Since launching Cloud IPQ last year, we have focused on the intersection of cloud computing and intellectual property with a particular emphasis on the increasing number of patent lawsuits filed against cloud computing companies and their customers. In our first post – Introducing Cloud IPQ: Enhancing Your Cloud IP IQ – we highlighted the increase in cloud patent lawsuits filed by NPEs/Patent Trolls, and the trend of high-volume plaintiffs filing multiple lawsuits against customers of cloud providers. That has certainly proven true, and frankly we expected that the majority of our coverage would center on such Patent Troll lawsuits. However, we have witnessed some relatively surprising trends beyond just Patent Troll litigation, namely, an increase in competitor-based lawsuits as well as more focus and assertion against open source software by patent trolls and operating companies alike. Other legal dynamics have also emerged that arguably increase the IP risk profile in the cloud, including court decisions and administrative actions that are now restoring patent power after many years of decline in the strength and certainty of U.S. patent rights.
With regard to competitor cases, cloud security is a hot space currently. Our first post on a competitor patent case involved Cloud Security Firm Zscaler and the two lawsuits filed against it by industry incumbent Symantec. We then reported on another cloud security lawsuit where Symantec's Blue Coat Systems unit found itself on the receiving end of an infringement complaint filed by Finjan, a prolific patent plaintiff in the security technology space. While the Symantec v. Zscaler cases are ongoing, Finjan settled with Symantec for $65 Million with $45 Million more possible if Symantec acquires "certain entities" over the next four years. Although it seemed that Blue Coat/Symantec had the momentum after successfully appealing and knocking down a large jury verdict in Finjan's favor, such a large settlement is seen as a blockbuster in the current patent climate.
Cloud competitor cases have run the gamut from established players going after newer entrants, newer entrants going after industry giants, and mid-tier cloud companies fighting it out with their patent rights. BMC Software's patent strike against Cherwell Software's cloud services and Citrix's infringement and unfair competition complaint against Avi Network's cloud application delivery platform are current examples of established cloud players using their patents to ward off emerging competition. And both cases involve executive ship-jumping and have a strikingly similar fact pattern in that the CEO of the smaller competitor was a former executive of the larger company. On the flip side, relative newcomer Centripetal Networks filed a patent lawsuit against networking solutions giant Cisco and in so doing put virtually its entire patent portfolio on the line by alleging infringement of 10 patents (out of its total portfolio of 14 patents). In the mid-tier cloud player space, Ribbon Communications filed two more lawsuits against Metaswitch in a seeming relentless pursuit to land a knockout blow in the cloud communications space. Ribbon filed those two lawsuits even after winning an $8.8 Million damages award against Metaswitch in earlier litigation. This is real money for a company of any size, and makes Ribbon's ongoing patent campaign against Metaswitch all the more interesting given Ribbon's annual revenues of $330 million and net loss of $35 million in its FY17.
Open Source Software is also in the cross-hairs of numerous cloud patent lawsuits. It's no surprise that patent trolls would target open source, such as Sound View's lawsuit against Fidelity directed at multiple OSS components, including jQuery, OpenStack Object Storage (Swift), Apache HBase, and Apache Storm. Sound View also filed earlier lawsuits against open source software usage of major cloud service providers such as Hulu, Facebook, Twitter and LinkedIn. What did come as a surprise were the patent infringement allegations that Citrix leveled against Avi Networks' cloud application delivery platform for OpenStack, mentioned above in the context of cloud competitor lawsuits.
In an example of a cloud services customer-targeted patent litigation campaign that also implicates open source software, a wave of lawsuits filed this year by PersonalWeb alleged infringement by over 50 could-based companies for the use of Ruby on Rails open source software implementing the HTTP web protocol as well as Amazon Simple Storage Service ("S3"). Essentially, these are lawsuits targeting Amazon customers including Airbnb, Square, Reddit, Venmo, and Kickstarter, just to name a few. Amazon filed a declaratory judgment action against PersonalWeb in February 2018 in response to this flood of customer suits. Other examples of cloud customer patent campaigns include GEMSA's patent onslaught against AWS customers such as Airbnb, Netflix, Spotify, Uber and Zillow (among many others), and Uniloc's patent complaints against cloud computing users such as health service providers, gaming companies, and financial services companies.
Overall, patent litigation filings in 2018 are on par, if somewhat below, the number of filings in 2017. However, in a fascinating development, the number of Non-NPE patent lawsuits exceeds the number of NPE lawsuits for the first time in years, according to Unified Patents Q1 2018: Patent Dispute Report.
While the consensus view is that the last several years (particularly post-Alice) have been tough going for NPEs, that trend seems to be changing. Both the Federal Circuit and the USPTO have taken recent positions on software patent eligibility that may make it harder to challenge software patents under Section 101 early in litigation and reject software applications during prosecution. And the USPTO also just announced an administrative ruling that is viewed as favorable to patent owners in Inter Partes Review proceedings. On the other hand, the change in patent venue law and subsequent decline of the Eastern District of Texas as the hotbed of patent litigation ushered in by TC Heartland will require NPEs to continue evolving their litigation tactics. On balance, however, these actions collectively may increase the strength and certainty of U.S. patents after a decade of what many see as an assault on patent value by the Supreme Court in particular. If that trend holds, and patents become more powerful (and valuable), cloud patent lawsuits will certainly be on the rise in the coming years.


For most of my time on the Online Left I thought it was pretty much generally accepted that the Democrats were not quite as Good as they should be, and while pragmatism (gotta get elected, gotta raise money, gotta deal with bad apples in your own party, etc...) was acknowledged - and debated - it wasn't especially controversial to say things like "politicians are influenced by big donors and corporate interests more than they should be" even if what to do about that was not always agreed upon.
Now this is apparently a crazy BernieBro view and pointing out that, say, 5000000 emails in a row from the DCCC informing us that we're all going to die unless I contribute $5 to their latest scam, might not actually be a strategy designed to win elections, is just proof that people are employed by Jill Stein and also, too, Russia.
The powers that be are not all Good. Some are better than others. This didn't use to be controversial. That Trump is Bad does not change this.


text:
Q: how to remove correlated or duplicated variable or individuals in r I have the following type (however very large number of variables and ind) data:
    mydf <- data.frame (Inv = 1:6, varA = c(1,1,1,  0,1,1),
   varB = c(1,0,1,  0, 1,1), varC = c(1,0,0,  0,1,1), varD = c(1,1,1,  0,1,1),
     varE = c(1,0,1,  0, 1,1), varF = c(1,1,1,  0, 1,1))

mydf
  Inv varA varB varC varD varE varF
1   1    1    1    1    1    1    1
2   2    1    0    0    1    0    1
3   3    1    1    0    1    1    1
4   4    0    0    0    0    0    0
5   5    1    1    1    1    1    1
6   6    1    1    1    1    1    1

I want to do all one to one comparsion (both variables and individuals / subjects) and keep only one if they are duplicated and name of duplicated individuals / variables to different file as log:
For example in above data:
Among variables:
varA is exactly same as varD and varF - so I will just keep varA only in new data

mydf$varA == mydf$varE
[1]  TRUE TRUE  TRUE  TRUE  TRUE  TRUE

varB and varE has exactly same data - so I will just keep varB
varC is unique

Among Inv( i.e. subjects):
1, 5 and 6 are same  -> so just keep 1

Thus the resulting output file is
        mydf <- data.frame (Inv = 1:4, varA = c(1,1,1,  0),
       varB = c(1,0,1,  0), varC = c(1,0,0,  0))
  Inv varA varB varC
1   1    1    1    1
2   2    1    0    0
3   3    1    1    0
4   4    0    0    0

I would be able to find duplication probably by correlation matrix:
cor(mydf[,-1])
          varA      varB      varC      varD      varE      varF
varA 1.0000000 0.6324555 0.4472136 1.0000000 0.6324555 1.0000000
varB 0.6324555 1.0000000 0.7071068 0.6324555 1.0000000 0.6324555
varC 0.4472136 0.7071068 1.0000000 0.4472136 0.7071068 0.4472136
varD 1.0000000 0.6324555 0.4472136 1.0000000 0.6324555 1.0000000
varE 0.6324555 1.0000000 0.7071068 0.6324555 1.0000000 0.6324555
varF 1.0000000 0.6324555 0.4472136 1.0000000 0.6324555 1.0000000

Ca we automate this process ?

A: This should do the trick:
dat <- mydf[-1]
cMat <- abs(cor(dat)) >= (1 - .Machine$double.eps^0.5)
whichKeep <- which(rowSums(lower.tri(cMat) * cMat) == 0)
cbind(mydf[1], mydf[whichKeep + 1])

  Inv varA varB varC
1   1    1    1    1
2   2    1    0    0
3   3    1    1    0
4   4    0    0    0
5   5    1    1    1
6   6    1    1    1

A: This is a highly dubious statistical undertaking but it's not hard to identify the elements in that correlation matrix that are 1 and which are in the upper triangle.
cmat <- cor(mydf[,-1])
hicorr <- which(row(cmat) < col(cmat) & cmat==1, arr.ind=TRUE)[,"col"]
hicorr
#varA varB varA varD
#   4    5    6    6

mydf[ , -hicorr]
  Inv varA varB varF
1   1    1    1    1
2   2    1    0    1
3   3    1    1    1
4   4    0    0    0
5   5    1    1    1
6   6    1    1    1

A: You could also use findCorrelation from the caret package:
findCorrelation(x, cutoff = .90, verbose = FALSE)

where the output is a vector of indices denoting the columns to remove.

A: I understand you are trying to perform feature selection/dimensionality reduction. In this case take a look at FSelector package from CRAN. In particular, there are several correlation-based filters, e.g. linear.correlation(formula, data). See this for details.

redpajama_set_name: RedPajamaStackExchange


The following is from the Nov. 15 City Council Meeting. All members were present.
Laguna Board of Realtors President Gary Boison Check presented a $6,987.26 check to the city for the installation of air conditioning in the holding pens at the Laguna Beach Animal Shelter.
The funds were proceeds from the annual Pet Parade and Chili Cook-off, hosted by the board's Charitable Assistance Fund.
•Marian Pack spoke again about safety issues associated with the aging San Onofre nuclear plant. She urged the council to follow San Clemente's lead and send letters to California Sens. Barbara Boxer and Dianne Feinstein and the Nuclear Regulatory Commission asking that the plant's license not be extended for another 20 years when it reaches its 40-year permit period in 10 years.
Councilwoman Verna Rollinger said that she would like an agenda item exploring actions taken by San Clemente and actions Laguna could take to be better informed on what to do in an emergency.
The Disaster Preparedness Committee is looking into emergency plans and will shortly submit a full report, according to Councilwoman Elizabeth Pearson. Mission Hospital is the designated disaster center, Pearson said.
Pearson said she is pleased with the committee's work and what they are doing to ensure adequate information on what needs to be done before, during and after a disaster and the organization of neighborhoods.
•Gallery owner Mark Chamberlain said that Faye Chapman and Alan Boinus had approached him some time ago to put on an exhibit at BC Space in recognition of the city's participation in National Hunger and Homelessness Awareness Week.
The result was an exhibit featuring Chapman's photographs of Laguna's homeless population, along with works by other artists, to run through Dec. 22.
Donations of $361 raised at the show were presented to the city with the hope of opening the Alternate Sleeping Location for the homeless on really cold days, Chapman said.
Chapman also thanked the artists who contributed bowls to the Hunger Bowl fundraiser, the restaurants that filled them and the buyers, which included Councilman Kelly Boyd and Deputy City Clerk Lizette Chel.
Boinus asked for donations of coats and socks to the Laguna Relief & Resource Center. Chapman said women's shoes are also needed.
•Fred Talarico distributed CDs showing a video of construction at 2425 S. Coast Hwy. and expressed dismay about the project. He said nothing has been done to address safety concerns or the removal of public parking due to construction vehicles.
Boyd said that Caltrans had issued a permit that would be available the following day for parking related to the construction.
•Adoption of a resolution to submit the Open Space Parcel Tax Initiative measure to the voters in the Nov. 6, 2012, General Municipal Election.
•A five-year lease for the Friends of Hortense Miller Garden for the operation and maintenance of the property and house.
The council approved a $295,000 allocation to fund six proposed Complete Streets projects. Complete Streets is a concept that proposes to minimize the role of the automobile in Laguna Beach, emphasize a pedestrian-oriented environment and promote the notion that city streets should be safe for use by everyone — cyclists, bikers, walkers and four-wheel vehicles.
A city task force came up with 80 recommendations after 33 months of research and meetings, from which 24 were proposed to the city staff, including the North Laguna Bike Route, which has been implemented.
"The staff selected six, with only two from the top 10," task force chairman Les Miklosy said at the council meeting.
6. $5,000 to hire a transportation planning consultant to evaluate the feasibility of implementing "complete streets" elements on Glenneyre Street.
A $15,000 contingency fund was set aside as a contingency for the completion of the projects.
The council approved an ordinance amendment that requires residents and tenants in a 100-foot radius of proposed development project be notified of a public hearing.
Attorneys Gene Gratz and Larry Nokes spoke against the amendment. Both said the noticing puts an additional financial burden on applicants and tenants are more mobile than property owners.
Council members Pearson and Boyd voted against the amendment.
The council's next regularly scheduled meeting will be held Dec. 6, beginning with a closed session at 5 p.m. The public meeting is scheduled to begin at 6 p.m. in the City Council Chambers, 505 Forest Ave.


text:
They call it a modern-day "tent of Abraham," a series of four cost-free restaurants—three in Brooklyn and one in Queens—where indigent Jews in need of kosher meals can sit at small, cloth-covered tables and be served by waiters five nights a week.
Excecutive director Alexander Rapaport stands behind a table full of steaming food before opening on a Wednesday night in April. Photos by Natalie Keyssar.
Called Masbia, Hebrew for satisfy, the free restaurants are a joint project of the Metropolitan Council on Jewish Poverty, UJA Federation, and the Masbia movement. The first site opened in Borough Park in 2005. The idea, says Executive Director Alexander Rapaport, came from a man named Mordechai Mandelbaum. "Mr. Mandelbaum is an extraordinarily good person," he begins. "To call him crazy good is not an exaggeration. His house is like a soup kitchen. He constantly has guests, people with nothing, who stay with him and his wife. I used to study Talmud with him—he's now in his 60s—and as we'd study we'd also schmooze. During one of our schmoozes the idea came up to launch a more formal, systematic way of feeding the hungry. We came to believe that a place like Masbia was sustainable, that better-off people would share their good fortune with us. Mr. Mandelbaum gave us the initial seed money and we opened in April, 2005."
The first night, he continues, eight people were served. Six months later, the restaurant was feeding between 100 and 120 patrons each evening. Since then, Masbia has continued to grow. A Flatbush restaurant opened in November; Williamsburg and Rego Park restaurants opened in March. Together, the four sites currently provide dinner to more than 500 people a day.
The idea to run Masbia as a fee-free restaurant—and operate without means-testing its patrons—is unprecedented. What's more, in treating the poor as if they are deserving of respect, it demonstrates that social service programs can offer compassionate help without being punitive.
Brooklyn is the capital of Jewish poverty," says William E. Rapfogel, CEO/Executive Director of the Metropolitan Council on Jewish Poverty [AKA Met Council]. "When we released our most recent report five years ago, 30 percent of Brooklyn's Jewish community was living at or below the poverty line." That, of course, was before the economic recession. At that point an estimated 250,000 Jews in the metropolitan area were living in penury, less than $22,350 a year for a household of three.
While Met Council offers a comprehensive range of social programs to low income people—from subsidized housing to job training—alleviating hunger has become an increasingly prominent part of the organization's work. Ten years ago, Rapfogel says, "We provided 200 people a month with food vouchers or bags of groceries. In October 2009, we gave out 15,000." Rapfogel cites an even more staggering statistic: During the Jewish New Year last fall, 120,000 pounds of apples and 18,000 jars of honey were distributed.
"Every single day we see people who've lost their jobs. Many have maxed out their credit cards, spent down their savings, and in some cases have raided their pension funds. We see them when they're on the verge of being evicted or foreclosed," Rapfogel continues. "These are people who had a very good lifestyle and were financially stable until three, six, nine months ago. Now they're at risk of losing everything. We see adult children moving back in with parents and adults who can no longer help their elderly relatives. The free restaurants are a place of respite for them, a place that helps them stretch whatever money they have."
"Part of why I come to Masbia is because it's social," Malcah, a 73-year-old widow, admits. "A neighbor told me about it a few months ago and I've been coming ever since. I don't cook anymore. I can't walk much and have trouble schlepping groceries, so this place is a godsend. The bus stops right in front. I've made friends here so I don't always have to eat alone. Plus," she laughs, "the price is right."
On a balmy April evening, a range of patrons sits and eats at Masbia-Flatbush: A young divorced mother and her two sons; a single 40-something college student; a group of teenagers; and numerous middle-aged and elderly adults.
Debby Miller and another volunteer prepare a plate of food for one of the restaurant's patrons.
Sixty-one-year-old Moshe says that he comes to Masbia two or three nights a week because he can no longer work full-time. "I have a job with a car service but my legs go numb and my back starts to hurt after a few hours of sitting," he says. "After two or three hours I have to stop and take a break, move around. I don't have health insurance so I haven't been able to check it out. Coming here helps because I need kosher food. My rent is $500 a month; I can earn that working a few days a week. As long as I eat at Masbia I can make ends meet."
Steve, a disabled 54-year-old, comes to Masbia every night it is open, five times a week. "I got a big cut in my food stamps, from $97 a month to $56. I have no idea why. I can't feed myself on $56—one trip to the grocery store and it's gone since kosher food is so expensive. It's nice here. Clean. The people who wait on me are courteous and the food is good." The night we meet, Steve's table is laden with a wonderful-smelling supper: A bowl of soup, a soft drink, plus a heaping plate of broiled chicken, mixed vegetables, roasted potatoes, and bread.
After finishing his meal, Steve—like several other diners—walks a few feet to a small box where voluntary donations can be left. Rapaport reports that contributions typically range from a quarter to a few dollars. He makes sure to emphasize that no one is pressured to give—all food is completely free but any offering is gratefully accepted.
Masbia is a restaurant which serves healthy meals to those in need free of charge in its new location on Coney Island Ave in Brooklyn, New York.
So how does Masbia raise the $30,000 a week it takes to run four cost-free restaurants?
According to Rapaport, the group is reaching out to the community to restart a tradition from the European shtetl [ghetto] that was largely forgotten once people settled in the U.S. "People used to share the simcha," he says, "providing a meal for the poor on a day of celebration. In the old days people would make a special meal to feed to poor in their village on their wedding day or the day before they got married. Sometimes they'd even give the poor a small stipend. We're reigniting that idea. People can donate the cost of one night's meal at Masbia—approximately $6.00 a plate—or can have their wedding reception at Masbia and feed their guests as well as anyone else who comes in on that particular night."
The idea, he says, is beginning to catch fire.
Aaron J. Sender, Masbia's Community Nourishment Coordinator, is particularly excited about blurring the line between patron and donor. Sender, a Brooklyn native, spent several years working on an organic farm in northern California. While there, he says he began dreaming about starting a pay-what-you-like restaurant. Then, during a 2009 visit to New York, he met Alex Rapaport. He's been working at Masbia ever since. "Food is so primal," he says. "It's one of the few things that everyone alive uses. On a very basic level, it connects people."
His goal, he continues, is to make Masbia as community-supported as possible, with public schools, Yeshivas, and neighborhood gardeners eventually growing and then donating fruit and vegetables to the program. Although he acknowledges the enormous financial contributions of hedge fund manager Larry Robbins and entrepreneur and poker champ Henry Orenstein in setting up the restaurants, he hopes that people from all walks of life—Jewish and non—will visit Masbia and contribute what they can.
"I think it's important to encourage people to come to Masbia when they want to eat out and by doing so, show that they support the idea of poor people eating well," he says.
Sender is also adamant that Masbia not look like a typical soup kitchen. "No one needs to come through the system to eat here," he says. He is revolted by the idea of poor people having to reveal their whole lives to get help. "I mean, you have to be fingerprinted to get food stamps. You have to fill out a ton of forms. You have to meet with a worker," he says, his exasperation obvious. "That's not what we do here. We want people to feel alive when they sit down and have a meal brought to their table by a waiter, served on real dishes. We want them to be able to step out of being needy and be treated with dignity."
Masbia serves free Glatt kosher meals to anyone who shows up. The restaurants are open from 4:00-9:00 p.m., Sunday through Thursday. Locations: 1372 Coney Island Avenue; 4114 14th Avenue; 65 Lee Avenue in Brooklyn; and 9808 Queens Blvd. in Rego Park. Contributions can be sent to Masbia, Post Office Box 191181, Brooklyn, NY 11219-9902
Eleanor J. Bader
Eleanor J. Bader is a teacher, writer, and activist. She writes the monthly Stoking Fire column on rhrealitycheck.org, and also contributes to feministreview.org, ontheissuesmagazine.com, The Progressive and other progressive, feminist publications and blogs.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: C-Free and MySQL connect Sorry if my English is bad, I tried to connect to mysql using cfree 5 and it returns [Error] D:\C-FREE~1\mingw\bin\ld.exe: cannot find -lD:\DoAn1\lib\libmysql.lib but I already linked it . Could anyone tell me what I should to do to fix this problem?
Problem Screenshot

A: the -l parameter used to set the library name with a filename and set the library path with -L parameter.
Example:
-lmysql -LD:\DoAn1\lib


APHA Annual Meeting Blog: Ready, set, walk!
This blogger can't even imagine how many collective steps are taken at an APHA Annual Meeting. A million? A trillion? To the moon and back? Okay, that's probably a bit of a stretch. But this is for sure: Anyone who's been to an APHA Annual Meeting knows the value of a good pair of walking shoes.
This year, however, we've decided to find out just how many steps we take during an Annual Meeting with APHA's first fitness challenge! The APHA 2015 Walking Challenge is as easy as this: sign up, sync up your step-counting device and go about your usual Annual Meeting activities. (Note: You'll have to sync up your device each day to get an accurate count.) The Walking Challenge officially begins tomorrow, Sunday, Nov. 1, and the winner with the most steps will be announced Tuesday, Nov. 3, at 5 p.m. Central time. Join the challenge today!


text:
Shocking Verbs, Lawless Nouns
Fiction, Poetry and Prose from the Bucks County Writers Workshop
Editor & (Last Shift at Dooney's Pizza Shack)
Cat Ladies of the Apocalypse
(Riders of Harvestland)
It's time to turn the "man and his dog wandering through a dystopian world" trope on its head, and tell the stories about cats and their women in appropriately dystopian settings. Cat Ladies of the Apocalypse wanders through the broken world narrative in stories of survival, technology, magic, and sheer human determination. But – like their cats – the authors frequently slipped out the upstairs window or knocked things off of countertops, and their stories are stronger for it. In these pages, cats help their women find a better life, protect and defend and fight for the women who have sheltered and protected them – and vice-versa. Because these are not only stories about surviving troubled times, but of the relationships between the women and their cats, between the members of communities both small and large, and how they thrive, even in the face of catastrophic loss. Cat Ladies of the Apocalypse – it's time to take them seriously!
The Jersey Devil: A Collection of Utter Speculation
(Seeking Monsters)
Authors Susan Tulio, River Eno, LCW Allingham, H.A. Callum, and Melissa D. Sullivan
The Jersey Devil mystery has endured for centuries, birthed from within the untamed acres of the Pine Barrens. Is it a monster? A man? Or a hoax? What does it want and where does it hide? Five stories explore the origins of a mystery that preys upon our most primal fears.
In a future fighting to recover from environmental catastrophe, a young girl's attempt to prove herself to her parents leads to a startling discovery.
A close encounter leaves two men obsessed, fifty years apart, but the twisted path they follow leads them away from the futures they imagined.
A reclusive rancher spins a web of intrigue for a reporter looking to change her luck. Can they find happiness together or will his secret destroy her?
Sisters seeking solace find themselves immersed in a land of myth, discover magik weaving in and out of their world.
A young woman fighting the vicious rumors that plague her small colonial town turns to her fiancé for help, but the perpetrators of the vicious hoax are more powerful than she imagined.
Is it utter speculation? Or do one of these stories hold the key to the truth behind this enduring legend?
The Lost Colony of Roanoke: A collection of Utter Speculation
(Goodwife Dare)
Authors River Eno, Dan Kinter, Susan Tulio and LCW Allingham
This message, carved into a tree, was the only clue to the fate of the colonists on Roanoke Island. For centuries, stories have circulated, theories were explored. But to this day, no one is really sure what happened to those first English settlers in the Americas.
These stories may offer a clue:
A road trip through dystopian United States brings adventure and danger, and perhaps answers about the mysterious disappearance of the settlers.
Gods of different worlds converge but their people may not blend as well as they do.
A history teacher's quest to liven up her lessons leads to a startling discovery about love, roots and the fate of the missing colonists.
A young mother thrust into leadership struggles to establish peace between the English and the Natives of Roanoke. But a greater threat lurks in the dark forests that may consume them all.
Four stories from four authors, all exploring the fate of the missing colonists. Are any of them the answer to the greatest mystery of the United States?
The Collections of Utter Speculation
© L.C.Allingham and lcwallingham.wordpress.com, 2015. Unauthorized use and/or duplication of this material without express and written permission from this site's author and/or owner is strictly prohibited. Excerpts and links may be used, provided that full and clear credit is given to L.C.Allingham and lcwallingham.wordpress.com with appropriate and specific direction to the original content. Most of the images displayed on this blog are the work of the talented photographers at unsplash.com


Fly from New York to spectacular St. Lucia for only $259!
WOW Air landing in Newark! Non-stop flights from New York to Iceland just $198!
New York to Manila, Philippines for only $510!
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New York to many European cities from only $430!


This is the APIv3 documentation for Django Packages. It is designed to be language and tool agnostic.
This API is limited to read-only GET requests. Other HTTP methods will fail. Only JSON is provided.
© Copyright 2010-2017, Audrey Roy Greenfeld, Daniel Roy Greenfeld and contributors. Revision 51793758.


Is it possible to use the voicemail light as an indicator of missed calls ? This indication of missed calls is much more noticeable. On most phone manufacturer uses a light indication of missed calls . I'm in the system also have a Snom 3xx and 8xx phones that have LED indication , and now I have a problem with users who get Digium phones , with much less conspicuous indication of missed calls . Is there a possibility to turn this type of indication of missed calls ?
You'd have to watch for a call and then use http://phones.digium.com/methods/digiumsetledstate to toggle it to the state and color you like. You'd probably also want to watch the call events (http://phones.digium.com/phone-api/refe ... callevents) to see if an incoming call (http://phones.digium.com/phone-api/refe ... vent-types) reaches the CONNECTED state (http://phones.digium.com/phone-api/refe ... tus-object).


When I was visiting New York last year, there were all kinds of interesting stores I was coming across. There were so many discount warehouses for just about any product or item you could think of. I had seen everything from discounted couches, beds, food, you name it! The one that really stuck out was this place called House Of Air Conditioners. That was quite an odd name, and I thought it was kind of funny. But, out of being curious I went in. And I could not believe what I was seeing! There were all kinds of HVAC equipment being sold for the lowest prices I ever heard of! They had window air conditioning units for as cheap as 70 dollars each. Then they even had some deals on central heating and air conditioning systems for only 50 dollars down payment. The total was only two hundred dollars to own it. This included free installation from a professional heating and cooling company that was part of their outlet. I sure wish I lived in New York City! I would have hopped on these ultra cheap air conditioning deals in a second! There was no way I could take advantage of these over the top heating and air conditioning system deals living in the south. Even though I knew this, I decided to ask the sales rep about it. He laughed at me and thought I was joking. Ahh well, it didn't hurt to inquire about getting this deal in my state. I do know when I get home, I am going to have a look around to see if there is any of these deals locally!


How to Apply Machine Learning To Solve Business Problems The truth is that there has been a lot of buzz around the concept of machine learning and various conferences, venture investments and business related queries have come up but even after all this most technology managers have a hard time figuring out how machine learning will help them to solve their business hurdles. In addition there is the fact that new artificial intelligence buzz words are coming up very frequently thus getting a hold of what applications are viable for one's business becomes an even more complex task and sifting through the names to figure out which ones are just plain exaggerations is also quite overwhelming. Even in the face of all that it is still possible for business to benefit immensely from machine learning and some of those benefits are listed below. One of the applications is making business predictions that are so complicated that they need the help of machine learning and one such instance is face detection given the fact that it is hard to write a set of rules that enables machines to detect all kinds of faces (given the variations in skin colour, fascial hair and angles of view). However it is quite possible to write an algorithm that can detect faces and the goodness is that the tools needed to create such algorithms are open source. The other application is in filtering spam emails because whereas some spam filters work by a certain set of rules that blocks all kinds of emails that have ever been flagged as spam. In most cases the filtering work is based on the content of the inbox that is relevant to each user and due to the vast amount of data that is generated then machine learning can come in to be quite handy. The truth is that tastes in movies and music vary widely across each person and these preferences change with time. That is why big companies now use machine learning to gather data from user ratings and other online engagements on various products so that they can be able to predict what any given user with watch, buy or listen to next time.
The other application is in speech recognition since there is no single combination of sounds that uniquely signals human speech and how people pronounce words vary immensely. Despite these variations machine learning can be used to identify speech patterns and convert the speech to text and some of these applications are used by Transcription companies the world over.
The other area of application is credit card fraud detection which is a rapidly evolving field and only a small portion of fraud detection are done using hard and fast rules. New protocols are also coming up and fraud detection systems need to detect such patterns in real time thus machine learning is applied to detect such cases as soon as they happen.


text:
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Published On: Wed, Jan 15th, 2014
Business | By Riz Baig
Japan Says It Will Approve Fukushima Operator's Revival Plan
Japan's trade ministry said it would approve on Wednesday a revival plan for the utility responsible for the Fukushima nuclear disaster, Tokyo Electric Power Co, its second attempt at restoring battered finances.
The plan hinges on Tokyo Electric (Tepco) restarting its Kashiwazaki Kariwa nuclear plant to cut fossil fuel costs, a contentious issue staunchly opposed by the local governor.
An earlier plan by Tepco outlining a revival after its Fukushima plant was hit by a massive earthquake and tsunami in 2011, triggering triple meltdowns at the site, had to be torn up because it could not restart Kashiwazaki.
Japan's trade minister Toshimitsu Motegi is scheduled to meet Tepco president at 1700 JST on Wednesday to officially approve the plan.
Tepco is also pledging to cut costs by reducing fuel spending and forming partnerships with other utilities, as well as upgrading fossil fuel plants, the Nikkei newspaper reported on Wednesday. The company is majority owned by the government after an earlier bailout.
Tepco's previous revival plan revolved around a Kashiwazaki restart in early 2013. The new plan envisages a restart of two reactors at the station in July and the utility hopes all seven reactors will go online by fiscal 2016.
The disaster at Fukushima, the worst nuclear crisis since Chernobyl in 1986 eventually brought about the halt of all nuclear power plants in Japan so they could be vetted under tougher new standards.
Opposition to atomic power remains strong in the country and is set to become a major issue in an election next month for governorship of metropolitan Tokyo, which owns a stake in Tepco.
Most candidates are opposed to restarting nuclear power plants and one, former prime minister Morihiro Hosokawa has received the strong backing from Junichiro Koizumi, one of Japan's most popular leaders, who ruled between 2001 and 2006.
In the nearly three years since the disaster, the utility has been plagued by a string of setbacks at the Fukushima station north of Tokyo, including leaks of highly radioactive water last year, prompting the government to step in with more support.
The local governor in Niigata, where Kashiwazaki is located, has been a vocal opponent of Tepco's management and has questioned whether the company has the ability to operate a nuclear station, following the failings in its preparation and response to the disaster. He has publicly called for the Tepco's liquidation.
Tepco is aiming to have all seven reactors at the Niigata plant, the world's biggest atomic station, operating by fiscal year 2016.
(Reporting by Mari Saito and Kentaro Hamada; Editing by Aaron Sheldrick and Jeremy Laurence)
How to Make Your Business More Efficient


text:
\section{Introduction}
\label{sec:introduction}

The
 {\em holographic scaling\/} of the Bekenstein-Hawking
 black hole entropy~\cite{Bekenstein_entropy}:
\begin{equation}
 S_{\rm BH} \propto {\mathcal A}
 \;  ,
  \label{eq:holo_entropy}
 \end{equation}
  i.e., its proportionality with the area ${\mathcal A}$
of the event horizon,
is one of the central results of contemporary theoretical physics
and a litmus test for quantum theories of gravity~\cite{BH-thermo_reviews}.
In essence, its apparent universality suggests that the horizon encodes information at the quantum
level~\cite{thooft:85,BH-thermo_reviews,Frolov-Fursaev}
(in the form of a holographic principle~\cite{holographic}).

In the search for patterns that may elucidate this intriguing property,
several lines of research have focused on the existence of a
near-horizon conformal symmetry~\cite{strominger:98,gov:BH_states,gupta:BH}
as a ubiquitous property of
black hole thermodynamics.
Most remarkably, in Refs.~\cite{carlip:near_horizon,solodukhin:99},
the thermodynamics is explicitly connected with
the underlying near-horizon conformal field theory
through the Cardy formula.
Alternatively,
within a general framework inspired by
the brick-wall model~\cite{thooft:85},
conformal quantum mechanics
(CQM)
has been directly linked with
the thermodynamic properties~\cite{BH_thermo_CQM,semiclassical_BH_thermo}
 by considering a field
 ($D \geq 4$)
\begin{equation}
S
=
-
\frac{1}{2}
\int
d^{D} x
\,
\sqrt{-g}
\,
\left[
g^{\mu \nu}
\,
\nabla_{\mu} \Phi
\,
\nabla_{\nu} \Phi
+
m^{2} \Phi^{2}
+  \xi R \Phi^{2}
\right]
\; ,
\label{eq:scalar_action}
\end{equation}
 in a given gravitational background.
However, a complete understanding of the predominant role of
conformal quantum mechanics in the
determination of the thermodynamic properties
of black holes has not yet been achieved.
This paper provides further strong
evidence of the conformal characterization of black hole thermodynamics by
focusing on a critical question regarding the entropy, while
the more complex
issue of the role played by
conformal symmetry in the Hawking effect is left for a forthcoming publication.
In effect,
the apparent central role of conformal quantum mechanics
is highlighted by the conformal determination of the Hawking temperature;
however,
the angular-momentum sums in semiclassical
treatments of quantum field theory
also seem to be an essential ingredient in the derivation
of the Bekenstein-Hawking entropy
$ S_{\rm BH} $.
Thus, the question we now formulate is
{\em whether the area law
[Eq.~(\ref{eq:holo_entropy})]
for the entropy is a direct consequence
of the statistical counting of the angular degrees of freedom\/},
independently of the attendant near-horizon conformal symmetry.
In other words,
is the scaling of the entropy critically constrained by the conformal nature of
the radial degree of freedom, above and beyond the
 role played by the angular degrees of freedom?
Or, by contrast, does the conformal symmetry
merely provide circumstantial evidence
rather than logical necessity for
the holographic nature of the entropy?
This issue is further motivated by early back-of-the-envelope phase-space
arguments~\cite{Susskind-angular_94,Susskind_Lindesay,quantum-loop-gravity},
which appear to suggest the accidental nature of the conformal symmetry,
with $ S_{\rm BH} $ in Eq.~(\ref{eq:holo_entropy}) arising from the
straightforward angular summation of degrees of
freedom.
However,
such arguments rely on implicit
assumptions regarding the radial degrees of freedom---incidentally,
related observations were made in Ref.~\cite{Marolf_thick-horizon},
where additional issues concerning the nature of the horizon were
also addressed.
Specifically, as we explicitly show in the next sections,
the radial contribution to the phase-space counting
provides a complementary condition,
with ``radial conformality''
of the near-horizon physics
guaranteeing the {\em preservation of
holographic scaling\/}---a property that
may be dubbed  {\em conformal tightness.\/}
What comes out from this analysis is further
evidence of the
governing role played by conformal quantum mechanics:
first, in the direct determination of the Hawking
temperature; and second,
as the central result of our paper,
in the statistical counting of the relevant near-horizon
degrees of freedom---this
provides a realization of the goal outlined in this introduction.

 Our proofs rely on properties of the metric,
with the details of the
argument involve understanding the apparent origin of the
  entropy scaling from the angular degrees of freedom in
  Sec.~\ref{sec:area_law},
  followed by a more precise characterization
  of conformality in Sec.~\ref{sec:conformal_tightness},
and
  further context in Sec.~\ref{conclusions}.
Additional topics in the appendices include the
computation of the near-horizon spectral integrals
 (Appendix~\ref{sec:spectral-integral})
 and alternative
  phase-space arguments  (Appendix~\ref{sec:phase-space}).

\section{Origin of the Area Law: Angular Degrees of Freedom?}
\label{sec:area_law}

Our stated goal is to
highlight the ``conformal emergence'' of
the area scaling of the black hole entropy,
by considering a quantum field probe
with action~(\ref{eq:scalar_action})
in a gravitational background of the form
\begin{equation}
 ds^{2}
=
- f (r) \,  dt^{2}
+
\left[ f(r) \right]^{-1} \, dr^{2}
+ r^{2} \,
 d \Omega^{2}_{(D-2)}
\;
\label{eq:RN_metric}
\end{equation}
[in which
$
d \Omega^{2}_{(D-2)}
$
is the metric on the unit $(D-2)$-spheres,
$S^{D-2}$,
that foliate the spacetime manifold
 and
the metric conventions of Ref.~\cite{misner_thorne_wheeler} are used].
Equation~(\ref{eq:RN_metric})
represents a static-coordinate description or generalized
Schwarzschild chart for a generic static geometry.
The corresponding family of spacetimes includes the
Reissner-Nordstr\"{o}m geometries in $D$ spacetime dimensions~\cite{mye:86},
and possibly extensions with a cosmological constant.
In Section~\ref{sec:conformal_tightness},
we generalize this family to emphasize
that the conformal nature of the thermodynamics appears to be a fairly
universal property of black holes.
In principle, the basic
results of {\em conformal tightness\/}
could be further
generalized to axisymmetric stationary spacetimes, as will be discussed
in a forthcoming paper, but the
properties of static
spacetimes considerably simplify our analysis.
This is because,
for static spacetimes,
there exists a coordinate frame in which the timelike coordinate $t$
is associated with the Killing vector field
$\mbox{\boldmath  $\xi$ }\!    = \partial_{t}$
that is hypersurface orthogonal~\cite{frolov_novikov}, and
for which the metric can be adapted to
$
 ds^{2}
=
g_{00} \,  dt^{2}
+
\gamma_{ij}
 d x^{i} dx^{j}
$,
where
$g_{00}$
and the spatial metric $\gamma_{ij}$
are time independent; this includes
Eq.~(\ref{eq:RN_metric}) and the generalization
of Sec.~\ref{sec:conformal_tightness}.
As is well known,
the Killing vector
$\mbox{\boldmath  $\xi$ }\! $
permits the introduction of
a Killing horizon ${\mathcal H}$
where
the surface gravity $\kappa$
is defined in purely geometric terms~\cite{frolov_novikov},
\begin{equation}
\kappa
 \stackrel{(\mathcal H)}{=}
\frac{1}{2}\sqrt{
-
\frac{1}{2}
\nabla_{\alpha} \xi_{\beta}
\nabla^{\alpha} \xi^{\beta}
   }
\label{eq:surface_gravity_general}
\; .
\end{equation}
In addition,
for the particular metrics~(\ref{eq:RN_metric}),
\begin{equation}
\kappa
 \stackrel{(\cal H)}{=}
\sqrt{ \nabla_{\alpha} V (x)  \nabla^{\alpha} V(x)  }
\stackrel{(\mathcal H)}{=}
\frac{1}{2} \,  f'_{+}
\, ,
\label{eq:surface_gravity}
\end{equation}
in which
$V(x) \equiv \sqrt{ - \xi^{\mu} \xi_{\mu}} =
\sqrt{f(r)}$
and all the derivatives
are evaluated at $\mathcal H$,
in terms of
$f_{+}' = f' ( r_{+} )$,
using the standard notation $r=r_{+}$ for the
outer event horizon, identified as the
largest root of the equation $f(r)=0$.
In the nonextremal case: $f'_{+}  \neq 0$,
which is the focus of this work,
the surface gravity has a well-defined nonzero value
(while the extremal will be tackled elsewhere, in view
of its subtle analytical properties~\cite{Frolov-Fursaev}).

It should be noticed that the Hawking temperature can be computed
for the metrics~(\ref{eq:RN_metric})
from the removal of the conical singularity,
with the result
\begin{equation}
\beta =  \frac{ 2 \pi}{\kappa}
\label{eq:temperature-gravity_connection}
\end{equation}
for the inverse temperature;
see
Sec.~\ref{sec:conformal_tightness}
for a more detailed account for
a generalized class of metrics.
As a general methodology,
the relations~(\ref{eq:surface_gravity})
and (\ref{eq:temperature-gravity_connection})---properly
generalized in
Sec.~\ref{sec:conformal_tightness}---will be systematically used
to simplify the interpretation of derived statements;
consequently, the focus of the calculations will be on
addressing the questions stated in Sec.~\ref{sec:introduction},
especially on the
conformal nature of the entropy.

 To shed light on the origin of the holographic
expression~(\ref{eq:holo_entropy})
we will often invoke {\em scaling relationships\/}
arising from the relevant physics,
e.g.,
$\kappa \simeq f_{+}' $.
While  the complete analytical results are displayed
in Subsec.~\ref{sec:area-law_computation},
the scaling equations
will help to elucidate the
role played by the different degrees of freedom
in the ensuing holographic behavior.

\subsection{Origin of the Area Law: Detailed Computation}
\label{sec:area-law_computation}

In this section we propose to identify the origin of the
area law for the entropy through the bookkeeping
of the relevant degrees of freedom.
A complete analysis takes us to the statistical counting
leading to the thermodynamics of the
scalar field $\Phi$ with action~(\ref{eq:scalar_action}).
The standard procedure starts with
resolution of the field
in terms of creation and annihilation operators,
which involves a complete set of orthonormal modes
that satisfy the Klein-Gordon equation.
For the static metric~(\ref{eq:RN_metric})
in generalized Schwarzschild coordinates $(t, r,\Omega )$,
the time coordinate can be separated via a Fourier resolution
by means of the Killing vector
$\mbox{\boldmath  $\xi$ }\! $,
such that the corresponding eigenmodes of positive
frequency $\omega$ satisfy
$\mbox{\boldmath  $\xi$ }\! \Phi = - i \omega \, \Phi$.
The spatial part of each mode is further rewritten
in Liouville normal form~\cite{BH_thermo_CQM}
by the substitution
$
\phi
(r,\Omega )
=
Y_{l m } \big(  \Omega \big)
\,
u (r)
\,
[f(r)]^{-1/2}
\,
r^{-(D-2)/2}
$,
where
the angular dependence of the ultraspherical harmonics
$Y_{l m } \big( \Omega \big)$
[with eigenvalues  $\lambda_{l,D}
=l(l+D-3) $]
supplements the normal radial behavior
\begin{equation}
u''(r)
+
\mathfrak{I} (r; \omega, \alpha_{l,D}  )
\,
u (r)
=0
\;  ,
\label{eq:Klein_Gordon_normal_radial}
\end{equation}
with
$\mathfrak{I}$
interpreted as the negative
of the  effective gravitational potential
$V_{\rm eff} (r)$.
Defining
$\alpha_{l,D}
=
\lambda_{l,D}  + (D-3)^{2}/4=
\left[
l + (D-3)/2
\right]^{2}
$
and
$\Theta= \omega/2 \kappa $,
the {\em dominant near-horizon
orders of the effective interaction\/}
$\mathfrak{I}   $
are
\begin{equation}
\mathfrak{I}  (r; \omega, \alpha_{l,D}  )
\stackrel{(\mathcal H)}{\sim}
\left(
\Theta^{2}
+
 \frac{1}{4}
\right)
\frac{1}{ x^{2} }
-
\frac{ \alpha_{l,D}  }{2 \kappa \,  r_{+}^{2}}
\frac{1}{x}
\;
\label{eq:neg-BH-potential}
\end{equation}
[up to terms $O(x)$],
where
$x= r -r_{+}$
and
hereafter we use the symbol
$ \stackrel{(\mathcal H)}{\sim} $
for the near-horizon expansion
with respect to the variable $x$.
In particular,
the SO(2,1) conformal
symmetry~\cite{jackiw&alfaro}
under scrutiny is displayed by the leading
term
in Eq.~(\ref{eq:neg-BH-potential}),
which amounts to
the scale invariant potential
$V_{\rm eff}
 \stackrel{(\mathcal H)}{\sim}
\left.
V_{\rm eff}
\right|_{\rm CQM}
=
-
\left( \Theta^{2} + 1/4 \right)/x^{2}
$,
i.e., it
has same degree of homogeneity as the
derivative term in Eq.~(\ref{eq:Klein_Gordon_normal_radial}).
The ``conformal parameter''
$
\Theta
$
characterizes the effective coupling strength of
conformal quantum mechanics~\cite{anomaly_qm_ISP_anyD,renormalization_CQM}
and determines the Hawking temperature~\cite{Hawking_temperature}
in a unique manner~\cite{padmanabhan_HawkingT}.
However, the explicit form of the
effective gravitational
potential in Eq.~(\ref{eq:neg-BH-potential})
also reveals that the naive power counting
with respect to $1/x$ breaks down due to
the existence of a competition
between the two leading terms in the near-horizon expansion:
\begin{quotation}
\noindent
(i)
The purely conformal potential
$\left. V_{\rm eff} \right|_{\rm CQM} \propto - 1/x^{2}$,
which carries the radial degrees of freedom,
 is strictly dominant
for low angular momenta
$\alpha
\equiv
\alpha_{l,D} $.

\noindent
(ii)
The centrifugal potential term
 $ \alpha/\left( f r^{2} \right)
\propto 1/x$,
which carries the angular degrees of freedom,
becomes comparable to the conformal interaction
for high angular momenta $\alpha$.
\end{quotation}
Evidently,
this {\em competition is the key\/} to answering the
questions posed in Sec.~\ref{sec:introduction},
{\em regarding the relative roles played
by the angular and radial degrees of freedom.\/}

With the above considerations in mind, we seek to reformulate the
thermodynamics within this framework, in purely conformal terms.
In this program, the critical components
of the entropy calculation involve the sequential
computation of the following
quantities: the spectral function, the entropy,
and Planck-scale renormalization.

First,
the computation of the spectral
function $N(\omega)$ from the near-horizon effective gravitational
potential~(\ref{eq:neg-BH-potential})
can be accomplished either
by WKB or phase-space methods~\cite{BH_thermo_CQM,semiclassical_BH_thermo},
and also via the heat-kernel approach~\cite{holographic_scaling_I}.
For our presentation in this section,
the competition between the angular-momentum degrees
of freedom and the leading near-horizon (conformal) effective
potential
is most clearly seen in its one-dimensional WKB form,
which furnishes the asymptotically exact ``spectral integral''
\begin{equation}
 N(\omega)
\stackrel{(\mathcal H)}{\sim}
\frac{ \Theta}{\pi \, \Gamma (D-2) }
\int_{0}^{ \alpha_{\rm max} (a) }
d \alpha \,
\alpha^{D/2- 2}
 \,
\int_{a}^{ x_{1} (\alpha) }
\frac{dx}{x}
\,
\sqrt{ 1
-
\frac{ \alpha }{2 \kappa r_{+}^{2} \, \Theta^{2} }
\, x }
\;
\label{eq:WKB_number_of_states}
\end{equation}
as the near-horizon limit of the
WKB algorithm~(\ref{eq:semiclassical_number_of_states})
further discussed in Appendix~\ref{sec:spectral-integral}.
In Eq.~(\ref{eq:WKB_number_of_states}),
$a$ is a coordinate cutoff defining the
left turning point
of the potential~(\ref{eq:neg-BH-potential})
with ``zero effective energy,''
while
$x_{1} = r_{1} - r_{+} $
is the root of the radicand associated with its
right turning point:
$x_{1} (\alpha)
=
2 \kappa r_{+}^{2} \, \Theta^{2} /\alpha$
and the
angular momentum upper bound
$\alpha_{\rm max} (a)
=
2 \kappa r_{+}^{2} \, \Theta^{2}/a
$
corresponds to the
left-right
turning-point
orientation
 $a \leq x_{1} (a)$.
An alternative phase-derivation is presented in
Appendix~\ref{sec:phase-space}.
The ``conformal measure''
$dx/x$ of the radial integration
in Eq.~(\ref{eq:WKB_number_of_states})
plays a key role in our derivation and is related
to the scale-invariant radial measure
used in Ref.~\cite{Marolf_thick-horizon}.

Second,
the computation
of the entropy itself can be performed
for the field~(\ref{eq:scalar_action})
with the canonical-ensemble
spectral rule~\cite{BH_thermo_CQM,semiclassical_BH_thermo}
\begin{equation}
S
=
- \int_{0}^{\infty}
d \omega
\,
\ln (1 - e^{-\beta \omega})
\,
\left[
\left(
\omega \frac{d}{d \omega} + 2
\right)
\frac{dN(\omega)}{d\omega}
\right]
\; ,
\label{eq:entropy_formula}
\end{equation}
which amounts to a thermal free-field distribution
of independent modes at inverse temperature $\beta$,
with canonical partition function
$
Z=
\exp \left[
 - \int_{0}^{\infty}
d N ( \omega)
\,
\ln (1 - e^{-\beta \omega})
\right]
$.
The details of the spacetime curvature background,
carried by the effective gravitational potential,
are embodied in the nontrivial spectral function $N(\omega)$
of Eq.~(\ref{eq:WKB_number_of_states}).

As a final step,
the evident near-horizon divergence
$
N (\omega)
 \stackrel{(\mathcal H)}{\propto}
a^{-(D/2-1)}
$
of Eq.~(\ref{eq:WKB_number_of_states}).
calls for regularization,
e.g.,
in terms of a short-scale coordinate regulator $a$
to be introduced as the lower limit in
the divergent integral
$\int_{a} dx $.
The ensuing ``ultraviolet
catastrophe'' of the spectral number function
$N(\omega) $
can be viewed as arising from
the ultraviolet singular nature of conformal quantum
 mechanics.
As first proposed in Ref.~\cite{thooft:85},
for the metrics~(\ref{eq:RN_metric}),
a geometric radial distance from the horizon,
 \begin{equation}
h_{D} =
\int_{ r_{+}}^{r_{+}+a} ds
\stackrel{(\mathcal H)}{\sim}
\sqrt{ \frac{2 a}{\kappa  } }
\; ,
\label{eq:brick_wall_geometrical_elevation_regularized}
\end{equation}
yields an invariant answer that reveals new spacetime physics
at the Planck scale $\ell_{P} \simeq h_{D}$.

The three essential steps listed above will now be used to show
the {\em conformal tightness\/} of the black hole entropy.
Equation~(\ref{eq:WKB_number_of_states})
can be evaluated
as shown in Appendix~\ref{sec:spectral-integral},
leading to
the spectral function
 \begin{equation}
 N(\omega)
\stackrel{(\mathcal H)}{\sim}
\frac{2}{\pi
\,   (D-2) \, \Gamma (D-2) }
\,
B \left( \frac{D-2}{2} , \frac{3}{2} \right) \,
\Theta^{D-1}
\,
\left( 2 \kappa \, r_{+}^{2}  \right)^{ (D-2)/2 }
\,
a^{-(D/2- 1)}
\; ,
\label{eq:WKB_number_of_states_2}
\end{equation}
where
$B(p,q)$ is the beta function.
Consequently,
after the
replacement~(\ref{eq:brick_wall_geometrical_elevation_regularized})
is made,
the relevant spectral-function scaling becomes
\begin{equation}
 N(\omega)
\simeq
\biggl(
\frac{\omega}{  2 \kappa} \biggr)^{D-1}
\;
\left(
\frac{
{\mathcal A}_{D-2} }{ \left[ h_{D} \right]^{D-2} }
\right)
\; ,
\label{eq:WKB_number_of_states_scaled}
\end{equation}
by the explicit introduction of the area
${\mathcal A}_{D-2} $ from factors $r_{+}^{D-2}$;
the ratios highlight the dimensionless nature of
$ N(\omega)$.
The numerical proportionality prefactor
in
Eq.~(\ref{eq:WKB_number_of_states_scaled}),
as needed for a replacement of the symbol
$\simeq$ by an equal sign,
can be written as
\begin{equation}
\widehat{\mathcal N}_{D}
=
\frac{ 2^{D-1} \, B \left( (D-2)/2 , 3/2 \right) }{
\pi \,   (D-2) \, \Gamma (D-2) \, \Omega_{D-2} }
\; .
\label{eq:WKB_number_of_states_prefactor}
\end{equation}

Finally,
the entropy~(\ref{eq:entropy_formula})
corresponding to Eqs.~(\ref{eq:WKB_number_of_states_scaled})
and (\ref{eq:WKB_number_of_states_prefactor})
is given by
\begin{equation}
S
\stackrel{(\mathcal H)}{\sim}
{\mathcal S}_{D}
\,
S_{\rm BH}
\,
\left(
\frac{ 2 \pi}{ \kappa \beta}
\right)^{D-1}
\; ,
\label{eq:entropy_brick_wall}
\end{equation}
which
consists of three distinct factors:
the numerical prefactor
${\mathcal S}_{D}$,
the Bekenstein-Hawking entropy
$S_{\rm BH}
\left(
\frac{1 }{4}
\,
{\mathcal A}_{D-2}
\right)
$
and the temperature factor
$\left( 2 \pi/
 \kappa \beta  \right)^{D-1}$.
While the latter
can be straightforwardly
set equal to unity due the Hawking inverse-temperature
identification~(\ref{eq:temperature-gravity_connection}),
the latter
involves some additional algebra
leading to
$
{\mathcal S}_{D} \left[ h_{D} \right]^{(D-2)}
=
  D \zeta (D) \Gamma (D/2-1)  \pi^{1-3D/2}/2^{D-2}
$
which shows it is a number of order unity,
if and only if the elevation $h_{D}$
is of the order of the Planck length $\ell_{D}$.
Thus, for the 't Hooft assignment~\cite{thooft:85}
\begin{equation}
 h_{D}
=
\frac{1}{2}
\left[
 D \zeta (D) \Gamma (D/2-1) \pi^{1-3D/2}
\right]^{1/(D-2)}
\; ,
\label{eq:brick_wall_geometrical_elevation}
\end{equation}
which is a form of geometric renormalization,
the entropy reduces to holographic result
$S_{\rm BH}$.

\subsection{Origin of the Area Law: Scaling Arguments}
\label{sec:area-law_scaling}

We would like to provide further insight into
the {\em scaling\/}
relations that
constitute the core of the
holographic scaling of the entropy.
Unlike the results of the previous subsection,
we will omit numerical factors,
assume naturalness, and enforce dimensional relations
defined up to
dimensionless factors of order one.

First,
 the value of the Hawking temperature
$T_{H} \equiv \left( \beta_{H} \right)^{-1}$
in addition to its connection with the surface gravity via
Eq.~(\ref{eq:temperature-gravity_connection})
implies the temperature--inverse-length chain of scaling relations
\begin{equation}
\left( \beta_{H} \right)^{-1}
\simeq
\kappa
\simeq
f_{+}'
\simeq
\frac{1}{r_{+}}
\; .
\label{eq:temp-curvature_scaling-chain}
\end{equation}
 Hereafter, the symbol
 $\simeq $
  stands for a binary scaling relation
 in the sense of dimensional analysis,
 governed by
the following criteria:
(i)  completeness of relevant scale dependence;
(ii)
a ``naturalness'' condition satisfied
by its numerical coefficients
 (i.e., with their
order of magnitude being
not too different
 from order one).
Such dimensional equality
 captures the scale homogeneity of any quantity with respect to all other
{\em relevant physical or geometrical parameters\/}
for black hole thermodynamics.
Most importantly,
the scaling chain of Eq.~(\ref{eq:temp-curvature_scaling-chain})
is essentially
characterized
by the mass of the black hole---with one
additional qualifier for the last relation
$ f_{+}'
\simeq
1/r_{+}$,
which
is only a rough estimate for black holes away from extremality.

The one additional ingredient not
displayed in Eq.~(\ref{eq:temp-curvature_scaling-chain})
is the spectral scaling
with respect to the inverse temperature $\beta$,
i.e., the dimensional relationship between
$\beta$ and a characteristic frequency $\omega$.
For the thermodynamic functions,
this scaling
is driven by
the thermal distribution,
e.g., as in
 Eq.~(\ref{eq:entropy_formula})
for the entropy.
Thus, the temperature dependence is
fully controlled by the statistical function
$L(\xi) =
-
\ln (1 - e^{-\xi} )$,
through the Boltzmann factor
 $e^{-\xi} $.
As this dependence is
carried by a function
of $\xi = \beta \, \omega$
alone,
the dominant frequency
is given by the dimensional equality
\begin{equation}
\omega_{\rm dominant}
\simeq
\beta^{-1}
\; ,
\label{eq:temp-frequency_relation}
\end{equation}
which further supplements
the chain of scaling relations~(\ref{eq:temp-curvature_scaling-chain}).
This
can be seen from the following
fairly argument for a generic thermodynamic function
\begin{equation}
{\mathcal T}
=
 \int_{0}^{\infty}
d \omega
\,
L (\beta \omega)
\,
\hat{\mathcal T}_{\omega} (\beta)
\!
\left[
\frac{ dN(\omega) }{ d\omega }
\right]
\; ,
\label{eq:generic_thermo-function}
\end{equation}
where
the operator
$\hat{\mathcal T}_{\omega}(\beta)$
governs the specific properties of the
thermodynamic function
${\mathcal T}$.
In general, ${\mathcal T}$.
is a differential operator with respect to $\omega$ and possibly
$\beta$-dependent,
as follows by the replacement of any
thermal derivatives
$\partial/\partial \beta =
\omega \partial/\partial \xi $
and
integration by parts
with respect to $\xi$; this effectively, replaces all
such derivatives by their frequency counterparts,
thus yielding the operator
$\hat{\mathcal T}_{\omega}(\beta)$.
For example,
 for the free energy ${\mathcal T} = F$,
$\hat{\mathcal T}_{\omega}= - 1/\beta$;
and for the entropy
${\mathcal T} = S$,
$\hat{\mathcal T}_{\omega}=   2 + \omega \, d/d \omega$.
As a result, it is
the interplay of the three factors in
Eq.~(\ref{eq:generic_thermo-function})---with the third
one being the density of states---that yields the
corresponding thermodynamic potential function.
The scaling statement~(\ref{eq:temp-frequency_relation})
follows by enforcing the replacement $\xi = \beta \omega$,
with $\xi \simeq 1$ being the dominant part of the
integral---this procedure is elaborated upon below
for the all-important entropy function.

In particular,
the characteristic dimensional
scaling of the entropy~(\ref{eq:entropy_formula})
can be seen
by rewriting it in the form
\begin{equation}
S
=
- \int_{0}^{\infty}
d \xi
\,
\ln (1 - e^{-\xi})
\,
\left[
\left(
\xi \frac{d}{d \xi} + 2
\right)
\frac{dN(\xi/\beta)}{d\xi}
\right]
\; ,
\label{eq:entropy_formula_dimensionless}
\end{equation}
and recognizing that the dominant value of the integral
arises from $\xi \simeq 1$.
As all the factors in
Eq.~(\ref{eq:entropy_formula_dimensionless})
are dimensionless and of order one, except for
the spectral function $N(\omega)$,
it follows that the final outcome is a number
governed by naturalness times the spectral function itself
evaluated at $\xi/\beta \simeq 1/\beta$; thus,
the dimensional relation is given by the shorthand expression
\begin{equation}
S
\left.
\simeq
N(\omega) \right|_{\omega \simeq \beta^{-1}}
\; .
\end{equation}
Consequently,
the scaling~(\ref{eq:WKB_number_of_states_scaled})
is directly transferred from the spectral function
to the entropy, with the additional
simplification that
the Hawking-temperature identification is made.
Once this is recognized,
the entropy acquires the factorized structure
\begin{equation}
 S
\simeq
\underbrace{
\left(
\begin{array}{c}
{\rm ``Extra}
\\
{\rm Scaling}
\\
{\rm Factor''}
\end{array}
\right)
}_{ \mbox{\small RADIAL}}
 \;
 \times
 \;
 \underbrace{
 \begin{array}{c}
\\
\; \; \;
{\mathcal A}
\; \; \; .
\\
\\
\end{array}
 }_{ \mbox{\small ANGULAR}}
 \label{eq:entropy_angular-radial-components}
\end{equation}
The structural simplicity of this result provides the
natural context to address the relative roles played
by the angular and radial degrees of freedom
for the area law of the entropy.
Moreover,
for the general-relativistic black holes described by the
standard metric~(\ref{eq:RN_metric}):
(i)
the near-horizon radial physics is conformal;
and (ii)
the ensuing scale-invariant nature of the radial degree of freedom
implies the {\em trivial scaling\/}
\begin{equation}
\left.
\left(
\begin{array}{c}
{\rm ``Extra}
\\
{\rm Scaling}
\\
{\rm Factor''}
\end{array}
\right)
\right|_{{\rm CQM}; \; \beta \equiv \beta_{H} }
\simeq
1
 \; ,
 \label{eq:angular-radial_entropy-resolution}
\end{equation}
which obviously generates the area law in
Eq.~(\ref{eq:entropy_angular-radial-components}).
Therefore,
Eq.~(\ref{eq:angular-radial_entropy-resolution})
which is thus traced back to conformal quantum mechanics,
involves a preservation of holographic scaling---the radial
degree of freedom does not spoil
the area law suggested by
Eq.~(\ref{eq:entropy_angular-radial-components}).

In conclusion, our procedure shows that
the separation of degrees of freedom leads to the
factorized form of the entropy~(\ref{eq:entropy_angular-radial-components}),
which, in turn,
generates the area law via a trivial
radial scaling.
Insofar as Eq.~(\ref{eq:angular-radial_entropy-resolution})
relies on the scale-invariant
nature of the near-horizon radial degree of freedom,
this result can be attributed to
the role played by conformal quantum mechanics.
Thus,
the {\em conformal tightness\/}
of Eq.~(\ref{eq:holo_entropy}) is established.

\section{Conformal Tightness of Black Hole Thermodynamics}
\label{sec:conformal_tightness}

In this section we provide additional evidence
for the {\em conformal tightness\/} of the area law
by expanding the family of spacetime metrics
and specifying the generalized conditions for
Eqs.~(\ref{eq:entropy_angular-radial-components})
and (\ref{eq:angular-radial_entropy-resolution})
to hold true.
For this purpose,
we simply relax the form of Eq.~(\ref{eq:RN_metric});
as a first step, we consider
the generic spherically symmetric metrics,
while in subsequent works we will show extensions
for axisymmetric metrics.

\subsection{Thermodynamic Landscape of Black Hole Solutions}

For the analysis of black hole solutions
{\em vis-\`{a}-vis\/}
thermodynamic and conformal properties,
we simply relax the form of Eq.~(\ref{eq:RN_metric})
into
\begin{equation}
 ds^{2}
=
- A (r) \,  dt^{2}
+
B (r) \, dr^{2}
+ r^{2} \,
 d \Omega^{2}_{(D-2)}
\; ,
\label{eq:off-shell_metric}
\end{equation}
which is the most general spherically symmetric metric.
This geometry
is described via
a further generalization of the Schwarzschild
coordinates of Eq.~(\ref{eq:RN_metric}), i.e.,
for coordinates
consisting of a time $t$ measured by an observer at
asymptotic infinity
and with
$r$ being an  ``area radial coordinate''
[such that a  $(D-2)$-sphere
of radius r has proper area
${\mathcal A} = \Omega_{D-2} \, r^{D-2}$].
As such, the new family
encompasses a larger
class of black hole metrics than
Eq.~(\ref{eq:RN_metric});
in addition, it may be regarded
as a generic  ``off-shell metric''
(not necessarily a solution to the general-relativistic field equations)
conceived to characterize
the conformal tightness property.
Within the family of metrics~(\ref{eq:off-shell_metric}),
one may consider the hypersurfaces with  $r= $ constant
and decrease $r$ gradually from asymptotic spacelike infinity,
until the value $r=r_{+}$ is reached for which the
hypersurface is everywhere null.
As $\partial_{\mu} r$ is normal to these
hypersurfaces, the inverse metric element $g^{rr} =
g^{\mu \nu}  \partial_{\mu} r
\partial_{\nu} r $ vanishes, so that
$
 g^{rr} (r_{+})= 0$
and
\begin{equation}
\left.
B (r)
\right|_{\mathcal H} = \infty
\; ,
\label{eq:roots_g^rr}
\end{equation}
which is a boundary
where ingoing timelike paths cannot go back to infinity;
thus, it is an event horizon ${\mathcal H}$.
Therefore, in this approach one identifies the roots of
$g^{rr}(r) = B^{-1}(r)$
and will select the largest one,
defining ${\mathcal H}$ and leading to the analysis
of the behavior of the fields in its neighborhood.

The metric~(\ref{eq:off-shell_metric}) leads to a Klein-Gordon
equation for the modes
\begin{equation}
\left(
g^{tt}
\partial_{t}^{2}
+
g^{rr}
\partial_{r}^{2}
+
\ldots
\right)
\Phi
=
0
\; ,
\label{eq:Klein-Gordon_curved-0}
\end{equation}
which
can be recast in a normal, Schr\"{o}dinger-like  form
\begin{equation}
{\mathcal A} \, \Phi
=
\omega^{2} \, \Phi
\; ,
\label{eq:Klein-Gordon_curved}
\end{equation}
after the usual Fourier frequency decomposition,
 $\partial_{t}^{2} \rightarrow - \omega^{2}$, leading to the
effective  near-horizon gravitational potential
 $ V_{\rm eff} (x) $
 shown below.
 To determine the relevant near-horizon physics,
 including $ V_{\rm eff} (x) $
 and other functions and parameters,
 we perform the near-horizon expansions,
 assuming that the metric coefficients
 admit the leading expressions
 \begin{eqnarray}
 A (r)
&  \stackrel{(\mathcal H)}{\sim} &
 c_{t} x^{p_{t}}
  \;  ,
\label{eq:A-exp_off-shell}
\\
\left[ B(r) \right]^{-1}
&   \stackrel{(\mathcal H)}{\sim}  &
 c_{r} x^{p_{r}}
\label{eq:B-exp_off-shell}
  \; ,
  \end{eqnarray}
where $c_{t}, c_{r} > 0$
from generic metric-signature
conditions while
\begin{equation}
p_{r} > 0
\label{eq:p-r_horizon-condition}
\end{equation}
 conforms to the existence of
of a horizon via Eq.~(\ref{eq:roots_g^rr}).
In this approach, $p_{t}$ and $p_{r}$
may be regarded as a free parameters to explore
{\em the ``thermodynamic landscape''
of possible black holes\/};
as it turns, the following analysis
inevitably brings us back to conformal quantum mechanics.
 The main  results are outlined in the next few subsections.

\subsection{Modified Effective Potential}
\label{sec:eff-potential_CT}

The
near-horizon gravitational potential
takes the form
 \begin{equation}
 V_{\rm eff} (x)
 \stackrel{(\mathcal H)}{\sim}
 - \frac{ \omega^{2} \, B (r) }{ A (r) }
\stackrel{(\mathcal H)}{\sim}
- \frac{ \lambda_{\rm eff} }{ x^{p} }
\; ,
\label{eq:eff-potential_off-shell}
 \end{equation}
 which involves the effective coupling
\begin{equation}
 \lambda_{\rm eff}
 =
 \frac{\omega^{2} }{ c_{t} c_{r} }
 \; ,
\label{eq:nh-potential-coupling_off-shell}
 \end{equation}
 and power-law exponent
 \begin{equation}
 p
 =
  p_{t}  + p_{r}
  \; .
\label{eq:nh-potential-exponent_off-shell}
 \end{equation}
 By comparison with the radial
 derivative terms of the normal-reduced
equation~(\ref{eq:Klein-Gordon_curved}),
 the scale-invariant case defining
conformal quantum mechanics
 occurs when
\begin{equation}
 \left. p \right|_{\rm CQM}
=   p_{t}  + p_{r}
= 2
\; .
\label{eq:CQM-exponents_off-shell}
 \end{equation}

\subsection{Geometrical Radial Distance}
\label{sec:radial-distance_CT}

From the radial part of the metric,
$d \rho^{2} = B(r) \,  dr^{2}$,
it follows that
\begin{equation}
\rho
  \stackrel{(\mathcal H)}{\sim}
\int \frac{ 1}{ \sqrt{ c_{r} x^{p_{r} } } }
\, dx
  \stackrel{(\mathcal H)}{\sim}
 \frac{ 1}{ \sqrt{ c_{r} }}
\,
\frac{  x^{1 -p_{r}/2 } }{
1 -p_{r}/2 }
\; ,
\label{radial-distance_off-shell}
\end{equation}
 whence the geometric elevation is
 \begin{equation}
h_{D}
  \stackrel{(\mathcal H)}{\sim}
 \frac{2}{2 - p_{r} }
 \,
 \frac{ 1}{ \sqrt{ c_{r} }}
\,
 a^{1 -p_{r}/2 }
\label{eq:elevation_off-shell}
\; .
\end{equation}
This generalizes
Eq.~(\ref{eq:brick_wall_geometrical_elevation_regularized})
and permits the computation of spectral
functions in terms of a geometric,
coordinate-invariant quantity.
Notice that this procedure breaks down for $p_{r} \geq 2$.
As a result,
\begin{equation}
p_{r} < 2
\;
\label{eq:p-r_upper-bound}
\end{equation}
is a necessary condition
for consistency of the framework.

\subsection{Hawking Temperature and Conical Singularity}
\label{sec:Hawking-temperature_CT}

  The two-dimensional Euclidean metric
$ \left.
 ds^{2}
\right|_{ d \Omega = 0 }
$
(restricted to the time-radial sector)
admits the near-horizon approximation
  \begin{eqnarray}
 \left.
 ds^{2}
\right|_{ d \Omega = 0 }
  & = &
   A \, dt_{E}^{2}
  +
B \, dr^{2}
  \; ,
  \\
   &   \stackrel{(\mathcal H)}{\sim}  &
    c_{t}
    \,
    x^{p_{t}}
    \,
    dt_{E}^{2}
   +
   \frac{ 1 }{ c_{r} }
   \,  x^{-p_{r}}
   \,  dx^{2}
    \\
    &   \stackrel{(\mathcal H)}{\sim}  &
    \left(
\frac{ 2 - p_{r} }{2}
\right)^{ \frac{ 2 p_{t} }{ 2 - p_{r} } }
    c_{t}
\,
   \left( c_{r}  \right)^{ \frac{  p_{t} }{ 2 - p_{r} } }
   \,
    \rho^{ \frac{ 2 p_{t} }{ 2 - p_{r} } }
    \,
        dt_{E}^{2}
      +
      d \rho^{2}
  \; ,
  \label{eq:nh-Euclidean}
    \end{eqnarray}
 which takes a conical form
$
 \left.
 ds^{2}
\right|_{ d \Omega = 0 }
=
d \rho^{2} +
 \nu^{2}
\rho^{2} dt_{E}^{2}
$,
ultimately leading to the Hawking temperature,
 {\em if an only if\/}
  \begin{equation}
  \left.
p \right|_{\rm thermal}
= p_{r} + p_{t} =2
  \; .
\label{eq:Hawking-exponents_off-shell}
  \end{equation}
Moreover, the factor $\nu$
in the Euclidean-time part
of the metric---which defines the conical angular
deficit---becomes
$\nu = \left( p_{t} \sqrt{ c_{r} c_{t} } \right)/ 2$.
In particular,
removal of the conical singularity
with a periodic time involves a rescaled
angular-type coordinate $\chi$
of period $2 \pi$,
such that
$
d \chi
=
\nu
\,
    dt_{E}
 $;
correspondingly,
the inverse Hawking temperature becomes
 \begin{equation}
 \beta
 =
 \frac{ 4 \pi }{ p_{t}  }
 \,
 \sqrt{
 \frac{1}{
 c_{t} \, c_{r}
 } }
 \; .
 \end{equation}
 The condition~(\ref{eq:Hawking-exponents_off-shell})
brings us back to the case of
conformal quantum
mechanics~(\ref{eq:CQM-exponents_off-shell}),
within  a generalized class of metrics.
This
{\em conformal resilience\/}
 of the metric confirms:
\begin{itemize}
\item
(i) the conformal character of the near-horizon physics;
\item
(ii) the concomitant conformal role played by the
Hawking temperature.
\end{itemize}
Furthermore,
the re-emergence of this conformality
further suggests the need for a deeper characterization
of this intriguing property---a topic that
will be expanded in a forthcoming paper.

\subsection{Surface Gravity}
\label{sec:surface-gravity_CT}

From the general definition~(\ref{eq:surface_gravity_general}),
based on  the timelike Killing vector
$\mbox{\boldmath  $\xi$ }\!    = \partial_{t}$,
the surface gravity can be computed
geometrically from the limiting
form of the metric coefficients
at the horizon.
In particular, for metrics diagonalized in the
$(t,r)$
sector,
the relation
\begin{equation}
\kappa
 \stackrel{(\mathcal H)}{=}
\sqrt{
-
g^{tt}  \, g^{rr} \,
\left(
\frac{g_{tt,r} }{2}
\right)^{2}
}
\label{eq:surface_gravity_diagonal-t-r-metrics}
\;
\end{equation}
follows by straightforward algebra.
In addition,
for the particular family of metrics~(\ref{eq:off-shell_metric}),
\begin{equation}
\kappa
 \stackrel{(\cal H)}{=}
\sqrt{
\frac{ \left( A' \right)^{2} }{ 4 AB}
}
 \stackrel{(\cal H)}{=}
\frac{ p_{t}
\,
\sqrt{ c_{r} c_{t}  }
}{2}
\; ,
\label{eq:surface-gravity_off-shell}
\end{equation}
in which
$\kappa$
has a finite and nonzero value
provided that the necessary condition $p_{t}+ p_{r} =2$
be satisfied.
This condition is
identical to
Eq.~(\ref{eq:Hawking-exponents_off-shell})
for consistency of the Hawking temperature,
i.e.,
 equivalent to the removal of the conical singularity.
The argument above verifies that $\kappa$ is inextricably linked to
the Hawking temperature,
and that the expected
relation~(\ref{eq:temperature-gravity_connection})
is indeed maintained.

\subsection{Angular Momentum}
 \label{sec:angular-momentum_CT}

 The Klein-Gordon equation has the form
  \begin{equation}
\left(
- \frac{1}{ A }
\,
\partial_{t}^{2}
+
\frac{1}{ B }
\,
\partial_{r}^{2}
+
\frac{1}{r^{2} }
\,
\partial_{\Omega}^{2}
+
\ldots
\right)
\Phi
=
0
\; ,
\label{eq:Klein-Gordon_curved-0_with-L}
\end{equation}
where the additional omitted terms possibly include the mass term,
the coupling to the Ricci scalar,
 and ``extra terms'' involved in the
normal reduction process~\cite{semiclassical_BH_thermo}.
These terms are irrelevant with respect to the near-horizon
expansion, with the only exception of the leading
form of the extra terms, which provides the critical coupling of
conformal quantum mechanics.
For our
current purposes, the third term in
Eq.~(\ref{eq:Klein-Gordon_curved-0_with-L})
displays the form taken by the angular degrees
of freedom of the scalar field; in turn,
 this implies that the near-horizon
 effective centrifugal term (to be combined with the reduced
operator $\partial_{r}^{2}$)
is
 \begin{equation}
 V_{L}
  \stackrel{(\mathcal H)}{\sim}
  \frac{ B }{    { r_{+}^{2} } }
\,
\alpha_{l,D}
  \stackrel{(\mathcal H)}{\sim}
    \frac{ \alpha_{l,D} }{    c_{r} \,  r_{+}^{2}   } \,
    x^{- p_{r} }
    \; .
\label{eq:nh-angular-potential_off-shell}
    \end{equation}

Notice that, once the CQM condition~(\ref{eq:CQM-exponents_off-shell})
is adopted, the dominance of the
singular term defining conformal quantum mechanics
is guaranteed for
$p_{r} < 2$, which is the same
restriction~(\ref{eq:p-r_upper-bound})
 found for geometric
consistency of the radial distance.
Moreover,
it is the integration with respect to the angular
parameter $\alpha_{l,D}$,
when Eq.~(\ref{eq:nh-angular-potential_off-shell})
is used for statistical counting,
that produces the
holographic area factor in the
entropy~(\ref{eq:entropy_angular-radial-components}).

\subsection{Spectral Density and Entropy}
\label{sec:spectral-density_CT}

The spectral function that generalizes
Eq.~(\ref{eq:WKB_number_of_states_2})
is computed in a similar manner
by any of the techniques mentioned in Sec.~\ref{sec:area_law}
and further developed in Appendix~\ref{sec:spectral-integral}.
This requires the use of the
effective potentials $V_{\rm eff}$
and $V_{L}$
given by Eqs.~(\ref{eq:eff-potential_off-shell})
and
(\ref{eq:nh-angular-potential_off-shell}).
Notice that, if $p \neq 2$,
the radial measure in
Eq.~(\ref{eq:WKB_number_of_states_aux-comp_generalized})
fails to be the
scale invariant $dx/x$ that ultimately led to the
``extra scaling factor''~(\ref{eq:angular-radial_entropy-resolution})
to be trivial, with a holographic entropy.
By contrast,
the ensuing modified spectral function and entropy
derived from Eq.~(\ref{eq:WKB_number_of_states_aux-comp_generalized})
have additional scale factors associated with
the  horizon curvature and radius (large scales in comparison
with the Planck length);
and ultimately
the trivial scaling~(\ref{eq:angular-radial_entropy-resolution})
breaks down, along with the area law for the entropy.
This can be seen by performing
the integrals in
Eq.~(\ref{eq:WKB_number_of_states_aux-comp_generalized});
thus,
from Eq.~(\ref{eq:WKB_number_of_states_dimensionless}),
the scaling of the spectral function takes the form
\begin{equation}
    N (\omega)
\stackrel{(\mathcal H)}{\propto}
\left(
 \sqrt{ \lambda_{\rm eff} }
\right)^{D-1}
\,
c_{r}^{D/2 -1}
    \,
a^{ - (q-1)}
\;
{\mathcal A}_{D-2}
  \; .
\label{eq:spectral-number_off-shell}
\end{equation}
The exponent
$q$, given in Eq.~(\ref{eq:modified_q-exponent}),
ultimately generates the breakdown of the
 scaling relations
and
 holographic scaling,
unless a precise combination of $p_{t}$ and
$p_{r}$ conspires to restore the conformal theory.
Once the replacement
of $a$ by the geometric elevation is
made via
Eq.~(\ref{eq:elevation_off-shell}),
the scaling of the spectral function becomes
\begin{equation}
 N(\omega)
\simeq
\biggl(
\frac{\omega}{  2 \tilde{\kappa} } \biggr)^{D-1}
\;
\left(
\frac{
{\mathcal A}_{D-2} }{ \left[ h_{D} \right]^{D-2} }
\right)
\;
\times
\left[
\left( h_{D} \right)^{2}
\,
c_{r}
\right]^{ \frac{(1-q)}{(2-p_{r})} + \frac{D-2}{2} }
\; ,
\label{eq:WKB_number_of_states_scaled_generalized}
\end{equation}
where
$\tilde{\kappa} =
 p_{t} \, \sqrt{ c_{r} c_{t}  }/2
$
is a convenient parameter, while
the last factor introduces nontrivial scaling,
 as anticipated.
In effect,
once this spectral function is replaced in
the entropy algorithm~(\ref{eq:entropy_formula}),
the ``extra scaling factor''
from the spectral function
survives in the form
\begin{equation}
\left.
\left(
\begin{array}{c}
{\rm ``Extra}
\\
{\rm Scaling}
\\
{\rm Factor''}
\end{array}
\right)
\right|_{ \beta \equiv \beta_{H} }
\simeq
\left[
\left( h_{D} \right)^{2}
\,
c_{r}
\right]^{ \frac{(1-q)}{(2-p_{r})} + \frac{D-2}{2} }
\; ,
\label{eq:extra-scaling_generalized}
\end{equation}
and prevents the delivery of a holographic entropy.

The argument above proves that deviations from conformal
quantum mechanics would lead to a breakdown of the simple
area law for the entropy.
In effect,
the removal of the spurious
scaling~(\ref{eq:extra-scaling_generalized})
is guaranteed if and only if the relation
$(1-q)/(2-p_{r}) + (D-2)/2 = 0$ is satisfied; when
 combined with the definition of $q$,
Eq.~(\ref{eq:modified_q-exponent}),
a linear relation between
$p_{t}$
and
$p_{r}$
is obtained, which yields
the CQM condition~(\ref{eq:CQM-exponents_off-shell}).
Even though this is a first level of conformal tightness,
the dimensions of $c_{t}$ and $c_{r}$
also depend on the chosen exponents $p_{t}$ and $p_{r}$,
with $[\tilde{\kappa}] = \left[ {\rm length} \right]^{- p/2}$:
this introduces yet additional scaling modifications
that depend on the size of curvature and horizon
radius.
However, at an even deeper fundamental level,
the thermodynamics is more tightly constrained by the fact
that
the parameter
$\tilde{\kappa} $ is not a surface gravity---simply put,
neither the surface gravity nor the temperature
can be consistently defined, as discussed in
Subsecs.~\ref{sec:Hawking-temperature_CT}
and \ref{sec:surface-gravity_CT}.

In short,
once the restriction~(\ref{eq:Hawking-exponents_off-shell})
is enforced for consistency,
the conformal theory applies with $p=2$,
and possibly a modified
angular-momentum term as in
Eq.~(\ref{eq:nh-angular-potential_off-shell}).
In that theory,
the
scaling~(\ref{eq:extra-scaling_generalized})
becomes trivial and holography is restored.
In effect,
regardless
of the value of
$p_{r} <2$ in
Eq.~(\ref{eq:nh-angular-potential_off-shell}),
the basic entropy expression~(\ref{eq:entropy_brick_wall})
will still hold true---possibly with a modified numerical prefactor that
carries  the value
$q = 1+ (D/2 -1) p_{t}$
(instead of $D/2$).
This concludes the proof of the conformal nature of the thermodynamics.
However, as we will see next, the theory has an even tighter
form of conformality.

\subsection{Conformal Resilience}
\label{sec:conformal-resilience_CT}

The values of $p_{t}$ and $p_{r}$
constrained by the arguments
of the previous subsections
include:
a horizon restriction~(\ref{eq:p-r_horizon-condition}),
a thermal restriction~(\ref{eq:Hawking-exponents_off-shell}),
and the geometric distance and CQM vs. angular-momentum
restriction~(\ref{eq:p-r_upper-bound}).
When the near-horizon expansions are
analytic, with integer exponents, the set of restrictions
uniquely selects the original case of conformal quantum mechanics,
 with $p_{t}=p_{r} =1$.
 The corresponding metric is
a modification of Eq.~(\ref{eq:RN_metric}),
with a similar analytic structure,
\begin{equation}
ds^{2}
=
- C(r) f(r) dt^2 + D(r)
[f(r)]^{-1} dr^2 + r^{2}
\, d \Omega^{2}_{(D-2)}
\;  ,
\label{eq:generalized_Schwarzs_metric}
\end{equation}
and supplemented with the restrictions: firstly that
 \begin{equation}
f(r)
\stackrel{(\mathcal H)}{\sim}
f'_{+} x
\;
\label{eq:scale-factor_nonextremal}
\end{equation}
(with $f'_{+} \neq 0$
defining a nonextremal black hole solution);
and secondly,
that
both $C$ and $D$ should tend to constant, nonzero values
\begin{equation}
C \stackrel{(\mathcal H)}{\sim} C_{*}
\; \; \; \; \; \; \; \;
D \stackrel{(\mathcal H)}{\sim} D_{*}
\;
\label{eq:metric-prefactors}
\end{equation}
at the horizon.
Alternatively,
by redefining the radial coordinate
via $r^2 \rightarrow r^2 D(r)$,
this generalized family takes the form
\begin{equation}
ds^{2}
=
- {C}(r) f(r) dt^2 + {D}(r)  \left\{
[f(r)]^{-1} dr^2 + r^{2}
\, d \Omega^{2}_{(D-2)}
\right\}
\;  ,
\label{eq:generalized_Schwarzs_metric2}
\end{equation}
with the conditions~(\ref{eq:scale-factor_nonextremal}) and
(\ref{eq:metric-prefactors})
to be enforced just as before.
It should be noticed that
$r$ is no longer the area coordinate
in Eq.~(\ref{eq:generalized_Schwarzs_metric2})
[except in the case when
${D}(r)  = 1$];
as customary,
we have kept the same symbol $r$, even though
the new $r$ is a different coordinate.

In short, as shown
throughout this section,
{\em the modified metrics
do not change the essence of the
conformal character or the fundamental
 thermodynamic results.\/}
As a simple illustration of this generalization, let us consider
the stringy solution
corresponding to a five-dimensional extended-supergravity
and non-extremal
metric~\cite{stringy_metric,ortin}
\begin{equation}
ds^{2} =
- \left(
1 -\frac{r_{0}^{2} }{ r^{2}}
\right) \,
F^{-2/3} dt^2
+
F^{1/3} \,
\left[
\left(
1 - \frac{r_{0}^{2} }{ r^{2}}
\right)^{-1}
dr^2 + r^2 d \Omega_{(3)}^2
\right]
\; ,
\label{eq:susy_metric}
\end{equation}
which falls into the class~(\ref{eq:generalized_Schwarzs_metric2}),
with the obvious identifications
$f = 1 - r_{0}^{2}/r^{2}$,
${C} = F^{-2/3}$,
and
${D} =  F^{1/3}$
[with $F(r) $
having a finite
value dependent on the free
parameters related to the three charges and mass].
This model can be further extended to $D$ dimensions~\cite{ortin},
\begin{equation}
ds^{2} =
- \left(
1 -\frac{r_{0}^{d-3} }{ r^{d-3}}
\right) \, F^{-(d-3)/(d-2)}
dt^2
+
F^{(1/(d-2)}
 \, \left[ \left(
1 - \frac{r_{0}^{d-3} }{ r^{d-3}}
\right)^{-1}
dr^2 + r^2 d \Omega_{(d-2)}^2
\right]
\; ,
\end{equation}
following the same pattern~(\ref{eq:generalized_Schwarzs_metric2});
and similar considerations could be applied to other geometries.
The general results derived in this paper
not only exhibit the conformal resilience property of the metric
but also provide a simple computational framework
for the evaluation of all the relevant thermodynamic quantities.

\section{Outstanding Issues and Conclusions}
\label{conclusions}

This paper
has presented an array of analytical and
scaling arguments that further suggest the fundamental
nature of the SO(2,1) conformal
symmetry for black hole thermodynamics.
In essence,
even though the angular degrees of freedom directly
yield the area scaling of the entropy,
 it is the conformal nature of the radial
degree of freedom that preserves
this remarkable holographic scaling.
Additional steps in this program include
verifying the conformal tightness
of holographic scaling for higher orders of the
heat kernel expansion and additional implications
for brick-wall type models.
In addition, the extension
of the metrics
of Subsec.~\ref{sec:conformal-resilience_CT}
to the case of axisymmetric spacetimes
will be discussed elsewhere.

Given the generality and scope of the conformal
framework for black hole thermodynamics,
our arguments highlight
the need to explore this symmetry at a deeper level. This
exploration would
entail finding the geometrical meaning of the symmetry
{\em vis-a-vis\/} the near-horizon approximation
and possibly finding a connection with the approach
of Ref.~\cite{carlip:near_horizon,solodukhin:99}.

\acknowledgments{
One of us (C.R.O.) would like to thank
the Kavli Institute for Theoretical Physics for generous support and for
providing a wonderful environment during the completion of this work.
Fruitful conversations with
Don Marolf
are also acknowledged.
This work was partially supported
by the National Science Foundation under Grants
No.\ 0602340  (H.E.C.)
and  No.\ 0602301 and PHY05-51164 (C.R.O.), and
by the University of San Francisco Faculty Development Fund
(H.E.C.).}

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