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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 Amer |