\top G \\phi}{\\sigma^2}}\\big)^2 \\big(\\frac{1}{\\sigma^d} {\\mathsf{g}}(\\sigma^{-1} y)\\big)\\,.$$ Integrating this quantity with respect to $y$ yields a bound on the $\\chi^2$ divergence. ",
"Let $\\xi \\sim {\\mathcal{N}}(0, I_d)$ and observe that $$\\begin{aligned}\n\\chi^2(\\theta, \\phi) & \\leq {{\\rm I}\\kern-0.18em{\\rm E}}\\Big[e^{\\frac{\\|\\theta\\|^2}{2 \\sigma^2}} \\big(e^{-\\frac{\\|\\theta\\|^2}{2 \\sigma^2}}{{\\rm I}\\kern-0.18em{\\rm E}}\\big[ e^{\\frac{\\sigma \\xi^\\top G \\theta}{\\sigma^2}}\\big|\\xi\\big] - e^{-\\frac{\\|\\phi\\|^2}{2 \\sigma^2}}{{\\rm I}\\kern-0.18em{\\rm E}}\\big[ e^{\\frac{\\sigma \\xi^\\top G \\phi}{\\sigma^2}}\\big|\\xi\\big]\\big)^2 \\Big]\\\\\n& = {{\\rm I}\\kern-0.18em{\\rm E}}\\Big[ e^{\\frac{\\|\\theta\\|^2}{2 \\sigma^2}} \\big(e^{-\\frac{\\|\\theta\\|^2}{ \\sigma^2}}e^{\\frac{\\xi^\\top (G + G')\\theta}{\\sigma}} - 2 e^{-\\frac{\\|\\theta\\|^2 + \\|\\phi\\|^2 }{2 \\sigma^2}} e^{\\frac{\\xi^\\top (G \\theta + G' \\phi)}{\\sigma}}\\\\\n&\\le 2{{\\rm I}\\kern-0.18em{\\rm E}}\\big[ e^{\\frac{(G'\\theta)^\\top G \\theta}{\\sigma^2}} - 2 e^{\\frac{(G'\\phi)^\\top G \\theta}{\\sigma^2}} +e^{\\frac{(G'\\phi)^\\top G\\phi}{\\sigma^2}}\\big]\\end{aligned}$$ where $G' \\in {\\mathcal{G}}$ is an independent copy of $G$ and we used the bound $e^{\\|\\theta\\|^2/(2\\sigma^2)} \\le 2$ that holds for $\\sigma \\geq 1$ and $\\|\\theta\\|^2 \\leq 1$.\n\nThe random variables $G \\theta$ and $G \\phi$ have moment generating functions that converge in a neighborhood of the origin, hence $$\\begin{aligned}\n\\chi^2(\\theta, \\phi) & \\le \\sum_{m \\geq 0} \\frac{2}{\\sigma^{2m} m!} {{",
"\\rm I}\\kern-0.18em{\\rm E}}\\left[((G'\\theta)^\\top G \\theta)^m - 2 ((G'\\phi)^\\top G \\theta)^m + ((G'\\phi)^\\top G\\phi)^m\\right] \\\\\n& = \\sum_{m \\geq 0} \\frac{2}{\\sigma^{2m} m!} ",
"\\|{{\\rm I}\\kern-0.18em{\\rm E}}[(G \\theta)^{\\otimes m}]\\|^2 - 2 \\langle {{\\rm I}\\kern-0.18em{\\rm E}}[(G \\theta)^{\\otimes m}], {{\\rm I}\\kern-0.18em{\\rm E}}[(G \\phi)^{\\otimes m}]\\rangle + \\|{{\\rm I}\\kern-0.18em{\\rm E}}[(G \\phi)^{\\otimes m}]\\|^2 \\\\\n& = \\sum_{m \\geq 0} \\frac{2}{\\sigma^{2m} m!} ",
"\\left\\| \\Delta_m\\right\\|^2 \\\\\n &\\leq 2\\sum_{m =1}^{k-1} \\frac{\\|\\Delta_m\\|^2}{\\sigma^{2m}m!} ",
"+ 24 \\sum_{m \\geq k} \\frac{2^{m}}{\\sigma^{2m}m!} {",
"\\varepsilon}^2\n \\leq 2\\sum_{m =1}^{k-1} \\frac{\\|\\Delta_m\\|^2}{\\sigma^{2m}m!} ",
"+ 24 e^{2} \\frac{{\\varepsilon}^2}{\\sigma^{2k}}\\,,\n$$ where the penultimate inequality follows from Lemma \\[lem:tensor\\_bound\\] in Appendix \\[supp\\]. ",
"The bound follows upon applying the inequality $D(\\theta {\\,\\|\\,}\\phi) \\leq \\chi^2(\\theta, \\phi)$ [@Tsy09].",
"\n\nWe now turn to the lower bound. ",
"Recall that the Hermite polynomials $\\{h_k(x)\\}_{k \\geq 0}$ satisfy the following three properties [@Sze75]:\n\n1. ",
" The function $h_k(x)$ is a degree-$k$ polynomial.",
"\n\n2. ",
" The functions $\\{h_k\\}_{k \\geq 0}$ form an orthogonal basis of of $L_2(\\gamma)$, where $\\gamma$ denotes the standard Gaussian measure on ${{\\rm I}\\kern-0.18em{\\rm R}}$, with $\\|h_k\\|_\\mu^2 = k!$.\n\n3. ",
" If $Y \\sim {\\mathcal{N}}(\\mu, 1)$, then ${{\\rm I}\\kern-0.18em{\\rm E}}[h_k(Y)] = \\mu^k$.\n\nGiven a multi-index $\\alpha \\in {{\\rm I}\\kern-0.18em{\\rm N}}^L$, define the multivariate Hermite polynomial $h_\\alpha$ by $$h_\\alpha(x_1, \\dots, x_L) = \\prod_{i=1}^L h_{\\alpha_i}(x_i)\\,.$$ The multivariate Hermite polynomials form an orthonormal basis for the space ${{\\rm I}\\kern-0.18em{\\rm R}}[x_1, \\dots, x_L]$ of $L$-variate polynomial functions with respect to the inner product over $L_2(\\gamma^{\\otimes L})$.\n\nGiven $y \\in {{\\rm I}\\kern-0.18em{\\rm R}}^L$ and $m \\geq 1$, denote by $H_m(y)$ the order-$m$ symmetric tensor defined as follows. ",
"The $(i_1, \\ldots, i_m)$th entry of $H_m(y)$ is given by $\\sigma^m h_\\alpha(\\sigma^{-1}y_1, \\dots, \\sigma^{-1}y_L)$, where $\\alpha \\in \\{0, \\ldots, m\\}^L$ denotes the multi-index associated to $(i_1, \\ldots, i_m)$: $\\alpha_l=|\\{j\\in [m]\\,: i_j=\\ell\\}|, l \\in \\{0,\\ldots, m\\}$. Property 3 of the Hermite polynomials implies that if $Y \\sim {\\mathcal{N}}(\\mu, \\sigma^2 I)$, then ${{\\rm I}\\kern-0.18em{\\rm E}}[H_m(Y)] = \\mu^{\\otimes m}$.\n\nFix $k \\geq 1$, and consider the degree-$k$ polynomial $$t(y) = \\sum_{m=1}^{k} \\frac{\\langle \\Delta_m, H_m(y)\\rangle}{(\\sqrt 3 \\sigma)^{2m} m!}\\,.$$ Note that, if $Y \\sim {\\mathrm}{P}_\\zeta$, then $${{\\rm I}\\kern-0.18em{\\rm E}}[t(Y)] = {{\\rm I}\\kern-0.18em{\\rm E}}\\left[\\sum_{m=1}^{k} \\frac{\\langle \\Delta_m, {{\\rm I}\\kern-0.18em{\\rm E}}[H_m(Y) | G]\\rangle}{(\\sqrt 3 \\sigma)^{2m} m!}\\right] = \\sum_{m=1}^{k} \\frac{\\langle \\Delta_m, {{\\rm I}\\kern-0.18em{\\rm E}}[(G\\zeta)^{\\otimes m}]\\rangle}{(\\sqrt 3 \\sigma)^{2m} m!}\\,.$$ Thus if $Y \\sim {\\mathrm}{P}_\\theta$ and $Y'\\sim {\\mathrm}{P}_\\phi$, we get $${{\\rm I}\\kern-0.18em{\\rm E}}[t(Y)] - {{\\rm I}\\kern-0.18em{\\rm E}}[t(Y')] = \\sum_{m=1}^{k} \\frac{\\|\\Delta_m\\|^2}{(\\sqrt 3 \\sigma)^{2m} m!} ",
"=: \\delta\\,.$$ Lemma \\[lem:tensor\\_bound\\] in Appendix \\[supp\\] implies that $\\delta \\leq 4$. Moreover, by Lemma \\[lem:hermite\\_variance\\_bound\\] in Appendix \\[supp\\], the variances of both $t(Y)$ and $t(Y')$ are bounded above by $e\\delta$. Applying Lemma \\[lem:testing\\_bound\\] in Appendix \\[supp\\] therefore yields $$D(\\theta \\, \\| \\, \\phi) \\geq \\frac{\\delta^2}{4 e \\delta + \\delta^2} \\geq \\frac{\\delta^2}{(4e+4)\\delta} > \\frac{1}{15} \\sum_{m=1}^{k} \\frac{\\|\\Delta_m\\|^2}{(\\sqrt 3 \\sigma)^{2m} m!}\\,.$$ Since $k \\geq 1$ was arbitrary and the summands are nonnegative, letting $k \\to \\infty$ yields the claim.",
"\n\n[^1]: Following an earlier version of this paper, [@AbbBenLee17] considered a version of multi-reference alignment when the distribution of $G$ is not uniform and showed that strictly better rates can be obtained in some cases.",
"\n\n[^2]: Our focus in this work is on statistical properties rather than on computation. ",
"In a companion paper [@PerWeeBan17], we propose and analyze a computationally efficient estimator for multi-reference alignment.",
"\n"
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"Many LGBT youth who are rejected by their own families often encounter prejudice, harassment and abuse when they shift to a foster home. ",
"Photo: FIle\n\nMany LGBT youth who are rejected by their own families often encounter prejudice, harassment and abuse when they shift to a foster home. ",
"Photo: FIle\n\nNew research claims that homeless youth that are LGBTQ are twice as likely to face homelessness than straight youth, and those who are homeless are at a significantly higher risk of violence and death compared to straight youth.",
"\n\nThe study, from Chapin Hall at the University of Chicago, included a phone survey of 26,161 youth across over 22 counties, as well as 215 in-depth interviews with LGBTQ-identified homeless kids. ",
"This is the second in a series of briefs done as part of the Voices of Youth Count project, attempting to link evidence and action in an effort to end youth homelessness.",
"\n\nLGBT youth were shown to face a higher percentage of risky conditions on the streets, including being 15% more likely to be physically harmed, 10% more likely to self-harm, and 18% more likely to exchange sex for basic needs.",
"\n\nDiscrimination both within and outside their family was also far more likely, with 64% of LHBTQ youth reporting such from their family, and 60% outside of it. ",
"This compared to only 37% of straight youth reporting either.",
"\n\nFurther, race also played a part, with youth who are black or multiracial and LGBTQ being 16% more likely to be homeless, versus only 4% for white, straight youth.",
"\n\nMatthew Morton, the principal Investigator for Voices of Youth Count, wants to offer up a silver lining. “",
"The findings are a bit grim, but hopefully it will galvanize communities to help these young people.",
"\n\nThis is a very resilient population, and many expressed hope that they can rise above their circumstances,” he continued. “",
"The message here is really from the youth themselves – if we listen to them and offer help where it’s needed, they can make great progress.”",
"\n\nIn the first Voices of Youth Count Study, from 2017, it was discovered that one in thirty 13-17 year olds had been homeless at some point in the year before."
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"This is a personal fitness & running inspiration blog. ",
"I am an active runner who loves to share my experiences--the good & the bad. ",
"Please connect with me if you have questions or would like to know more!",
"\n\nCategories\n\nMay 7 Weekly Workouts [week 245]\n\nJust not feeling it last week. ",
"I am hoping that I will gain some momentum this week, even while traveling!",
"\n\nWeek 245\n\nSunday, 4.29:Rested, but walked a ton around Seattle!Monday, 4.30: Was going to run, but the gym was packed this morning at the hotel. ",
"Instead I did a back+bicep workout which was good!Tuesday, 5.1: Got to the gym early enough to run 1 mile on the treadmill!! ",
"It felt really good, even if my time was a little slower at 10:45 :)Wednesday, 5.2:Ran 1 mile on the treadmill again in 10:40!! ",
"And then we walked a ton more around Seattle before heading home :)Thursday, 5.3: I did not get up for an early morning workout and then had a bit of an emotional meltdown after work....so I skipped my kettlebell class, too. ",
"I needed the break.",
"Friday, 5.4: Skipped today.",
"Saturday, 5.5:Ran the Derby Day 5K in 36:32--I knew it wouldn't be a great race, but I'm still glad I finished. ",
"I also ran/walked there and back for an additional 3ish miles!!"
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"GENOME ANNOUNCEMENT\n===================\n\nGuinea pig cytomegalovirus (GPCMV) is a model for congenital CMV infection, an important cause of disability in newborns ([@B1]). ",
"Strain 22122, isolated by Hartley in 1957 (ATCC, VR-682), is the only characterized GPCMV strain ([@B2]). ",
"Salivary gland (SG) extracts are pathogenic, while tissue culture (TC)-passaged viruses are significantly attenuated. ",
"Existing genomic sequences were derived from TC-adapted GPCMV ([@B3]) or ATCC virus (AB592928), which was also passaged multiple times in cell culture prior to submission to ATCC ([@B4]). ",
"Therefore, we sought to determine the sequence of pathogenic GPCMV, obtained originally from the ATCC strain 21222 stock but maintained exclusively by serial passage in animals. ",
"Sequencing was performed directly on DNA purified from the SG homogenate, with no intervening passage in TC.",
"\n\nApproximately 35 serial SG passages were made in animals over a 25-year period (1985 to 2010) in Cincinnati, OH, and Minneapolis, MN, in strain 2 guinea pigs. ",
"SG tissue was minced, homogenized, clarified by centrifugation, and stored at −80°C. ",
"350 PFU of this SG stock has been shown to produce efficient fetal infection and disease ([@B5]). ",
"Virions in the SG homogenate were pelleted by ultracentrifugation; treated with DNase, proteinase K, and RNase; phenol/chloroform extracted; alcohol precipitated; and sequenced on an Illumina Genome Analyzer. ",
"119,721,518 raw reads were obtained; reads under 70 bp were discarded. ",
"Of the remaining 83,249,482 reads, 30,674,158 matched AB592928, which was used as a template to assemble the full-length SG sequence. ",
"Average coverage was 9,609.",
"\n\nInitial comparison of the SG sequence to AB592928 suggested deletions within a ∼1.2-kb region of complex repeats thought to comprise the GPCMV origin of replication. ",
"Given the short lengths of the Illumina reads, these apparent deletions likely represented artifacts of sequence assembly, as within repeated sequences the assembly software could not orient overlapping reads to properly reconstruct the region. ",
"To address this concern, SG homogenate was PCR amplified using primers flanking the repeat region (5′-TGG GTG TGG GAG TGG CTT TG-3′ and 5′-TCG GTC TGG ATG CGT GTT G-3′) with Vent polymerase, and multiple independently isolated clones were sequenced. ",
"The size of the PCR product (∼1.2 kb) was consistent with that predicted from AB582828 and, following TA cloning and Sanger sequencing, this sequence was incorporated into the final SG sequence.",
"\n\nComparison of the AB592928 with SG-22122 revealed eleven differences: four nucleotide substitutions, one clarification of an ambiguous base, one nucleotide insertion, and five 1- to 2-nucleotide differences in poly(T) tracts. ",
"None of the differences altered codons in currently annotated open reading frames.",
"\n\nWe annotated an alternative start codon for *gp148*, a major histocompatibility complex (MHC) I homolog. ",
"This removed a cluster of sixteen 7-bp repeats near the 5′ end of the previous annotation, resulting in a shorter protein of 350 amino acids (aa). ",
"Other putative open reading frames (ORFs) are similar between the SG-22122 and AB592928 sequences, with the exception that analysis of SG-22122 revealed two additional previously unannotated gp138 family ORFs (gp138.1, gp138.3). ",
"All three members of this gp138 ORF cluster were noted to bear predicted structural similarity to immunoglobulin variable (IgV) domains from the Ig superfamily of proteins, suggesting a potential role for these gene products in immune evasion/immune modulation.",
"\n\nIn summary, the sequence of the pathogenic SG genome reveals surprisingly few differences from ATCC stock. ",
"A similar level of stability has been noted upon whole-genome sequencing for murine CMV ([@B6]). ",
"For GPCMV, attenuation following tissue culture passage may be related to deletions in key pathogenesis genes ([@B7]). ",
"Additional studies are required to elucidate the molecular basis of GPCMV attenuation following TC adaptation.",
"\n\nNucleotide sequence accession number. {#",
"h1}\n-------------------------------------\n\nThe genome sequence for this strain has been deposited with GenBank under the accession number [KC503762](KC503762).",
"\n\n**Citation** Yang D, Tamburro K, Dittmer D, Cui X, McVoy MA, Hernandez-Alvarado N, Schleiss MR. ",
"2013. ",
"Complete genome sequence of pathogenic guinea pig cytomegalovirus from salivary gland homogenates of infected animals. ",
"Genome Announc. ",
"1(2):e00054-13. ",
"doi:10.1128/genomeA.00054-13.",
"\n\nThis work was supported by NIH grants CA019014, HD044864, and HD038416.",
"\n\nHelpful discussions with Martin G. Myers, Galveston, TX, are gratefully acknowledged (M.R.S.).",
"\n"
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"A federal appeals court handed Florida Gov. Rick Scott a decisive victory this week when it temporarily blocked a federal judge’s ruling that urged the state’s executive clemency board to overhaul its “fatally flawed” process of restoring felons’ voting rights by Thursday.",
"\n\nWhy it matters: The order issued late Wednesday, which affects an estimated 1.5 million convicted felons, comes as reform advocates have been fighting to restore their voting rights for nearly two decades.",
"\n\nPeople with past felony convictions are permanently barred from voting, and they have to wait up to eight years to request Scott and the clemency board to consider restoring it.",
"\n\nFelony disenfranchisement laws affect about six million nationally, but Florida, which remains the most stringent, bans more people from voting than any other state.",
"\n\nTimeline:\n\nSeptember 2000: A federal class-action lawsuit against then-Gov. Jeb Bush, challenged the constitutionality of the ban, which plaintiffs said barred an estimated 600,000 people from voting. ",
"They argued that the prohibition dating back to 1868 was used to prevent newly enfranchised blacks from voting during the Reconstruction and Jim Crow era.",
"\n\nA federal class-action lawsuit against then-Gov. Jeb Bush, challenged the constitutionality of the ban, which plaintiffs said barred an estimated 600,000 people from voting. ",
"They argued that the prohibition dating back to 1868 was used to prevent newly enfranchised blacks from voting during the Reconstruction and Jim Crow era. ",
"The state denied that argument, and the U.S. Supreme Court had declined to hear the case.",
"\n\nApril 2007: Then-Republican Gov. Charlie Crist enacted clemency reforms that automatically restored voting rights to non-violent offenders. ",
"The new rules no longer required a hearing or petition, and the clemency board had 30 days to review and grant approvals. ",
"Crist (D), a U.S. Rep, wrote in an Orlando Sentinel op-ed last year that voting rights for 155,315 Floridians were restored in four years. ",
"An estimated 950,000 were disenfranchised.",
"\n\nMarch 2011: Scott unraveled the Crist-era clemency rules soon after taking office in 2011. ",
"He set a five-year minimum waiting period for nonviolent ex-felons and eight years for others. ",
"Those who were denied have to wait at least two years to re-apply.",
"\n\nScott has only approved 3,008 applications since 2011, a spokesperson at the state’s Commission on Offender Review told Axios this week. ",
"Meanwhile, there's a backlog of more than 10,000 applications awaiting review.",
"\n\nJanuary 2018: Organizers behind a November ballot measure announced that voters will decided whether to automatically restore voting rights for ex-felons, except for those convicted of murder or sex crimes. ",
"It needs at least 60% approval.",
"\n\nMarch 2018: A federal judge issued a scathing ruling that ordered the state to establish “robust and meaningful” new rules to determine when and how to restore ex-felons' voting rights. ",
"In February, the judge said the current \"unconstitutional\" process unfairly relies on the personal support of Scott.",
"\n\nApril 2018: Facing a looming deadline, Scott called an emergency late-night meeting Wednesday with board members to consider new rules. ",
"But it got cancelled after the Atlanta-based 11th U.S. Circuit Court of Appeals sided with him and ruled he doesn't have to immediately overhaul the process. ",
"The suit to overturn the ban was filed in March of last year.",
"\n\nThe big picture:\n\nFelon voting rights restoration has the potential to shift the makeup of the country’s largest battleground, which plays a deciding role in presidential elections.",
"\n\nObservers said Democrats would largely benefit because the prohibition disproportionately affects African-Americans, a group that overwhelmingly votes Democratic. ",
"In Florida, per the Sentencing Project, more than 1 in 5 African-Americans are affected.",
"\n\nThis story has been updated to reflect news developments."
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"Living With the TekRMD\n\nAdjusting to Usage\n\nAdapting to Opportunities\n\nIt all started when...\n\nBy using the Tek RMD you can dramatically improve the way you live in your indoor space. ",
"To get the very best out of it though, you may need to think a little bit differently and potentially make some changes to your indoor environment.",
"\n\nOne positive fact we can state right away is that the Tek RMD is both shorter and narrower than a wheelchair so that if you can navigate through your indoor environment with your existing wheelchair, you can certainly do that much more easily with the Tek RMD.",
"\n\nAfter some time you may find that you need to use your wheelchair much less indoors - you will be using your Tek RMD much more as it simply allows you to do more.",
"\n\nThe issue we need to consider is how best to get on and off the device. ",
"You can transfer from your wheelchair to the Tek RMD but in order to do so, generally you will need to have a wheelchair with at least tilt-up foot rests and removable sides. ",
"If you have fixed foot rests its likely that you can't get the Tek RMD close enough.",
"\n\nHaving visited lots of clients at home we have seen many scenarios that would actually make transfers from any device problematic. ",
"One of the really nice aspects of the Tek RMD is that transfers onto it can be a lot easier than transfers into a typical wheelchair. ",
"However, in order to make this a reality the current nature of your environment should be considered. ",
"Whether you are going to be using the Tek RMD at home or at an office it is recommended that a competent healthcare professional has recently undertaken an environment assessment of the premises. ",
"Are you clear about what types of situations you want to use the device in? ",
"Do you want to be able to transport the device? ",
"How will this be done?",
"\n\nTo get onto the Tek RMD you are going to need to bring it to you, perhaps using the remote control, and the \"tail\" of the device needs to go under the surface, (perhaps a bed or chair)you are sitting on. ",
"This means that there must be sufficient clearance underneath. ",
"For example, if the chair you are on has a base that extends right to the floor this is going to be an issue because you wont be able to safely get your feet onto the device and the seat cushion in place because of this base keeping the Tek RMD too far away from you.",
"\n\nAlso, if you try to transfer from a really soft surface you might have problems getting the cushion in place. ",
"When we visit people at home now to demonstrate the Tek RMD we will typically bring a slide sheet and a suitable chair so that lack of suitable equipment doesn't limit the demonstration.",
"\n\nWill you use the Tek RMD to transfer on/off the WC? ",
"If so we need to consider the transfer areas and turning circle that are available. ",
"Using the Tek RMD is is certainly possible to transfer on and off a combined wheeled/commode and shower chair - this can go over the WC and in and out of the chair."
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"Hyde v Wrench\n\nHyde v Wrench [1840] EWHC Ch J90 is a leading English contract law case on the issue of counter-offers and their relation to initial offers. ",
"In it Lord Langdale ruled that any counter-offer cancels the original offer.",
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"On 6 June 1840 Wrench wrote to Hyde's agent offering to sell the farm for £1000, stating that it was the final offer and that he would not alter from it. ",
"Hyde offered £950 in his letter by 8 June, and after examining the offer Wrench refused to accept, and informed Hyde of this on 27 June. ",
"On the 29th Hyde agreed to buy the farm for £1000 without any additional agreement from Wrench, and after Wrench refused to sell the farm to him he sued for breach of contract.",
"\n\nJudgment\nLord Langdale's judgment read:\nUnder the circumstances stated in this bill, I think there exists no valid binding contract between the parties for the purchase of this property. ",
"The defendant offered to sell it for £1000, and if that had been at once unconditionally there would undoubtedly have been a perfect binding contract; instead of that, the plaintiff made an offer of his own, to purchase the property for £950, and he thereby rejected the offer previously made by the defendant. ",
"I think that it was not afterwards competent for him to revive the proposal of the defendant, by tendering an acceptance of it; and that, therefore, there exists no obligation of any sort between the parties.",
"\n\nSee also\n Stevenson, Jacques & Co v McLean [1880] 5 QBD 346\n Butler Machine Tool Co Ltd v Ex-Cell-O Corp (England) Ltd [1977] EWCA Civ 9\n\nReferences\n\nFurther reading\n\nCategory:English contract case law\nCategory:English agreement case law\nCategory:1840 in case law\nCategory:1840 in British law"
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"/* Soot - a J*va Optimization Framework\n * Copyright (C) 1999 Patrick Lam, Patrick Pominville and Raja Vallee-Rai\n *\n * This library is free software; you can redistribute it and/or\n * modify it under the terms of the GNU Lesser General Public\n * License as published by the Free Software Foundation; either\n * version 2.1 of the License, or (at your option) any later version.",
"\n *\n * This library is distributed in the hope that it will be useful,\n * but WITHOUT ANY WARRANTY; without even the implied warranty of\n * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. ",
" See the GNU\n * Lesser General Public License for more details.",
"\n *\n * You should have received a copy of the GNU Lesser General Public\n * License along with this library; if not, write to the\n * Free Software Foundation, Inc., 59 Temple Place - Suite 330,\n * Boston, MA 02111-1307, USA.",
"\n */\n\n/*\n * Modified by the Sable Research Group and others 1997-1999. ",
" \n * See the 'credits' file distributed with Soot for the complete list of\n * contributors. ",
" (Soot is distributed at http://www.sable.mcgill.ca/soot)\n */\n\npackage soot.baf;\n\nimport soot.*;",
"\n\npublic interface PrimitiveCastInst extends Inst\n{\n public Type getFromType();\n public void setFromType(Type t);\n \n public Type getToType();\n public void setToType(Type t);\n}\n"
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"A foreign company supplies and installs a machine in Switzerland. ",
"According to past practices of the Swiss Federal Tax Administration (SFTA), this process undeniably qualified as a work contract for supply."
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"A cDNA encoding a chitinase from the epithelial cell line of chironomus tentans (Insecta, diptera) and its functional expression.",
"\nA cDNA coding for chitinase was isolated from Chironomus cells, which possesses conserved regions I and II characteristic for family 18 chitinases, a C-terminus enriched in Glu and Pro without the typical \"PEST-region,\" putative glycosylation sites, a reduced number of C-terminal cysteines, and no typical chitin binding domain. ",
"Northern blots revealed one specific signal with an apparent size of 2.3 kb. ",
"The cDNA was expressed in the baculovirus/Spodoptera system as a His-tag fusion protein, which was secreted as a functionally active enzyme into the medium and could be separated from endogenous viral and Spodoptera-specific chitinases."
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"SipA, SopA, SopB, SopD and SopE2 effector proteins of Salmonella enterica serovar Typhimurium are synthesized at late stages of infection in mice.",
"\nSalmonella pathogenicity island (SPI)-1 is essential for invasion of non-phagocytic cells, whereas SPI-2 is required for intracellular survival and proliferation in phagocytes. ",
"Some SPI-1 effectors, however, are induced upon invasion of both phagocytic and non-phagocytic cells, suggesting that they may also be required post-invasion. ",
"In the present work, the presence was analysed of SipA, SopA, SopB, SopD and SopE2 effector proteins of Salmonella enterica serovar Typhimurium in vitro and in vivo during murine salmonellosis. ",
"Tagged (3xFLAG) strains of S. enterica serovar Typhimurium were inoculated intraperitoneally or intragastrically to BALB/c mice and recovered from the spleen and mesenteric lymph nodes of moribund mice. ",
"Tagged proteins were detected by SDS-PAGE and immunoblotting with anti-FLAG antibodies. ",
"In vitro experiments showed that SPI-1 effector proteins SipA, SopA, SopB, SopD and SopE2 were secreted under SPI-1 conditions. ",
"Interestingly, it was found that S. enterica serovar Typhimurium continued to synthesize SipA, SopB, SopD and SopE2 in colonized organs for several days, regardless of the route of inoculation. ",
"Together, these results indicate that SPI-1 effector proteins may participate in the late stages of Salmonella infection in mice."
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"\n36 So.3d 98 (2010)\nMEANS\nv.\nSTATE.",
"\nNo. ",
"2D09-5833.",
"\nDistrict Court of Appeal of Florida, Second District.",
"\nMay 12, 2010.",
"\nDecision Without Published Opinion Belated Appeal dismissed.",
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[
"/*\n * Copyright 2018 Karl Dahlgren\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 */\n\npackage com.castlemock.web.mock.soap.mode;\n\nimport com.castlemock.core.basis.model.",
"ServiceProcessor;\nimport com.castlemock.core.mock.soap.service.project.input.",
"ImportSoapProjectInput;\nimport com.castlemock.web.basis.manager.",
"UrlManager;\nimport com.google.common.base.",
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"Highly superb strands of wire\n\nHere you will find how an artist makes intricate tree sculptures by twisting single hairs of wire. ",
"The concept is pretty fascinating and you will require having a deep take a look at the image gallery to comprehend how the art work is in fact made. ",
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"April 8, 2010\n\nApple today previewed its iPhone OS 4 software and released a beta version of the software to iPhone Developer Program members. ",
"The iPhone OS 4 beta release includes an updated Software Development Kit (SDK) with over 1,500 new Application Programming Interfaces (APIs) and over 100 new features that will be available to iPhone and iPod touch users this summer. ",
"New features include Multitasking for third party apps; Folders to better organize and access apps; improved Mail with a unified inbox, fast inbox switching and threaded messages; enhanced Enterprise support with even better data protection, mobile device management, wireless app distribution and more; Apple’s new iAd mobile advertising platform; and iBooks, the new ebook reader and online bookstore recently debuted on the iPad.",
"Read More\n\nThat's it from Apple's presentation. ",
"We will be doing detailed coverage of iPhone OS 4 later today. ",
"Check it out on TechzTalk.",
"\n\nA: If you want to kill an app... you don't have to. ",
"We architected something so the user doesn't have to be the custodian of applications. ",
"We'll give the apps in the foreground as needed. ",
"The user doesn't have to worry about that at all. ",
"The user just uses things. ",
"In multitasking, if you see a task manager, they you blew it. ",
"Users shouldn't have to ever, ever, ever think about that stuff.",
"\n\nQ: How do you close applications? ",
"And can people use stuff other than iAd and Game Center?",
"\n\nA: iPhone 3GS has been a huge hit, and iPod touch sales have also accelerated last year. ",
"So a lot of these products, the install base, are the most recent products. ",
"The old ones will still live the new features, but they'll just miss ones like multitasking. ",
"If that's an incentive to get people to upgrade to the new phone, that's fantastic.",
"\n\nQ: Are you concerned at all about multitasking and other features that wouldn't work for the older devices. ",
"Will this OS upgrade be enough to get people to upgrade their phones and renew their contracts.",
"\n\nA:Steve says we can enhance the App Store any time as its a sever side program, it doesn't depend of iPhone OS releases. ",
"We are working constantly to improve discoverability via features like Genius.",
"\n\nQ:How are you going to improve App Store for discoverability?",
"\n\nA:Even though we've been using these internally for sometime, you still have butterflies in your stomach the few weeks before launch. ",
"You never really know until you get it into customers hands, and they tell you what they think. ",
"The feedback we've got has been off the charts. ",
"We think this is a profound gamechanger. ",
"We think when people look back some number of years from now, they'll see this as a major event in personal computation. ",
"What's been really great for me is how quickly people I've never talked to before just telling me how much this product is... How much they think this product is going to change their life and what they do. ",
"People are getting it very quickly. ",
"Says Steve.",
"\n\nQ: Anything surprises you about the initial reaction to the iPad so far?",
"\n\nA: There is a porn store for Android to go to. ",
"You can download them, your kids can download them. ",
"That's a place we don't want to go. ",
"We're not going to go there.",
"\n\nQ: Are there any plans for you to run unsigned applications, like on Android?",
"\n\nSteve says multitasking was depended on hardware and that was the only limiting factor.",
"\n\nSteve is talking about iAd for iPhone platform and how it is much better than ads on the desktops. ",
"today.",
"\n\nA: Everything is possible.",
"\n\nQ: So widgets are possible on iPad?",
"\n\nA: We just shipped it on Saturday. ",
"And then we rested on Sunday.",
"\n\nQ: Why have you veered away from widgets on the iPad?",
"\n\nA: We think it works better with the push notifications as it's in place now. ",
"The fast-app switching is very complicated and more impressive than what we demoed.",
"\n\nQ: It there a reason why you won't be able to see a timeline in Twitter or an IM conversation in the background?",
"\n\nA: This isn't necessary. ",
"They're all HTML5 ads and people can choose any tools they want. ",
"As long as it's HTML5.",
"\n\nQ: Will Apple provide platform for ad development?",
"\n\nA: No\n\nQ: Is there going to be any change in Apple's position on Java and Flash in iPhone OS 4?",
"\n\nA: I think there will be some boundaries, like you can’t run any ad on ABC. ",
"There are some ads we’d rather not run, but the concept are that these advertisers will pay to run ads, and they’ll pay to run ads that are well received by the audience. ",
"I’m not sure there’s going to be anything but a light touch.",
"\n\nQ: Will Apple have approval process for ads?",
"\n\nA: The problem was there are a bunch of different social networks out there. ",
"So the goal was to build it into the platform, and everyone can be on the same gaming platform. ",
"We expect developers will build this into their games because they’ll have a wider audience.",
"\n\nQ: Will social gaming network supplant all the social gaming networks already available? ",
"and what will Apple do different?",
"\n\nA: No. ",
"It'll still be near the end of April.",
"\n\nQ: Will hugh ipad demand impact international launch?",
"\n\nA:I dont think it uses that much data.",
"\n\nQ: About Pandora?",
"\n\nA: I'm not sure that's a correct assumption.",
"\n\nQ: Multitasking will increase data usage, How will AT&T handle it?",
"\n\nSteve Jobs, Phil Schiller and Scott Forstall are on stage for Q&A.\n\nThis is for the presentation. ",
"Apple will have Q&A section after a short break.",
"\n\niPad will get it in Fall. ",
"The iPhone OS 4 will be available for iPhone 3G and 2nd gen touch but some features like multitasking won't work on those devices.",
"\n\nSteve is now naming each of the seven new features that were shown just now. ",
"He says there are over 100 new features for users.",
"\n\nThe ads can have content like video, games, images and even downloads. ",
"He is now showing NikeiD ad where you can design your own shoe. ",
"And now Target ad. ",
"The ads are very interactive and fun. ",
"I never thought I would say such a thing about Ads.",
"\n\nHe is demoing a news app with Toy Story 2 ad. ",
"When you click on the ad it loads within the app and lets you interact with the content in the ad.",
"\n\nMain features of iAd are Emotion + Interactivity, Ads keep you in your app, built into iPhone OS, Apple sells and hosts the ads and 60 percent revenue goes to developers.",
"\n\nSteve says iAd is about both interaction and emotion. ",
"It's about motion plus interactivity. ",
"The ads keep you in your app. ",
"Today when you click on a banner ad, it yanks you out of your app and throws you onto the advertisers web page. ",
"So people don't click on the ads.",
"\n\nSteve is showing interaction vs emotion graphic, and say most ads today are either interactive or emotion and they are not good at both.",
"\n\nHe says average user spend 30 mins everyday using apps and if they view ad every 3 minute it's 10 ads per device per day. ",
"Which is 1 billion ad impression per day.",
"\n\nThe seventh feature of iPhone OS is iAd. ",
"Steve says \"We have a lot of free or reasonably priced apps... we like that, but our devs have to find ways to make money. ",
"So our devs are putting ads into apps, and for lack of a better way to say it, we think most of this kind of advertising sucks.\"",
"\n\nThe sixth feature of iPhone OS 4 is Game Center. ",
"it will offer developers APIs for features like Social gaming network, Invite friends, Matchmaking, Leaderboards and achievements. ",
"In short Xbox Live for iPhone.",
"\n\nFourth feature is iBooks on iPhone. ",
"The app is very similar to iPad and Steve says \"We think customers will really enjoy this.\"",
"\n\nThe third feature is enhanced Mail app. ",
"It now offer unified inbox, multiple exchange accounts, fast inbox switching, threaded messages and open attachments with third party apps.",
"\n\nuser can have as many Folders as he wants and can keep them anywhere. ",
"Folders will now allow users to keep up to 2,160 apps.",
"\n\nThe second new feature on the iPhone OS is called \"Folder\". ",
"Steve says \"I want to make a folder with games in it. ",
"I have several games, so I just push my finger on an app, they start to jiggle, and I just drag one app on top of another and it instantly makes a folder. ",
"It also automatically names the folder based on the categories of these apps, or I can rename.\"",
"\n\nThe last two multitasking APIs are Task completion and Fast app switching. ",
"Some application like photo uploading takes time uses can switch application will the photo uploads and the Task completion and Fast app switching will take care of rest.",
"\n\nNext he talks about Push notifications and says that Apple has served over 10 billion notifications in past 9 months.",
"\n\nScott says that GPS uses fair bit of power which is ok for turn by turn navigation apps, but for location aware social networking apps Apple uses cell towers that are always connected.",
"\n\nThe third multitasking API is background location that can be used by applications like TomTom for turn by turn navigation.",
"\n\nScott is back on stage\n\nHe says Until today navigation away from the app meant that I would go offline. ",
"Now I go into the background, but I can still receive calls. ",
"he shows few screen shots of Skype working in background.",
"\n\nDavid Ponsford from Skype is on stage to demo VOIP background API for their application.",
"\n\nPandora uses background audio API one of the seven multitasking APIs. ",
"Now Scott is talking about VOIP background API.",
"\n\nPandora team is on stage and demoing multitasking for the app. ",
"Pandora can now be controlled from lock screen.",
"\n\nScott says Apple developed seven multitasking services as APIs and developers can access them depending on type of application. ",
"he is showing and example using Pandora.",
"\n\nScott Forstall is on stage to explain how Apple did this while preserving battery life and performance..\n\nThe UI puts all the apps that are running at the bottom of the screen like a Dock and you can access all the running apps by double clicking the home button.",
"\n\nDemo Time: \"I'm gonna launch mail, this has a URL - so I jump to safari - so I want to go back to mail... I double click the home button and a window raises that shows me all the apps that are running.\"",
"\n\nSteve says that Apple figured out how to implement multitasking without performance loss and uses a great UI to accomplish that.",
"\n\nApple is going to talk about 7 new features and the first one is Multitasking\n\nSo today we are giving a developers preview of iPhone OS 4. ",
"It has over 1500 new APIs, 100 new user features and framework for hardware accelerated math functions.",
"\n\nSteve says \"Now let's get to the iPhone.\" ",
"He says iPhone has won JD Power award three years in a row, has 64percent mobile browser user share, over 50 million iPhone sold and total of 85 million iPhone and iPod touch sold.",
"\n\nSteve is showing off apps specifically designed for iPad like Marvel, ABC, Epicurious and MLB.",
"\n\nWe will be doing live coverage of the iPhone OS 4 event that will be held on April 8th at 10 AM PST/ 1PM EST / 6PM London. ",
"Apple sent out invitations for the event yesterday, which will be held on Apple’s campus in Cupertino CA. ",
"Join us here on April 8th for live coverage on live.techztalk.com. ",
"Bookmark the link.",
"\n\nAustralian Macworld is reporting that Apple will be updating their MacBook Pro lineup next Tuesday, April 13th. ",
"The website received a tip from person familiar with Apple product cycles and inventory levels. ",
"The Macworld wrote:“The source said that an update to every MacBook Pro model was expected on Tuesday, 13 April, but was unsure about the MacBook Air.",
"”Read More\n\nApril 7, 2010\n\nAt the iPhone OS 4 event on Thursday, Apple will show the next generation of iPhone OS, which would include direct printing support, multitasking for third party apps, API for new iAd mobile advertising service and better support for game development.",
"Read More\n\nApril 6, 2010\n\nVerizon’s CEO Ivan Seidenberg said before Council of Foreign Relations that the company has told Apple it wants to carry the iPhone. ",
"He further added he has no information if Apple is working on such a device now. ",
"He decline to comment if and when the popular smartphone will be available for Verizon customers.",
"Read More\n\nAnandTech ran comparative benchmarks on iPad, iPhone 3GS and Nexus One to see how Apple’s new A4 SoC stacks up against the Samsung chip in iPhone and Qualcomm Snapdragon QSD8250 chip in Nexus One. ",
"The Snapdragon will also be used in many tablets coming later this year, so this comparison should give an idea of what we can expect performance-wise from those tablets. ",
"AnandTech ran two tests, in the first test they recorded load time for various websites and in the second test they used WebKit SunSpider JavaScript Benchmark tool.",
"Read More\n\nFor Apple, iPad was yet another successful product launch, as the device sold more than 300,000 units in first 15 hours. ",
"This prompted HP to release video of their upcoming iPad like tablet called HP Slate. ",
"The video highlights features of Slate that are missing from the iPad, like webcam, USB port and SD card slot. ",
"HP shows Skype running on the tablet, a photo gallery application, iTunes, multitouch and video recording.",
"Read More\n\nApril 5, 2010\n\nJust two days after iPad launch, Apple today sent out invitations to media outlets for an iPhone OS 4 event that will be held on April 8th. ",
"The invitation shows the shadow of number ‘4’ and says “Get a sneak peek into the future of iPhone OS.” ",
"No other information is provided on the invitation about the event.",
"It is rumored that iPhone OS 4 will bring multitasking for third party apps, better notification system and unified mailbox. ",
"We should find out more on April 8th. ",
"We will be doing live coverage of the iPhone OS 4 event. ",
"Please join us on April 8th at 10 AM PST/ 1 PM EST on live.techztalk.com.",
"\n\nApple today announced that it sold over 300,000 iPads in the US as of midnight Saturday, April 3. ",
"These sales included deliveries of pre-ordered iPads to customers, deliveries to channel partners and sales at Apple Retail Stores. ",
"Apple also announced that iPad users downloaded over one million apps from Apple’s App Store and over 250,000 ebooks from its iBookstore during the first day.",
"Read More\n\nApril 4, 2010\n\nBGR is reporting that iPad OS has some interesting references to upcoming Apple productions. ",
"The devices section in the filesystem shows new model references including iPhone3,2, iPhone3,3, iPod4,1 and iProd2,1. ",
"The current model number for iPhone 3GS is 3,1, for iPod touch is 3,1 and for iPad (aka iProd) is 1,1.Read More"
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"Q:\n\nParadigmatic vs syntagmatic relationship\n\nI was exploring some various aspects of corpus linguistics and studying different approaches to corpus research on the internet when I came across these phinomena of paradigmatic\nand syntagmatic relations. ",
"Can I get a satisfactory elucidation of both the concepts as well as how do the two technically and practically differ?",
"\n\nA:\n\nParadigmatics\nParadigmatic relations are relations of opposition and functional identity of linguistic elements (“OR - OR”), i.e. the paradigm combines sets of linguistic units that are similar according to one, and opposed according to some other criterion(s). ",
"Paradigmatics determines to which level of grammatical categoric hierarchy this or that language unit belongs, by analogy with units similar to it. ",
"For example:\n\nPhonemes b, ɪ, g, s, t are the building blocks of morphemes big and -est.",
"\nMorphemes big and -est can combine to form the lexeme biggest.",
"\nThe lexeme biggest is a part of the sentence That's the biggest fish I've ever seen!",
"\n\nSyntagmatics\nSyntagmatic relations can be defined as relations of compatibility of elements of the level in a sequence of speech events, i.e. combination of phonemes with phonemes, morphemes with morphemes, etc. ",
"Syntagmatics combines language units by their direct combination.",
"\nThe logical formula of syntagmatic relations is the formula \"AND - AND\", i.e. both one element and another element together, side by side, one after another, forming a chain of the same-order elements of a certain length.",
"\nExamples of syntagmatic relationships:\n\nb=ɪ=g=ɪ=s=t, bɪg=ɪst (phonetico-phonological level)\nbig=est, water=s, walk=ing (morphological level)\nteach=er, re=name (word-building sub-level)\ntall man, tall tree (lexical level)\nThe tree is tall. (",
"syntactical level)\n\n"
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"Pages\n\nRa Ranunkel, chairman of the cooperation council in Vollsmose, sounds the alarm and warns of an ethnic conflict in the crime-ridden district of Odense.",
"\n\n\"It started to get violent about a month ago,\" says the chairman, who for years has defended Vollsmose's reputation.",
"\"I always said that I'm proud to live in Vollsmose. ",
"I'm starting to get tired of it now. ",
"More and more of my neighbors come and say they're being attacked and robbed of their money. ",
"Or that their car was burned down. ",
"And all ask: 'Ra, can't you do anything?'. ",
"I have to admit, I feel powerless.\"",
"\n\nRa Ranunkel says that there's a hardcore gang of Somalis who commit attacks that are deliberately directed at Danes. ",
"And it's primarily the Palestinians who burn the Danes' cars and control Bøgetorvet (Bøge square).",
"\n\nHe thinks that the attacks are directed at Danes as a response to 'the Danish group of DPP-ers and racists' that in the past 5-6 years have been loudly voicing their views.",
"\n\n\"You noticed that something was brewing, and now the Somalis and Palestinians are striking again.\"",
"\n\nAn ethnic conflict, a racial conflict, is breaking out, and it's scary, says Ra Ranunkel, who before today's conference on Vollsmose's problems also sharply criticizes the Odense municipality's preventing efforts against youth gang-crime, which he calls 'totally ridiculous'. ",
"Fire them all and start from the beginning, he says."
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"\"fecal body odor\" - my story and cure\n\nI suffered from what this community calls \"fecal Body Odor \" from the age of 15-18/19. ",
"I actually think I found this message board all those years ago. ",
"I'm now 26 and 100% cured and never once have had a reoccurrence. ",
"I remember how my life was back then and how it is now and it is a complete turn around. ",
"When I finally figured out what my problem was and got to a place where I could sit in a room and have a normal conversation, be treated like a normal person, I literally cried of happiness.",
"\n\nStory:\nAround 15 years old, I started to get comments about odor. ",
"I wasn't sure if it was caused by me at first, but then after I kept hearing the comments, I became aware that it was me. ",
"Strangely enough, I couldn't smell it myself.",
"\n\nAs you can imagine or have experienced, the mental torment is almost unbearable. ",
"As a kid in high school, I was frequently in close proximity of a group of kids. ",
"After about a year of suffering through this, I became well known as \"the kid who smells like shit\". ",
"I had a small group of friends, who never really commented on it, but the signs were there - opening windows in winter, keeping a distance, etc.",
"\n\nSince I couldn't prove it myself, I pleaded with my parents to go to the doctor. ",
"It wasn't till I was older that we actually started going to the doctor, and even then I think they didn't believe me. ",
"So I was stuck with it. ",
"After reading through a couple of these posts, I see all the same patterns. ",
"OCD about cleanliness, charcoal, strange rituals, enemas, colognes/perfumes, etc. ",
"I even remember trying to use bleach to clean myself. ",
"Dumb idea.",
"\n\nI nearly always had a little bit of gas. ",
"Since I was afraid of the smell getting worse, I would hold it in and go to the bathroom to release whenever I could. ",
"I would never have any stains or actual shit/liquid, so I was perplexed as to what was happening to me. ",
"When I started going to the doctor, I got all sorts of random diagnoses - even one doctor said it might just be me growing up. ",
"What the f*** does that even mean?????",
"\n\nI remember one especially humiliating day in my life, my Science teacher called a random seat reassignment. ",
"Everyone got up, and she called kids to their seats one by one. ",
"I started getting worried it was about me. ",
"When I was one of the last 5 or so kids, I got so ashamed my face turned beet red. ",
"To make it even more humiliating for me, she left one seat open in front of me, one seat open behind me and then put the class trouble maker in the very last seat. ",
"A kid in the class asked why they changed seats, The teacher didn't answer. ",
"One of the kids said \"It's because x smells like shit\". ",
"It was agonizing. ",
"Since it had been a while suffering through this, and being publicly humiliated was a common occurrence at that point, I felt like I needed to escape. ",
"I went home and was ready to kill myself. ",
"I went to the bathroom, filled up the tub, and got in with the knife. ",
"I sat there for hours. ",
"I couldn't do it.",
"\n\nThe most tormenting part of this period of my life was the psychological torture. ",
"Since I couldn't smell it myself, I began to try to confirm or disprove with friends and only the closest of friends. ",
"All I wanted was to know. ",
"When I was working with a good friend of mine, at the end of the shift, he openly said \"Dude, you smell like ass.\" ",
"He told me about it and how everyone knew. ",
"He said it wasn't all the time, and sometimes it was much worse and sometimes it was just a little bit. ",
"I finally had someone be upfront with me about it after over a year of dealing with the issue.",
"\n\nMy symptoms:\nOdor - described as \"shit\", \"farts\", or \"it stinks in here\"\nInability to detect own odor\nGas/Bloating\nFoul smelling stool\nA weird bubbling in my lower right gut after eating\nConstipation\n\nMy (self found) cure:\nAt the end of it all, no doctor was able to help. ",
"I tried all sorts of diets, but it was one in particular that cured me. ",
"For 6 weeks straight, I ate nothing but boiled chicken breast, white rice with no seasoning, egg whites, and green pea pods (low in fructose). ",
"My symptoms slowly decayed until they were completely gone. ",
"I started reincorporating things into my diet, and noticed them come back. ",
"Through the process of elimination, I had nailed it down to gluten. ",
"two years of suffering - all because of gluten. ",
"After completely eliminating gluten from my diet, I was cured. ",
"This is much more difficult than it appears, and it can't be half-assed. ",
"If you want to be free of this - it has to be 100% adherence without any slip ups. ",
"I'm sure if you went through what I had to go through, you won't have trouble finding the motivation to do this.",
"\n\nMy life is on the right track. ",
"I moved as soon as I could, and began a new life. ",
"I graduated near the top of my class, and I'm now working in NYC at a top consulting firm. ",
"This wouldn't have been possible with the debilitating and strange disorder I had when I was younger. ",
"I don't think I would be alive today had I been suffering this long.",
"\n\nPlease, if you have not tried it, and the above story/symptoms sound familiar to you, you need to try a 100% strict gluten-free diet. ",
"I would also suggest following the \"zero-chance\" diet I followed above to maximize your body's ability to heal.",
"\n\nI felt the need to share my story and my cure. ",
"I don't expect it to be the cure all for all of you, but I know for a fact it will be the way for some of you. ",
"Good luck."
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"Removal of bone staples: a potential problem in revision surgery after ligament reconstruction.",
"\nTwo cases are presented in which an attempt to remove the staples used to fix an artificial ligament caused the front of the tibia to break. ",
"A mechanical study was performed to measure the force required to extract staples from bone cement, as well as from calf bone with predrilled holes. ",
"Using an extraction tool, it was possible to generate an average maximum force of 500 N by hand. ",
"At that load, all staples remained firmly anchored in cement and bone. ",
"The staples were impossible to remove even when the serrations in the staple legs were reduced by 75%. ",
"Accordingly, special care is necessary to avoid complications with staple removal."
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"We all remember the well known fable told by the ancient Greek slave and storyteller, Aesop, about the shepherd boy drafted to keep watch over a flock of sheep, and who, out of boredom, decided it would be great fun to fool the nearby villagers by pretending that a wolf was attacking the sheep. ",
"After the villagers rushed out to save the sheep a number of times based upon the boys fabrications they refused to believe him when the wolf actually did show up and began to consume the sheep. ",
"The obvious declared moral of the story is generally expressed as “There is no believing a liar, even when he is telling the truth.” ",
"However, there is, I submit, a second, unstated moral to the story as well. ",
"In spite of the shepherd boys prevarications the wolf was real. ",
"And the wolf did, in the end, show up to devour the poor sheep.",
"\n\nAll of which, however, raises an interesting question. ",
"Why are so many people so fearful of and ready to believe in an impending doomsday?",
"\n\nWhich brings us to the point of this article. ",
"For centuries various predictions have been made about the end of the world on certain dates by assorted means and on quite a number of occasions these predictions have been believed by a significant number of people. ",
"Obviously the world did not perish in late December of 2012, the most recent date proffered in a long tradition of doomsday predictions that failed to materialize.",
"\nMy first encounter with such notions came about in the late 1960s while still in high school. ",
"I had a poster on my bedroom wall entitled ‘Goodbye California’ and it showed the entire state of California breaking off from the North American mainland and sinking into the Pacific ocean, an occurrence which was widely believed to be impending by a considerable number of folks back then. ",
"A few years later, in 1973, with the approach of Comet Kahoutek, apocalyptic predictions were proliferating, causing fairly widespread consternation among the astronomically challenged. ",
"I next recall the date of November, 1978 being foretold as a time of extensive global disasters.",
"\n\nAsteroid 2012 DA14 is a cosmic reminder that catastrophes, in spite of all the over-hyped, pseudo-scientific predictions, are indeed real.",
"\n\nThen came the early 1980s and more predictions about a series of escalating disasters brought about by planetary alignments. ",
"The eruption of Mt. St. Helens in May of 1980 was interpreted as being the opening act of this apocalyptic succession. ",
"The culminating date was usually given as March, 1982, when all hell was going to break loose. ",
"When that date passed without incident others claimed that December 31, 1988 was actually the date. ",
"Of course that date came and went as well. ",
"Then there were the prophecies of Nostradamus, which foresaw July of 1999 as the arrival date for the ‘King of Terrors,’ inciting widespread fear among believers of a looming apocalypse. ",
"Only a year later came the turn of the Millennium and fears of a planetary pole shift in May of 2000 that was going to wipe out civilization. ",
"One book from the early 1980s which made this prediction went through multiple reprints up to the year 2000, but, needless to say, has not been reprinted since.",
"\n\nNot to mention Y2K, which was, perhaps, somewhat plausible. (",
"We can’t discount the fact that governments did spend over 100 billion dollars to secure computers against failure, which may have been partially responsible for averting the expected breakdown.) ",
"And finally we have the Mayan Apocalypse which even spawned a silly movie. (",
"With awesome special effects though). ",
"These are only a few of the failed predictions of which I am aware that have occurred in my lifetime and that of many readers of this article. ",
"There have been countless dozens of other such failed predictions stretching back through the centuries.",
"\nI think we could say that the batting average of those prognosticators of world destruction has so far been about zero. ",
"All of which, however, raises an interesting question. ",
"Why are so many people so fearful of and ready to believe in an impending doomsday? ",
"I think the answer to that question brings us to the unstated moral of Aesops fable. ",
"In spite of all the false alarms the wolf turned out to be very real.",
"\nAnd eventually the wolf showed up.",
"\n\nIf instead of trying to foresee the future of planet Earth we look to its past, we are confronted with a rather disquieting mass of evidence about the actual history of global change. ",
"While the prevailing paradigm for most of the 20th century was one of slow, gradualistic change, that interpretation has given way to a new model which recognizes that profoundly dynamic changes have occurred on a scale unprecedented in recent times. ",
"Every month now brings new scientific revelations about Earths cataclysmic history. ",
"Over the last 3 or 4 decades scientists from multiple disciplines have steadily documented the reality of great catastrophes in the history of the Earth. ",
"We now know that Earth has been subject to devastating asteroid and comet impacts too numerous to count. ",
"We know that gigantic volcanic eruptions have occurred sufficient to cause the sky to go dark for months at a time. ",
"We know that great tsunamis have repeatedly occurred that would make the Japanese and Indonesian disasters look trivial. ",
"We know that vast ice ages have gripped the planet over and over again. ",
"We know that mega scale floods that can only be described as biblical in scale have devastated large regions of the Earth’s surface. ",
"And we suspect that the majority of Earth’s extinct animal species succumbed to global disasters. ",
"And finally, we now know that numerous cataclysmic episodes have occurred during the several hundred millennia that we humans have been present on Earth.",
"\n\nIs it possible that racial memories of past tribulations suffered by the human species fuels the fear of future doomsdays? ",
"And given what we know about the frequency of past global disasters would it not be prudent to assume that at some future date we will again be faced with apocalyptic events? ",
"Recognition of Earth’s catastrophic history does not imply a fatalistic view of life. ",
"Rather it provides a dose of reality and a higher perspective on the human condition than is generally acknowledged by the occupants of the institutions of social power.",
"\n\nRecognition of Earth’s catastrophic history does not imply a fatalistic view of life. ",
"Rather it provides a dose of reality and a higher perspective on the human condition than is generally acknowledged by the occupants of the institutions of social power.",
"\n\nProfessor of zoology and human ecology Kenneth E. F. Watt, wrote in his 1974 book The Titanic Effect that “The magnitude of disasters decreases to the extent that people believe that they are possible, and plan to prevent them, and to minimize their effects.” ",
"Thanks to early warning systems in place in the Pacific Ocean, which were lacking in the Indian Ocean, the great Japanese tsunami of 2011 caused about 20,000 deaths whereas the death toll of the Indonesian tsunami of 2004 was on the order of 300,000.",
"\nAs if to underscore the point of this article, on February 15 came a close brush with disaster.",
"\n\nAsteroid 2012 DA14 will whiz by the Earth at a distance of only 17,000 miles, inside the orbits of many satellites. ",
"It is about the same size as the object which exploded over Siberia in 1908. ",
"Had it collided with Earth the resulting detonation would have been equivalent to that of the largest hydrogen bombs, enough to completely obliterate a metropolitan area the size of Atlanta.",
"\nAsteroid 2012 DA14 is a cosmic reminder that catastrophes, in spite of all the over-hyped, pseudo-scientific predictions, are indeed real.",
"\n\nShare:\n\nAuthor: Randall Carlson\nRandall Carlson is a master builder and architectural designer, teacher, geometrician, geomythologist, geological explorer and renegade scholar. ",
"He has 4 decades of study, research and exploration Into the interface between ancient mysteries and modern science, has been an active Freemason for 30 years and is Past Master of one of the oldest and largest Masonic lodges in Georgia. ",
"He has been recognized by The National Science Teachers Association for his commitment to Science education for young people.",
"\nHis work incorporates Ancient Mythology, Astronomy, Earth Science, Paleontology, Symbolism, Sacred Geometry and Architecture, Geomancy, and other arcane and scientific traditions. ",
"For over 25 years he has presented classes, lectures, and multimedia programs synthesizing this information for students of the Mysteries.",
"\nIt is his aspiration to affect a revival of lost knowledge towards the goal of creating the new world based upon universal principles of harmony, freedom, and spiritual evolution.",
"\n\nSign Up for SGI Newsletter!",
"\n\nEmail address:\n\nPlease Support this Work!",
"\n\nWe are independent media, totally supported by you. ",
"Here we unveil the ancient mysteries, exploring the interface between archaic wisdom and modern science and the infinite possibilities emerging through their synthesis. ",
"We invite you to participate in this journey as we map the forgotten landscapes of the human experience and present information which rewrites the history of the human species all together.",
"\n\nSince 2008 we have produced over 100+ hours of FREE educational media materials for hundreds of thousands of individuals throughout the world.",
"\n\nIf you would like to help us continue to produce independent media without commercial advertising please visit our store.",
"\nYour purchases, monthly subscriptions and donations help us to continue providing this paradigm shifting knowledge to the world."
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"I don’t really get all the hate and whining from some people about Javascript. ",
"I love Javascript. ",
"It won’t run everything or solve every problem but it is the closest we have come to “run everywhere” — please do NOT mention java.",
"\n\nWith JavaScript, you can write applications that work:\n\nin your browser (pretty much ALL of them) on your phone and tablet (e.g. React Native, Cordova) on your desktop (e.g. electron runs on mac, linux and windows) on your server i.e. node and even in your database e.g. mongo db\n\nYou don’t need to download a 4 gigabyte IDE to create a hello world, you don’t need to wait 30 seconds to compile and run your app, all you need is a text editor.",
"\n\nSure javascript has flaws but so does every other language out there but nothing comes close — not even by a mile — to the reach and flexibility of javascript.",
"\n\nI love the innovation (and explosion) of javascript tools, frameworks and libraries. ",
"I know it can get tiring but it’s fun and guess what, you don’t have to learn ALL of them.",
"\n\nWith that said, can we please stop creating “languages” on top of javascript and please stop using them in your projects. ",
"We already have enough: TypeScript, Elm, Coffeescript and others. ",
"I highly recommend you do NOT use any of them in your projects.",
"\n\nI will explain why but first let me say how much I appreciate the diversity in choices and most importantly how thankful I am to the people that create these awesome tools and “languages”.",
"\n\nI was a huge coffeescript fan. ",
"I decided to use it for a project a few years back and as the project grew I quickly realized that it was the worst decision I made and six months into the project we converted everything over to Javascript. ",
"Here are some reasons why that was a bad decision:\n\nI had to setup another build step to “compile” it back to javascript — not a huge deal but it can be a pain some times. ",
"It was hard to debug since what you see in the browser is different than what you have in your coffeescript files — for the most part, it is very readable but you still have to perform a mental translation. ",
"The biggest problem by far is the learning curve for every new developer that joined the team — yet another thing they had to learn to ramp up Most code samples online were 99% in javascript The team started losing some of their javascript skills, slowly, javascript code started to look foreign and we wanted to see everything in coffeescript. ",
"We didn’t keep up with all the advancement in javascript e.g. ES6 — arrow functions, deconstruction and so on. ",
"We couldn’t just put a few lines of code directly in a script tag to test some stuff and if we did, we had to do it in javascript and later “translate” it back to coffeescript.",
"\n\nAlthough, I didn’t play much with TypeScript or Elm, I would assume they would have similar problems.",
"\n\nRE: TypeScript — really cool concept and I get some of the benefits but why are we forcing a dynamic language to “act” static? ",
"C# is jumping through hoops to act more dynamic and now we are jumping through hoops trying to make javascript act more static. ",
"WTF!!!",
"\n\nTL;DR"
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"Thus far, as a medical observation system for, when performing an operation of a minute part of the brain, the heart, etc. ",
"of a patient that is an object to be observed, observing the minute part, an optical microscope system including a magnification optical system that magnifies the minute part has been known (e.g. see Patent Literature 1). ",
"When performing an operation using the microscope system, an operator such as a medical doctor (the user) performs the operation while observing the surgical site via an eyepiece.",
"\nFIG. ",
"10 is a diagram schematically showing a situation in which an operator performs an operation using a conventional optical microscope system. ",
"As shown in FIG. ",
"10, an operator 401 performs an operation while observing the surgical site of a patient 402 via an eyepiece 502 of a microscope unit 501. ",
"Hence, when the operation time is increased, the burden on the eye of the operator 401 is increased, and also the burden on the body of the operator 401 due to maintaining the same posture is increased.",
"\nAs a technology to solve such a problem of the optical microscope system, a video microscope system including an imaging means that images a minute part such as a surgical site is known (e.g. see Patent Literature 2). ",
"FIG. ",
"11 is a diagram schematically showing a situation in which an operation is performed using a conventional video microscope system. ",
"As shown in FIG. ",
"11, an operator 401 performs an operation while observing, with a monitor 602, an image of the surgical site of a patient 402 captured by an imaging unit 601. ",
"By such a video microscope system, the burden on the eye and body of the operator in a long-time operation can be greatly reduced as compared to the case of the optical microscope system."
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"Methadone-induced respiratory depression in the neonatal guinea pig.",
"\nRespiratory depression, the most serious side-effect of opioid treatment, is well documented for morphine, the most commonly used opioid in neonatal care. ",
"Less is known about methadone, a clinically relevant opioid analgesic, especially during neonatal development. ",
"This study was undertaken to determine the neonatal respiratory effects of methadone. ",
"We hypothesize that methadone is equipotent to morphine, compared to our previous morphine results in the same animal model, but has a much longer duration of action, due to its longer elimination half-life. ",
"Neonatal guinea pigs (3-14 days old) randomly received a single subcutaneous dose of methadone or saline. ",
"Using a non-invasive plethysmographic method, we measured ventilatory and metabolic parameters before injection and at intervals for 32 hr after injection while pups breathed \"room air\" or 5% CO(2) gas mixtures. ",
"Methadone-induced depression of ventilation was most evident during 5% CO(2) challenge. ",
"The onset of drug effects was within 15 min for all ages and doses, but the duration of action decreased with age. ",
"While the depth of methadone-induced respiratory depression did not depend on pup age, the control of breathing was different in 3-day-old pups, where inspiratory time increased fourfold; twice that of older pups. ",
"We conclude that methadone induces a naloxone reversible respiratory depression in guinea pig neonates and, in the very young, causes an abnormal breathing pattern due to changes in respiratory timing. ",
"Methadone is more potent than morphine with respect to neonatal respiratory depression, but surprisingly, the duration of methadone action was not longer than morphine."
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"Q:\n\nClicking a Link in td with Watir\n\nI'm trying to click an image in with a Watir script. ",
" Here's the source code for what I'm trying to click:\n<td align=\"right\">\n <a onclick=\"confirmAndSubmit('Exit application?', '",
"saveAndExit')\"\n onmouseover=\"style.cursor='hand'\">\n <img src=\"/images/exitBlue.gif\" align=\"bottom\"></img>\n </a>\n</td>\n\nWatir seems to be unable to locate the element based on my code. ",
" It seems like I have to write sort of a complex statement with the arguments. ",
" I've tried \n$b.td(:img, \"src='/images/exitBlue.gif'\").click\n\nas well as\n$b.element(:css, \"td[onClick='confirmAndSubmit('Exit application?', '",
"saveAndExit')']\").click\n\nbut nothing seems to locate the data in the table. ",
" Any suggestions on how to go about clicking this?",
"\n\nA:\n\nI found the solution with a bit of trial and error. ",
" As mentioned in the comments, the image was inside nested tables. ",
" I targeted the div inside the first table and was able to get to it. ",
" The element.flash method was invaluable in troubleshooting. ",
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"Forging is a process by which the shape and physical properties of metal can be changed. ",
"The process involves placing a piece of metal (normally heated) between the halves of a die and forcing the die to close by impact or pressure. ",
"The operation causes a controlled plastic deformation of the metal into the cavities of the die. ",
"This flow of material results not only in a change in shape of the metal but also increases the density and uniformity of the metal, improves its grain structure, and causes a shape-conforming grain flow. ",
"The resulting workpiece has properties which are superior to those generated by other methods, making forging essential where high performance workpieces are required.",
"\nOne of the critical components of a forging system is the lubricant which separates the die from the workpiece. ",
"As with all lubricating situations, it is essential that this lubricant be effective to minimize wear of the extremely expensive forging dies and minimize expenditure of energy over a wide range and condition. ",
"Somewhat peculiar to the forging process, however, is that merely maximizing lubricity is not the only goal, since a certain degree of friction between the workpiece and die is essential to optimize the properties of the workpiece. ",
"This controlled lubricity is particularly important when it is necessary to fill deep impression dies.",
"\nAs modern demand for safer and more dependable machine structures increases, the forging art is being applied to more difficult materials, at higher temperatures and pressures to form more complex shapes. ",
"Although oil-based lubricating compositions, which are effective under these extreme conditions, have been developed, their properties are found to conflict seriously with national commitments to personal safety and protection of the environment. ",
"The oil-based lubricants are normally flammable and can ignite well below common operating temperatures. ",
"Normal operation results in billowing carbonaceous smoke which is unpleasant and sometimes toxic. ",
"Furthermore, cleaning of the workpieces and dies requires solvent washes that produce large quantities of rinse which, because of the economics of recycling, and desire to protect the environment can present serious disposal problems.",
"\nAttempts at avoiding the problems inherent in the use of oil-based lubricants have generally been directed toward water-based compositions. ",
"Early attempts, involving mixtures of graphite, clay minerals, and molybdenum disulfide, were found ineffective, because they did not sufficiently wet the hot metal surfaces to provide lubrication. ",
"They were also unacceptable due to the corrosion caused by the high temperature break-up of the components. ",
"Additives and substitutions which solved one problem often created another. ",
"For example, the addition of soaps to improve wetting often caused caking in cavities and increased smoke production and odor. ",
"These and other difficulties experienced with the prior art devices have been obviated in a novel manner by the present invention.",
"\nIt is, therefore, an outstanding object of the invention to provide a forging lubricant which effectively reduces the die wear and energy requirement of a forging operation.",
"\nAnother object of this invention is the provision of a forging lubricant which produces an extremely low amount of air pollution.",
"\nA further object of the present invention is the provision of a forging lubricant which is simple and easy to produce and which, in use, leads to relatively less expense in overall operating costs.",
"\nIt is another object of the instant invention to provide a forging lubricant which has a long storage life and is not adversely effected by reasonable storage conditions.",
"\nA still further object of the invention is the provision of a forging lubricant which is simple and safe to apply to the dies using spray equipment.",
"\nIt is a further object of the invention to provide a forging lubricant which is virtually non-flamable and does not give off undesirable vapors or corrosive byproducts during use.",
"\nIt is a still further object of the present invention to provide a forging lubricant which has the proper combination of lubricating properties to allow predictable and effective operation of forging technique over wide ranges of workpiece shape, temperature, pressure and material.",
"\nAnother object of the invention is the provision of a forging lubricant which has adequate insulating properties to prevent undesirable cooling of the workpiece by the dies.",
"\nAnother object of the invention is the provision of a forging lubricant which effectively prevents workpieces from sticking in the dies after the forging operation.",
"\nAnother object of the invention is the provision of a forging lubricant which is easily cleaned from the workpiece and dies, and which results in a rinse which can be easily cleaned to avoid pollution.",
"\nWith the foregoing and other objects in view, which will appear as the description proceeds, the invention resides in the combination and arrangement of steps and the details of the composition hereinafter described and claimed, it being understood that changes in the precise embodiment of the invention herein disclosed may be made within the scope of what is claimed without departing from the spirit of the invention."
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"\nA strategy for improving the efficacy of stem cell-based bone tissue engineering (TE) constructs is to combine bone morphogenetic protein-2 (BMP-2) with multipotent stromal cells (MSC). ",
"Previous studies on the potential cooperative effect of BMP-2 with human multipotent stromal cells (hMSCs) on bone formation in vivo have, however, shown contradictory results likely due to the various and/or inappropriate BMP-2 doses. ",
"Our results provided evidence that the addition of BMP-2 at low dose only was beneficial to improve the osteogenic potential of hMSCs-containing TE constructs, whereas BMP-2 delivered at high dose overcame the advantage of combining this growth factor with hMSCs. ",
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"Is Scott Pruitt plated in gold or spattered in mud?",
"\n\nIn a week where President Trump’s lawyer got raided by the FBI, Trump’s top homeland security adviser resigned, the president agreed to join an international response to an act of chemical warfare believed to have been conducted by the Syrian government, and news broke that the speaker of the House would not seek re-election — all before breakfast on Wednesday– it’s easy to forget that we ended last week expecting the imminent ouster of Environmental Protection Agency Administrator Scott Pruitt.",
"\n\nPruitt is still employed at the moment. ",
"But accusations that he received improper gifts from lobbyists, misspent public funds and engaged in questionable personnel practices have not gone away. ",
"And the swirl of scandal surrounding him — there are at least five open inquiries into Pruitt’s behavior, according to The Washington Post, and the Office of Government Ethics indicated Friday that it remains very interested in his alleged ethics violations— prompted a flurry of appraisals of his tenure at EPA. ",
"They were … confusing. ",
"He’s either a deregulatory master, praised by Trump and Republicans for his pro-business approach to environmental regulation and accused by liberals of destroying the environment. ",
"Or he’s just a spin master, and most of his accomplishments have been smoke and mirrors.",
"\n\nThe truth, of course, is a matter of perspective, and it all depends whether you take the short, long or longer view.",
"\n\nDespite his reputation for effective and prolific deregulation, much of what Pruitt has done is to prevent the implementation of Obama-era policy by simply delaying those policies — not dismantling them. ",
"Consider, for example, the EPA’s own list of deregulatory actions that were completed under Pruitt’s tenure, which it compiled as part of documenting the agency’s adherence with Trump’s 2-for-1 deregulation executive order requiring that for each new regulation added to the books, two must be removed. ",
"There are 24 actions on this list, but only two actually represent the complete and successful negation of an Obama-era environmental policy. ",
"In one case, the EPA withdrew its request for oil and gas companies to complete a survey about their equipment and the tools they were using to reduce greenhouse gas emissions. ",
"In the other, the agency rolled back a decision to increase air-quality-monitoring requirements on facilities that store and treat certain kinds of hazardous waste. ",
"Of the remaining 22 actions, 10 were delays of Obama-era proposals — mostly extending deadlines for when rules would go into effect. ",
"One implemented a rule written during the George W. Bush administration that the Obama EPA had tried to block. ",
"Three offered exemptions for ozone pollution rules to a handful of counties in Tennessee and Louisiana. ",
"Two were uncontroversial updates of standards. ",
"One made a minor amendment to product-labeling laws. ",
"Five implemented rules that had originally been put forward under the Obama administration.",
"\n\nOn the whole, Pruitt is getting less done than he would like you to believe. ",
"It’s not nothing. ",
"Those deregulatory actions made Pruitt’s EPA the most productive deregulator in the Trump administration, according to 2017 data compiled by the Office of Information and Regulatory Affairs. ",
"But as Pruitt has rushed to block as much as possible, the EPA has wound up issuing shorter, less detailed rulings that aren’t holding up well to legal challenges. ",
"Case in point: In May 2017, Pruitt issued a 90-day stay of an Obama-era regulation that sought to reduce methane emissions at landfills. ",
"But the stay was challenged in court and the EPA let the stay expire, allowing the regulation to take effect. ",
"As of now, the rule remains on the books, even as Pruitt’s EPA was sued by several states for failing to enforce it. ",
"It’s possible that his legacy could end up mimicking that of Reagan appointee Anne Gorsuch, who slashed the EPA’s budget and enforcement activities — then resigned under a cloud of ethics violations just 22 months into the job. ",
"Her work left little measurable impression on environmental quality.",
"\n\nBut while it’s possible to draw an analogy between Pruitt and Gorsuch, Donald Trump is no Ronald Reagan. ",
"A big part of why Gorsuch was unable to build a lasting legacy is that she was replaced by William Ruckelshaus, the EPA’s founding administrator, who undid many of her efforts aimed at dismantling the agency. ",
"Environmental protection, however, was much more of a bipartisan issue back then, and if Pruitt leaves his post, his replacement will probably look a lot like, well, Pruitt. ",
"Hostility to more environmental regulations and skepticism about comprehensive government efforts to combat climate change are GOP orthodoxy now, not just the views of one rogue administrator.",
"\n\nSure, the Senate would have to confirm anyone put forward as a replacement for Pruitt, and Republicans hold a very narrow majority in that chamber (51-49). ",
"But there’s no guarantee that Democrats would unanimously oppose an anti-environmental-regulation nominee. ",
"In fact, Democrats in coal country, like West Virginia Sen. Joe Manchin, might feel pressure to vote for someone like that, particularly in an election year. ",
"Manchin, who voted to confirm Pruitt, has already said that he’ll support Andrew Wheeler, a former coal lobbyist, for the role of EPA deputy administrator. ",
"And if even if Pruitt leaves and is not replaced quickly, it’s unlikely that the agency would go back to its more pro-regulation Obama-era approach, since some of Trump’s political appointees would likely remain at the EPA.",
"\n\nRemember when one-time chief strategist Steve Bannon left the White House? ",
"Bannon’s departure didn’t matter much in the end because the president himself adheres to nationalist/populist/conservative identity politics, so they remained an element of this policy-making even without Bannon there to push those views. ",
"This is even more true in Pruitt’s case. ",
"Essentially the entire Republican Party agrees with Pruitt on environmental policy, both inside and outside of the Trump administration. ",
"Trump agrees with him too.",
"\n\nIf the president gets rid of Pruitt (and his ethics problems), Trump will still find ways to annoy liberals and delight conservatives on environmental policy. ",
"And maybe this time he’ll get an EPA administrator who does more than delay. ",
"In other words, it’s possible to say that Pruitt isn’t the deregulation powerhouse the president has portrayed him as (the short-term view) and that his resignation wouldn’t exactly spell relief for liberals (the long-term view).",
"\n\nOnly from a still longer-term perspective is there truly good news for environmentalists. ",
"The very quirks of the EPA that allowed Pruitt to undo as much as he did may in turn undo his legacy. ",
"Congress delegates a great deal of authority to the EPA, which the agency uses to take broad laws — particularly the Clean Air Act and the Clean Water Act — and interpret them, creating the regulations that actually make the laws function. ",
"In the process, it decides how (or whether) those regulations will be enforced. ",
"That system gives the agency wide-reaching ability to determine the boundaries of its own mission and act without worrying about what Congress thinks. ",
"Unsurprisingly, therefore, Pruitt was not the first EPA administrator to be criticized for acting autocratically. ",
"His immediate predecessor, Gina McCarthy, faced similar accusations for her pro-regulation activity.",
"\n\nPruitt’s path to undoing much of McCarthy’s legacy (and that of her boss, then-President Barack Obama) was relatively straightforward. ",
"All he had to do was decide that McCarthy had incorrectly interpreted the law, or decide not to implement proposals made under her tenure.",
"\n\nBut this kind of power and autonomy can cut both ways. ",
"McCarthy learned that lesson when Pruitt moved EPA policy to the right. ",
"And Pruitt may learn it too — but probably not until someone else sits in the Oval Office."
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