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Do you have any doubts about leasing Chrysler Pacifica?
We'll dispel them! In case you don't know, the first gas and electric minivan on sale is Pacifica. It's a hybrid that has a 3.6 liter V-6, a 16-kWh battery pack and a system of two electric-engine to make 260 horsepower in general, and to produce 80 MPG driving through the city. According to Chrysler, he battery pack, which is under the second-range floor, could be refilled no more than 2 hours by 240-volt electrical сconnection. You are able to drive on the electric 30 miles. In addition, this mixed system uses a dual-engine gearbox, which is designed by Chrysler, and it unites two electric engines. These motors could be used for driving the lead wheels through clutches instead of giving over only to recapture power.
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We Do It with Landscaping.
Chatham Landscape Services is a full service design, installation and maintenance company founded in 1995 by Landscape Architect Scott Chatham. We have been the most honored company at the Georgia Urban AG Council Professional Landscape Awards every year since 2014.
Learn more about the principles that serve as the foundation for our services.
We are active members of the Atlanta Apartment Association, Community Associations Institute, Georgia Urban AG Council and National Association of Landscape Professionals. We have partnered with AAA on the Membership Committee and volunteer on its Can-Can Ball Committee to assist with raising money to support the Atlanta Food Bank.


text:
Smog May Harm Your Bones, Too
THURSDAY, Nov. 9, 2017 (HealthDay News) — Exposure to air pollution can increase the risk for osteoporosis and broken bones in older adults, a new U.S. study suggests.
Researchers analyzed data on 9.2 million Medicare enrollees in the Northeast/Mid-Atlantic area who had been hospitalized for fractures from 2003 to 2010.
The investigators found that even a small increase in exposure to air pollution particulate matter called PM2.5 was associated with an increase in fractures among older adults. PM2.5 is the label for fine inhalable particles with a diameter of 2.5 micrometers or smaller, according to the U.S. Environmental Protection Agency.
The researchers also examined eight years of follow-up among 692 middle-aged, low-income adults in the Boston health survey.
The findings showed that people who lived in areas with higher levels of PM2.5 and black carbon — a type of air pollution from vehicle exhaust — had lower levels of an important calcium and bone-related hormone, and greater decreases in bone mineral density than did those exposed to lower levels of the two pollutants.
The study was published Nov. 9 in The Lancet Planetary Health.
"Decades of careful research has documented the health risks of air pollution, from cardiovascular and respiratory diseases to cancer and impaired cognition [thinking skills], and now osteoporosis," said senior author Dr. Andrea Baccarelli. He chairs environmental health sciences at Columbia University's Mailman School of Public Health in New York City.
"Among the many benefits of clean air, our research suggests, are improved bone health and a way to prevent bone fractures," he said in a school news release.
While the study found an association between air pollution and bone problems, it did not prove a direct cause-and-effect relationship.
The U.S. National Institute of Arthritis and Musculoskeletal and Skin Diseases has more on osteoporosis.
THURSDAY, Nov. 9, 2017 (HealthDay News) — Exposure to air pollution can increase the risk for osteoporosis and broken bones in older adults, a new U.S. study suggests. Researchers analyzed data on 9.2 million Medicare enrollees in the Northeast/Mid-Atlantic area who had been hospitalized for fractures from 2003 to 2010. The investigators found that even a small increase in exposure to air pollution particulate matter called PM2.5 was associated with an increase in fractures among older adults. PM2.5 is the label for fine inhalable particles with a diameter of 2.5 micrometers or smaller, according to the U.S. Environmental Protection Agency. The researchers also examined eight years of follow-up among 692 middle-aged, low-income adults in the Boston health survey. The findings showed that people who lived in areas with higher levels of PM2.5 and black carbon — a type of air pollution from vehicle exhaust — had lower levels of an important calcium and bone-related hormone, and greater decreases in bone mineral density than did those exposed to lower levels of the two pollutants. The study was published Nov. 9 in The Lancet Planetary Health. "Decades of careful research has documented the health risks of air pollution, from cardiovascular and respiratory diseases to cancer and impaired cognition [thinking skills], and now osteoporosis," said senior author Dr. Andrea Baccarelli. He chairs environmental health sciences at Columbia University's Mailman School of Public Health in New York City. "Among the many benefits of clean air, our research suggests, are improved bone health and a way to prevent bone fractures," he said in a school news release. While the study found an association between air pollution and bone problems, it did not prove a direct cause-and-effect relationship. More information The U.S. National Institute of Arthritis and Musculoskeletal and Skin Diseases has more on osteoporosis. Let's block ads! (Why?) Original Article
"Decades of careful research has documented the health risks of air pollution, from cardiovascular and respiratory diseases to cancer and impaired cognition [thinking skills], and now osteoporosis," said senior author Dr. Andrea Baccarelli. He chairs environmental health sciences at Columbia University's Mailman School of Public Health in New York City.
"Among the many benefits of clean air, our research suggests, are improved bone health and a way to prevent bone fractures," he said in a school news release.

redpajama_set_name: RedPajamaCommonCrawl


Now PLDT Business Solutions, partnered with ENCASH, is taking SWUP technology to the countryside, making it possible for rural banks to maintain their own ATM network. A recent contract signing with 1 st Valley Bank, GM Bank, Bangko Kabayan and the Bank of Florida marked a milestone in the Philippines Microfinance, enabling far-flung regions in the country with accessible ATM. Banking customers in the province no longer to worry about commuting to the nearest city to deposit or withdraw money. The convenience of having an ATM in their own hometown and also to eliminates the need for cardholders to withdraw and carry around a large amount of cash. Clearly, when it comes to rural banking wireless is the way to go.
PLDT and ENCASH will be providing the 4 rural banks with SWUP-enabled ATMs that will be deployed in remote locations such as Arayat, Palayan City and Zaragoza . SWUP technology allows these ATMs to operate within the coverage of over 6,000 cell sites that serve 99.3% of the country's population. The unmatched wireless coverage will enable the 1 st Valley Bank, GM bank, Banko Kabayan, and the Bank of Florida to deploy their ATMs wherever their branches may be, creating more revenue opportunities for rural bank and enterprises alike and bringing state of the art banking convenience to the rural clientele.
In the long run, deploying wireless ATMs is about more than creating convenience for the rural banking customers. It's about enabling the countryside with the means for sustainable growth. As more and more rural banks opt to SWUP it, it's clear that wireless technology is more than just a trend. It's the catalyst for a change that is long overdue.


text:
UKIP leader compares EDL founder Tommy Robinson to Nelson Mandela and claims he will be seen as a 'hero'
Robinson was jailed earlier this year
Dan Bloom
UKIP's leader Gerard Batten has compared jailed Tommy Robinson to Nelson Mandela - who spent decades fighting to end apartheid in South Africa.
He praised the far-right EDL founder, who has accused Islamic extremists of "waging war on us" and was even banned from Twitter for breaching its policy on "hateful conduct," the Mirror reports.
He told the BBC's Newsnight: "In the great scheme of things, when these events are played out, he will be seen as on the right side of a struggle between good and evil."
Gerard Batten praised the far-right EDL founder
Mr Batten's comments will be seen by critics as further evidence of UKIP's dramatic shift to the right under his leadership and it comes just days after he spoke at a rally in support of Robinson.
Hundreds of people say they're going to the 'Free Tommy Robinson' march in Cambridge
UKIP banned EDL members from joining the party years ago.
Tommy Robinson, 35, was jailed for 13 months in May, after admitting contempt of court by publishing information that could prejudice an ongoing trial.
Tommy Robinson has been freed from prison after winning an appeal
Mr Batten admitted Robinson had been jailed "sometimes," but added: "Nelson Mandela was in prison several times in his career and he was actually convicted of terrorist charges and he was involved in the planning of killing innocent people.
"Tommy Robinson has never done that."
Why Robinson was jailed
The 'Muslim prison' myth
What happened at Cambridge protest
Tommy Robinson graffiti in Cambridge
Accused of comparing the two he said: "I was making the comparison that people who've been convicted of very serious crimes then go on to be seen as a hero."
But he added Robinson was on the "right side of the great cause" even though he had "never condoned" his behaviour.
Mr Batten's views appear dramatically more right-wing than those of ex-leader Nigel Farage, who this week made clear Robinson broke the law and once called Nelson Mandela a "human hero" and compared him to Jesus.
Protesters gathered in central London over the weekend to demand Robinson's release.
He launched an appeal against his 13-month jail term yesterday (July 18).


text:
Q: What is "edge percolation"? What is the meaning of the term "edge percolation"?
Context is graph theory, specifically, random graphs. In general, what does "percolation" mean in the context of random graphs?
Thanks.

A: Edge percolation (often called bond percolation) refers to edges being randomly removed from a graph (a lattice): typically, each edge has probability $p\in (0,1)$ of surviving, independent of other edges. In line with original meaning of percolation (fluid transit), one may think of the removed edges as closed, i.e., made impermeable. How far the fluid may spread in the graph depends on the size of connected component. One often sees a phase transition: emergence of a unique giant component when $p$ exceeds  some value $p_0$.
Some references: Wikipedia page, Pierre Nolin's lecture slides.

redpajama_set_name: RedPajamaStackExchange


Hotel Stansted has great pleasure to recommend Elsenham Golf & Leisure as their preferred local golf partner.
Golf is a popular game and is suitable for all ages and all abilities, with no limits on having to be a great golfer to play. There are a number of golf courses in and around the Stansted area, with a few listed below. Whether you are looking for a golfing weekend away, a quick round with friends or hitting some balls on a driving range then there is something to suit you and all located very close to Stansted airport.
This beautifully presented 9-hole golf course is situated on 2,800 yards of breathtaking Essex countryside. It has 11 driving range bays that are flood lit to practice your drive and swing, along with lessons if needed that are available from the pro that you can book in advance. You can enjoy practicing on the 3-hole practice golf course before you play with other facilities available including a pro shop, club rooms, bar, food and leisure facilities. This course is very close to Stansted airport and many of Hotel Stansted's feature hotels.
A long standing club that's design was influenced by the legendary James Braid. It is a traditional members club with a strict dress code and is used for PGA qualifying events. Situated 10 minutes from Stansted Airport, this 71 par course offers 6404 yards of challenging but fair golf, with obstacles to tackle such as trees and shrubs of various sizes. Other facilities at this course include a clubhouse with restaurant and bar, pro shop with club fitting service, along with golf pros available.
Set amongst the English countryside this 70 par, 18-hole golf course is the 3rd oldest course in Essex. It features tree lined fairways, fast greens and water hazards to keep you on your toes. This course is suitable for anyone from the amateur golfer to the more professional golfer and is considered to be one of the best courses in the country with its natural contours. A club house with bar, pro shop and changing rooms are other facilities on offer at this course.
Set amongst private park lands this 72 par, 18-hole champion course is set in 6510 yards of unique grounds. The course has been open since 1994 and was designed by Neil Coles MBE and David Williams. This course is considered to be one of the finest courses in the region. Along with the champion golf course is a modern clubhouse offering great food with a bar facility, driving range, golf buggies for hire, pro shop and changing rooms. The course is another golf course worth finding time to play a round of golf, when you are in the Stansted airport area.


Cinnamon is a likable protagonist: emotional without being too sappy, short-tempered at times, snarky but overall kind. Other characters include her friend Kara, her grandfather she's looking after, her rival in the competition, her new/old crush, her jerk ex-husband, and other townsfolk. They are each interesting without being too unbelievable.
Is this great literature? Of course not, but it has what I'm generally looking for in a cozy mystery: a goodhearted but realistic protagonist who stands up for herself and others, a mystery which is solved by the end in a satisfying manner, a cute setting, and a happy ending.
As I said above, this one is heavier on the romance and the interpersonal relationships and lighter on the mystery. I did really like Cinnamon's relationship with the ultimate villain; she was able to see the other person's position and sympathize in a way that humanized the whole story.
The very end is heavier on the sappiness, but it wasn't too bad.
It's harmless fluff, but enjoyable enough that if I see more of this author's work on sale, I'll probably pick some up. It's a perfect commuting read, or a comforting brain break from holiday stress.
Also, this author is supposedly coming out with an even more explicitly Christmas cozy mystery soon, so I'll definitely be on the lookout for that.


text:
News Sports TLH Limelight Opinion Obituaries E-Edition Legals
George Langford, Municode founder, 'father of Seminole Boosters,' dies at 95
Jeffrey Schweers
George Langford, the "father of the Seminole Boosters" who helped save the Florida State University athletic program in the 1970s and unceasingly supported FSU for five decades, died at his home Thursday night. He was 95.
He had been in declining health for years, his oldest son Lawton Langford said.
"His family was not with him at the moment of death, but we had a chance to be with him during the day," said Langford, chairman and CEO of Municorp., the publishing company founded by his father in 1951. "We had a long goodbye. There was nothing that needed to be said or hadn't been thought of."
Born Oct. 13, 1923, in Thomasville, Ga., George Robertson Langford was the youngest of four, all of whom died before him. After graduating from Thomasville High, Langford served under Gen. George S. Patton in World War II from 1942 to 1945.
After he was honorably discharged from the U.S. Army, Langford attended the University of Virginia, graduating in 1950 with a bachelor's and a law degree.
He began Municipal Code Corporation in 1951 and married Marian Corbin Lawton in 1954. Together they built Municode into the largest codifier of municipal codes in the U.S., with more than 2,000 clients. Langford was named Florida Entrepreneur of the Year in 1989 and inducted into the Institute of American Entrepreneurs.
Lawton Langford described his father as both a creator and savior. He co-founded Springtime Tallahassee and served as its second Andrew Jackson after Pete Rich. He was instrumental in starting Leadership Tallahassee and raised money to save the Monticello Opera House. He turned around Tall Timbers and made it what it is today.
"He was a consummate fundraiser who would excite people and get them to join in on his cause," Lawton Langford said.
Some other beneficiaries of his fundraising prowess include the Maclay School, the Tallahassee Chamber of Commerce, the United Way, St John's Episcopal Church and St Peter's Anglican Church
Most of all, Langford was an avid supporter of FSU, even though he was the only member of his family to not attend FSU. He was a founding member of the Seminole Boosters Board of Directors in the 1970s, serving an unprecedented three terms. He served six terms as the FSU Foundation chair and is one of only two people given lifetime membership on the FSU Foundation Board of Trustees.
He co-chaired FSU's first capital campaign and is credited with founding the Golden Chiefs.
"George earned admiration and respect not only for the countless ways he served FSU but also for his integrity and great love for this institution," FSU President John Thrasher said. "He has passed on that love to his family, and they also are very active with Florida State."
Langford received the Alumni Association's highest honor, the Bernard Sliger Award, and the Circle of Gold. He also received the Athletics Hall of Fame Moore-Stone Award, an Honorary Doctor of Letters, and the FSU faculty's Torch Award.
FSU Vice President for Advancement Tom Jennings said, "It is difficult to overstate his positive influence on so many aspects of FSU's athletic and academic programs."
Administrators and alumni turned to Langford during one of the university's darkest hours — when the football team won a total of three games in three seasons during the 1970s.
"Our program was in dire straits," said Andy Miller, who was 24 when Langford hired him in 1975 as executive director of the Seminole Boosters. "We had no money at all."
The athletic department was $600,000 in the red and had no prospect of raising more. Then-FSU President Stan Marshall threatened to shut down the department if money wasn't raised to clear the debt.
Everyone around the table agreed that Langford was the person who could help, Miller said. "He was the guy that really took on the challenge, meeting people and holding meetings. We pulled it out of the dump."
Langford fell back on his fundraising skills and raised the money needed to keep the program alive. And he was influential in hiring a new coach named Bobby Bowden.
Langford became a mentor to Miller, imparting his business leadership and fundraising skills on him.
"He taught me all the valuable information about business and relationships and dealing with people," Miller said.
In recognition of Langford's support and achievements, the Boosters created the George Langford Award to recognize people who made a huge commitment to FSU.
"I just can't say enough about him," Miller said. "He had way more impact on the university than anyone I know."
Langford was recognized for his unwavering commitment and dedication to FSU in 2000 when the Boosters named a grassy area outside Doak Campbell Stadium after him. The Langford Green underwent a massive renovation in 2017 after being used for years by fans, vendors, and ESPN's College GameDay hosts for years. The field even played host to country music star Brad Paisley.
Langford was one of the first patrons to give $1 million to FSU.
The family made an endowed gift of $600,000 to the College of Law at the request of then-dean Sandy D'Alemberte to create an endowed chair in municipal corporations, Lawton Langford said. The gift was matched by $400,000 from the Florida Legislature.
But the college never hired an endowed professor in municipal corporations, so his father called D'Alemberte, who by that time was president of the university to talk about moving the endowment to another department.
"Dad asked what department had never received a gift and it turned out to be the classics department," Lawton Langford said. "Can you imagine being the department head and getting a phone call about receiving a $1million gift?"
Langford is survived by two sons, Lawton Langford and George Robertson Langford Jr., their wives and five grandchildren. The family asks that donations be sent to St. Peter's Church or the Seminole Boosters.
Funeral arrangements are not yet confirmed.
Contact Schweers at jschweers@tallahassee.com. Follow him on Twitter @jeffschweers.
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The Conselho Político da Oposição (CPO) (or Political Opposition Council) is a political party in Angola which contested the 2012 general election.
The party flag is has a blue background with a white open book in the center. Above the book is the party acronym "CPO" in white. Superimposed over the book in the centre of the flag are are eighteen yellow stars which are arranged to form a circle.


Do NOT look up Birkenstocks on Urban Dictionary.
One of the most rewarding aspects of interior design is having the ability to fully customize various design elements to create a cohesive, distinctive space. Often we are dreaming up new finish options for custom cabinetry, sketching unique upholstery styles or printing fabrics in custom colorways. In many ways, the only limitation is your imagination.
What many find surprising is that we do so many custom rugs, too! While many people cringe at the word 'custom', the truth is that most things are made to order at more affordable pricing in an ever evolving market. Don't get me wrong, we buy plenty of vintage rugs to put in the mix. However, sometimes there are situations that call for a piece that fits the room or in many cases inspiration strikes and the design process begins.
Color- We can combine and use colors in combinations we would not readily find on the market.
Pattern- Designing rugs allows us to employ scale and pattern to work closely with the other selections in the room.
Size- No more digging through piles and piles only to find out that the rug you love is not in the right size for the room.
Price- We are now able to get more affordable options with custom designs than ever before.
Lead time- No more waiting 9-12 months for a rug….weavers are incredibly fast and we are getting really large pieces in under 5 months.
Quality- There is nothing like the feel of a hand knotted rug which will wear well and last many lifetimes. Don't get me wrong, a machine made carpeting is great for wall to wall applications and hand tufted rugs have a certain appeal particular to the project.
One of a Kind- The best part in designing custom rugs is that no two are alike. In a world where we are all trying to build our own identity having something that is one of kind is hard to find.
Take a look at some of the fun rugs we have designed below and the inspiration behind the final product. Hoping this helps you look at what is under your feet in a whole new light. It's as easy as 1-2-3!
This fun family room rug took flight beginning with a favorite fabric from Quadrille. The border is one of my all time favorites (so much detail). The rug has a high low weave which makes the pattern sing against the neutral field but it does not overpower the space.
See all those little feet in the last image? That's right, we put this supersized wool area rug in a basement rec room inspired by a Schumacher fabric that was used on the sectional. The truth is that hand knotted wool rugs are more durable than most products out there. Wool is easy to clean because it is a natural fiber and healthier in your home environment.
I am such a sucker for repeating patterns. This stair runner takes cue from the amazing kitchen backsplash (see more of this project on our site). The quatrefoil design is a carry over we employed throughout the spaces to build both pattern and color cohesion.
This is a wool rope rug- made in the USA. Sometimes our projects need a more homespun quality and I have loved using custom rope rugs in the past. The casual feel of the larger threads makes me swoon. This particular project will be featured in the June issue of House Beautiful and we cannot wait for you to see the whole house.
This home study had a very special color story that made finding a rug more difficult. It was important to me to build color cohesion from the Sister Parish fabrics and introduce the coral into more of the schematic. This basket weave design is an all time favorite of mine probably because I always loved practicing icing cakes with this design. Can you tell that everything in my life also revolves around food?
This formal living room rug is the size of a parking lot. In fact when they sent pictures of completion they had it rolled out in an empty lot with ten men needed for the weight. Most might see this rug as a big risk but the reality is that it is the glue in this room. As you can see in the inspiration board, there are so many colors and small scale patterns in the space that it begged for one piece that made them all work so well together. I love that no one in the world will have this same rug and that this family is making memories against a back drop that is unique to their home.
There is so much to know about rugs that we could talk about this for weeks. Jena Zlock from our office thought this detail in particular was worth sharing and I agree.
Hand-knotted rugs require a great deal of skill and time. Most of the artisans who fabricate our rugs are in Nepal, where the tradition of hand-knotted rugs is a time-honored art form. They are woven on a loom, and an average weaver can hand tie about 8,000-10,000 knots per day. To provide a frame of reference, a single square inch can vary from 60 to 100 knots per inch. A standard 8x10 rug is 11,520 square inches, which translates to 1,152,000 knots (at a 100-knot count)! That means it would take a single weaver at least 115 days to complete a single rug. And that timing does not include the time it takes prior to harvest, card, and dye the wool (In our case the wool is dyed to order, which allows us to fully customize the colors.). As such, a custom rug for us will typically take 12-14 weeks for production, and an additional few weeks to ship stateside.
Are there any aspects of design that you would like to learn more about? Feel free to comment below or send us a message at info@scwinteriors.net.


HomeAdvisor's Seventrust Deck Cost Guide lists prices associated with installing Seventrust ... including items such as reclaimed wood, sawdust, and discarded plastic bags.
If you're looking for 2018 breakdown for Cost of Composite Decking ... View other Decking Options & Costs: Composite, Wood, PVC & Metal Decking Choices.
Composite decking is created by mixing polyethylene with wood fiber, rice hulls or other fillers as ... created new products that encapsulate the composite material with a thin layer of PVC capstock for a ... Composite Decking Price Comparison.


We did portrait session with beautiful Juliette at Cornwall park with a lot of move, fun, ideas. And then Jonny arrive to wait while its all done.
But we decide keep shooting and made couple of shots with guys together. It was interesting experience with couple's dance, hug, kisses, real tender and love.


text:
\section{Introduction}
\label{intro}
The data from various neutrino oscillation
experiments analyzed in the context of three neutrino oscillations have revealed
five fundamental parameters by now
\cite{Gonzalez-Garcia:2014bfa,Capozzi:2013csa,Forero:2014bxa}. These include two
squared differences of neutrino masses and three mixing angles in the
Pontecorvo-Maki-Nakagawa-Sakata mixing matrix $U_{\rm PMNS}$. For any of the
normal or inverted ordering in the neutrino masses, their $3\sigma$ ranges can
be summarized as \cite{Gonzalez-Garcia:2014bfa}:
$$
0.270<\sin^2\theta_{12}<0.344~,~~~0.385<\sin^2\theta_{23}<0.644~,~~~0.0188<\sin^2\theta_{13}<0.0251$$
$$7.02 < \frac{\Delta m^2_{21}}{10^{-5}~{\rm eV}^2} < 8.09~,~~2.325 <
\frac{\Delta m^2_{31}}{10^{-3}~{\rm eV}^2} < 2.599~{\rm or}~-2.259 <
\frac{\Delta m^2_{32}}{10^{-3}~{\rm eV}^2} < -2.307$$
Here $\Delta m^2_{ij}
\equiv m_i^2-m_j^2$. Discrete symmetry based approaches have been quite widely
used in order to explain the special values of lepton mixing angles, see for
example recent reviews
\cite{Altarelli:2010gt,Altarelli:2012ss,Smirnov:2011jv,King:2013eh,
Ishimori:2010au}.
One assumes that the global symmetry group $G_f$ of the leptons is spontaneously
broken to the smaller symmetries $G_\nu$ and $G_l$ of the neutrino and the
charged lepton mass matrices respectively. The leptonic mixing can solely be
fixed from the choice of $G_l$ and $G_\nu$ in a given $G_f$
\cite{Lam:2007qc,Lam:2008rs,Lam:2008sh,Lam:2012ga,Lam:2011ag}. Possible choices
of $G_f$ leading to three non-degenerate neutrinos are extensively studied in
\cite{Toorop:2011jn,deAdelhartToorop:2011re,Hernandez:2012ra,Hernandez:2012sk,
Holthausen:2012wt,Parattu:2010cy,Fonseca:2014koa,Talbert:2014bda} and mixing
patterns are
analyzed. In a novel approach, it is shown that suitable choices of $G_f$ can
also lead to the cases with one massless neutrino
\cite{Joshipura:2013pga,Joshipura:2014pqa}, two or three degenerate neutrinos
\cite{Hernandez:2013vya,Joshipura:2014qaa} and two degenerate and one massless
neutrino \cite{Joshipura:2014qaa}. In an alternate approach, it is shown
recently in \cite{Joshipura:2015zla} that a massless neutrino with/without a
degenerate pair of neutrinos can arise if neutrino mass matrix is assumed to be
anti-symmetric under $G_\nu$.

After the clear evidence of nonzero $\theta_{13}$ and with the most recent data,
we now start to have an indirect indication of the sixth parameter, namely the Dirac
CP phase $\delta_{\rm CP}$ in the lepton sector. In fact the observed value of
$\theta_{13}$ and measured combination of $\theta_{13}$ and $\delta_{\rm CP}$ by
T2K long-baseline experiment \cite{Abe:2013hdq} are in good agreement if
$\delta_{\rm CP} \sim -\pi/2$ \cite{Strumia:2015dca,Elevant:2015ska}. This however is a mere
indication at present and more data will certainly provide clear picture in the
near future. Nevertheless, such a special value of CP phase may be indicative
signal of some hidden symmetries in the lepton sector. The current global fits of
neutrino oscillation data disfavours the maximal atmospheric mixing angle at
$1\sigma$ however it is in accordance with the data at $3\sigma$ in case of both
normal and inverted ordering in the neutrino masses. The ansatz and symmetries
of neutrino mass matrix predicting $\theta_{23}=\pi/4$ and $\delta_{\rm CP} =
\pm \pi/2$ have been proposed earlier in
\cite{Ma:2001dn,Harrison:2002et,Babu:2002dz,Grimus:2003yn}. In the simplest
case, the above prediction can be obtained if the Majorana neutrino mass matrix
in the diagonal basis of the charged leptons, namely ${\cal M}_{\nu f}$,
satisfies \begin{equation} \label{gmt} S_{23}^T {\cal M}_{\nu f} S_{23} = {\cal M}_{\nu f}^*,
\end{equation} where \begin{equation}\label{s23} S_{23}=\left( \ba{ccc} 1 & 0& 0\\ 0 & 0 & 1\\ 0 & 1 & 0
\end{array} \right)~.\end{equation} The symmetry transformation is a discrete $Z_2$ symmetry
corresponding to $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } interchange together with CP conjugation
\cite{Grimus:2003yn,Kitabayashi:2005fc,Farzan:2006vj,Joshipura:2009tg,
Gupta:2011ct,Grimus:2012hu,Mohapatra:2012tb,Feruglio:2012cw,Holthausen:2012dk,Nishi:2013jqa,Mohapatra:2015gwa}.
Such an ${\cal M}_{\nu f}$ leads to the relations among the elements of PMNS
matrix, $U \equiv U_{\rm PMNS}$:
\begin{equation} \label{gen-pred} |U_{\mu i}|=|U_{\tau
i}|~
~{\rm for}~ i=1,2,3~\end{equation}
and predicts $\theta_{23} = \pi/4$ and
$\sin\theta_{13} \cos\delta_{\rm CP}=0$, equivalently $\delta_{\rm CP} = \pm
\pi/2$ if $\theta_{13} \neq 0$. The relations in Eq. (\ref{gen-pred}) were first
proposed in \cite{Harrison:2002et} and we refer them as the results of
``generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry" in the leptonic mixing matrix\footnote{This is also
referred as ``$\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } reflection symmetry" in some literature
\cite{Harrison:2002et,Farzan:2006vj}.}.

We show that predictions in Eq. (\ref{gen-pred}) or generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry
arise on more general grounds and can follow without invoking CP and/or the $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu }
symmetry and can follow even if Eq. (\ref{gmt}) is not satisfied by ${\cal
M}_{\nu f}$. As we shall see, Eq. (\ref{gen-pred}) arises if neutrinos and the
charged lepton mass matrices are invariant under specific residual symmetries
contained in some discrete subgroups (DSG) of $O(3)$. The residual symmetries
contained in DSG of $O(3)$ can be used to get a neutrino mass matrix with
non-degenerate or partially degenerate spectrum with two of the masses being
equal. The generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry follows in both the cases. While the general
result that we derive holds for any DSG of $O(3)$, we shall discuss specific
examples of groups having three dimensional irreducible representation (irreps).
There are only three such groups, namely $A_4$, $S_4$ and $A_5$. All of which have
been widely discussed in the literature
\cite{Altarelli:2010gt,Altarelli:2012ss,Smirnov:2011jv,King:2013eh,Ishimori:2010au}
and we shall recapitulate some of the known results and present new examples
specifically in case of the partially degenerate neutrino mass spectrum.

The $A_5$ symmetry together with CP transformation has been studied recently in
\cite{Everett:2015oka,Li:2015jxa,DiIura:2015kfa,Ballett:2015wia} in order to
predict the
neutrino mixing angles and CP phases in the case of three massive Majorana
neutrinos. Our approach is different from these works as we do not impose CP explicitly
but discuss situations under which the generalized CP predictions arise
automatically. Also the choice of residual symmetry $G_\nu$ leading to
degenerate solar pair is not considered in the quoted works.

In the next section, we present our main result and discuss the emergence of
generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry from the DSG of $O(3)$. We then discuss specific
examples of the general result in Section \ref{example}. An explicit model based
on the $A_5$ group is constructed in the Section \ref{model}. Finally, we
summarize in the last section.

\section{Discrete subgroups of $O(3)$ and maximal $\theta_{23}$ \& $\delta_{\rm
CP}$}
\label{review}
We discuss the sufficiency conditions leading to
generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry predictions, Eq. (\ref{gen-pred}). Let $T_l$, $T_\nu$
and $S_{a\nu}$ with $a=1,2$ denote $3\times 3$ {\it real} orthogonal matrices
with the property \begin{equation}\label{con} T_l^n=T_\nu^m=S_{a\nu}^2=1~ ~{\rm with}~
n,m\geq 3~\end{equation} and $[S_{1\nu},S_{2\nu}]=0$, $[T_l,T_\nu]\not=0$,
$[T_l,S_{a\nu}]\not=0$. Let the Hermitian combination $M_lM_l^\dagger$ of the
charged lepton mass matrix $M_l$ satisfy
\begin{equation} \label{cl-invariance} T_l^\dagger
M_l M_l^\dagger T_l = M_l M_l^\dagger~\end{equation}
and neutrino mass matrix be invariant
under either $S_{a\nu}$ or $T_\nu$:
\begin{equation} \label{cn-invariance} {\rm (a)} ~~
S_{a\nu}^T M_\nu S_{a\nu}=M_\nu ~~~~{\rm or}~~~~{\rm (b)}~~ T_\nu^TM_\nu
T_\nu=M_\nu~.\end{equation} Then the resulting $U_{\rm PMNS}$ displays the exact
generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry with elements satisfying Eq. (\ref{gen-pred}). It is
clear that if $(T_l,~S_{a\nu})$ or $(T_l,~T_\nu)$ close to a finite group, then
the minimal such group would be a DSG of $O(3)$. Thus the DSG of $O(3)$ can
naturally lead to the generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry.

The case (a) in Eq. (\ref{cn-invariance}) corresponds to three
non-degenerate
neutrino masses and (b) to partially degenerate spectrum with two equal
neutrino masses. The neutrino mass matrix is invariant under a $Z_2\times Z_2$
symmetry in the case (a). This symmetry corresponds to changing the signs of any
two of the three neutrino fields in their mass basis. Such a symmetry is always
present if all three neutrinos are massive Majorana particles and
non-degenerate. If two
of the neutrinos are degenerate then the residual symmetry is bigger since one
can multiply the corresponding fields $\nu_1$ and $\nu_2$ by complex phase $\eta$
and $\eta^*$ respectively leaving their combined mass term invariant. The
residual symmetry in this case is $Z_m$ with $m\geq 3$ and implies a partially
degenerate spectrum which has been considered in detail in
\cite{Hernandez:2013vya,Joshipura:2014qaa}.

The proof of the above uses an important and well known result that matrices
diagonalizing symmetry operators of the mass matrices also diagonalize the
corresponding mass matrices themselves
\cite{Lam:2007qc,Lam:2008rs,Lam:2008sh,Lam:2012ga,Lam:2011ag}. Specifically, let
$V_l$ $(V_\nu)$ be $3\times 3$ unitary matrix diagonalizing the symmetry
operators $T_l$ ($S_{a\nu}$ or $T_\nu$). Then the matrices $U_l$ and $U_\nu$, diagonalizing $M_l M_l^\dagger$ and $M_\nu$ respectively, are
given by $U_l=V_l P_l$ and $U_\nu=V_\nu P_\nu$, where $P_l$ and $P_\nu$ are
arbitrary diagonal phase matrices. As a result, the elements of the
$U \equiv U_{\rm PMNS}$ matrix satisfy
\begin{equation} \label{UMNS}| U_{ij}| = | (U_l^\dagger U_\nu)_{ij}|=| (V_l^\dagger
V_\nu)_{ij}|~. \end{equation} Eqs. (\ref{cl-invariance},\ref{cn-invariance}) allow us to
determine the general form of $V_l$ and $V_\nu$. For this, we note that
eigenvalues of any unitary matrix satisfies
\begin{equation} \label{ev-su3} \lambda^3-\chi
\lambda^2+\chi^*\lambda -1=0~, \end{equation}
where $\chi$ denotes the trace of the matrix (or character)
and all the eigenvalues $\lambda$ satisfy $|\lambda|=1$. If $\chi$ is real then
one of the roots of the above equation is $\lambda_1=1$ and the other two are
given by $\lambda_{2,3}=\frac{1}{2}\left( \chi-1\pm \sqrt{(\chi-1)^2-4}\right)$.
This has only two real solutions of modulus one corresponding to $\chi=3$ and
$\chi=-1$. These respectively correspond to an identity element and elements of
order 2. The remaining solutions are non-real and complex conjugate to each
other. Such elements necessarily have order $\geq 3$. It follows that the
matrices $T_l$, $T_\nu$ satisfying Eq. (\ref{con}) have eigenvalues $\lambda_i =
(1, \eta, \eta^*)$ with $\eta \neq \pm 1$ and $|\eta|^2=1$ while $S_{a\nu}$ have
eigenvalues $(1,-1,-1)$.

Any $T_l$ with a pair of complex conjugate eigenvalues is necessarily
non-diagonal in the basis in which it is real and its eigenvalue equation is
given by
\begin{equation} \label{Tl-ev} T_l v_i=\lambda_i v_i~, \end{equation}
where $v_i$ are
eigenvectors. It follows from the eigenvalues of $T_l$ that $v_1$ can be chosen
real and $v_2 = v_3^*$. Thus, $V_l$ diagonalizing $T_l$ can be chosen to have a
general form
\begin{equation}\label{gen-vl} V_l=\left( \ba{ccc} x_1&z_1&z_1^*\\
x_2&z_2&z_2^*\\ x_3&z_3&z_3^*\\ \end{array} \right)~,\end{equation}
with real $x_i$ and complex
$z_i$. The corresponding matrix diagonalizing $M_lM_l^\dagger$ would be given by
$U_l=V_l P_l$. Next, we show that the matrix $U_\nu$ diagonalizing $M_\nu$ has
the form \begin{equation} \label{gen-vnu} U_\nu = O_\nu Q_\nu~\end{equation} in both the cases (a) and
(b), where $O_\nu$ is a real orthogonal matrix and $Q_\nu$ is a diagonal phase
matrix. Since $[S_{1\nu},S_{2\nu}]=0$, both $S_{a\nu}$ are diagonalized by a
common unitary matrix and since $S_{a\nu}$ and their eigenvalues are real, the
eigenvectors of $S_{a\nu}$ can also be chosen real. The same $O_\nu$ would
diagonalize the neutrino mass matrix also due to symmetry relation Eq.
(\ref{cn-invariance}). But the neutrino masses can be complex and $Q_\nu$ in Eq.
(\ref{gen-vnu}) corresponds to their phases. For the case (b), the matrix
$V_\nu$ that diagonalizes $T_\nu$ is formally the same as Eq. (\ref{gen-vl})
which diagonalizes $T_l$. This follows from the fact that both $T_l$ and $T_\nu$
are real and have a pair of complex conjugate eigenvalues. Thus we can write %
\begin{equation} \label{vnu} V_\nu=\left( \ba{ccc} u_1&u_1^*&w_1\\ u_2&u_2^*&w_2\\
u_3&u_3^*&w_3\\ \end{array} \right)\end{equation} with $w_i$ real. Note that the ordering of
eigenvectors is not determined from the symmetry arguments and we have chosen an
ordering in Eq. (\ref{gen-vl}) which would give generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry. Other
choices would correspond to $e$-$\tau$ or $e$-$\mu$ symmetries leading to the
predictions $|U_{ei}|=|U_{\mu i}|$ or $|U_{ei}|=|U_{\tau i}|$ respectively in
$U_{\rm PMNS}$. The ordering in $V_\nu$ in Eq. (\ref{vnu}) is however chosen
requiring that the degenerate pair of neutrinos corresponds to the solar
neutrinos pair. While $V_\nu$ diagonalizing $T_\nu$ is given above, the
diagonalizing matrix $U_\nu$ does not differ from it merely by a phase matrix as
in the case of non-degenerate neutrinos. The degeneracy in the first two masses
implies \cite{Joshipura:2014qaa}
\begin{equation} \label{unu} U_\nu = V_\nu
U_{12}R_{12}(\theta_X)P_{\beta_2}~, \end{equation} with \begin{equation} \label{u12} U_{12}=\left(
\ba{ccc} \frac{i}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\
\frac{-i}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\ 0&0&1\\ \end{array} \right)~,\end{equation}
$R_{12}$
denoting arbitrary rotation in the 1-2 plane by an angle $\theta_{X}$ and
$P_{\beta_2}={\rm Diag.}~(1,1,e^{i\beta_2/2})$. It then follows from Eqs.
(\ref{unu},\ref{u12}) that $U_\nu$ also has the same form as given by Eq.
(\ref{gen-vnu}). It is then straightforward to verify that $U_l=V_l P_l$ with
$V_l$ as in Eq. (\ref{gen-vl}) and $U_\nu$ as in Eq. (\ref{gen-vnu}) lead to
$U_{\rm PMNS}$ matrix satisfying Eq. (\ref{gen-pred}).

A neutrino mass matrix which is $Z_2\times Z_2$ symmetric can in general possess
non-trivial phases represented by $Q_\nu$ in Eq. (\ref{gen-vnu}). If these
phases are trivial and if $U_l$ is in the form of Eq. (\ref{gen-vl}) then the
Majorana neutrino mass matrix in the diagonal basis of the charged leptons is
given by
\begin{equation} \label{mnf} {\cal M}_{\nu f}\equiv U_l^T M_\nu U_l=\left(
\ba{ccc} X & A & A^*\\ A & B & C\\ A^*&C&B^*\\ \end{array} \right)~,\end{equation}
where $X$ and
$C$ are real parameters. This provides the most general solution of Eq.
(\ref{gmt}). The above $M_{\nu f}$ was first obtained
\cite{Ma:2001dn,Babu:2002dz} in the context of $A_4$ model with quasidegenerate
neutrinos. It was then argued in \cite{Grimus:2003yn} that this form can result
from a combined operation of the $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } and CP symmetry and leads to prediction of
the maximal $\delta_{\rm CP}$.

If $M_\nu$ is $Z_2\times Z_2$ symmetric but Majorana phases are non-trivial then
even with $U_l$ as in Eq. (\ref{gen-vl}) one does not get the above specific
form of Eq. (\ref{mnf}) but Eq. (\ref{gen-pred}) still holds. Thus
the combined
operation of CP and $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry is sufficient but not necessary to get the the
maximal $\theta_{23}$ and $\delta_{\rm CP}$.

It has been noticed before \cite{Ferreira:2012ri,Ma:2015gka,Ma:2015pma} that the
from given in Eq. (\ref{mnf}) follows if $V_l$ is given by \begin{equation} \label{uw}
V_l=U_\omega=\frac{1}{\sqrt{3}}\left( \ba{ccc} 1&1&1\\ 1&\omega&\omega^2\\
1&\omega^2&\omega\\ \end{array} \right)~\end{equation} with $\omega=e^{2\pi i/3}$ and if neutrino
mass matrix is real. The above form of $V_l$ is a special case of our general
form, Eq.(\ref{gen-vl}) and results when $T_l$ is identified with a $Z_3$ group
associated with cyclic permutations of three objects. A similar case is also
studied recently in the contexts of type II seesaw
\cite{He:2015afa,He:2015gba,Dev:2015dha}.

We end this section with some important remarks connected with the above result.
\begin{itemize}
\item If one were to replace $Z_n$ invariance of
$M_lM_l^\dagger$ also by a $Z_2\times Z_2$ symmetry then both $U_l$ and $U_\nu$
would be real upto a diagonal phase multiplication on right and $\delta_{\rm
CP}$ would be zero. If $Z_2\times Z_2$ invariance of $M_\nu$ in case of the
non-degenerate neutrinos is replaced by a single $Z_2$ then reality of $V_\nu$ and
hence the prediction of the generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry does not hold. An example
of this is found in a specific model \cite{Varzielas:2013hga} based on the $A_5$
group which uses a single $Z_2$ symmetry for neutrinos. As far as the degenerate
neutrinos are concerned, the order of $T_\nu$ is necessarily $>2$. Thus
all DSG of $O(3)$ giving degenerate neutrinos necessarily also give Eq.
(\ref{gen-pred}).
\item If neutrinos are degenerate then both the solar
angle
and $\delta_{\rm CP}$ are undefined. This is reflected by the presence of the
unknown angle $\theta_X$ in Eq. (\ref{unu}). But note that the relations in Eq.
(\ref{gen-pred}) hold even if $U \to U R_{12}(\theta_X) P_{\beta_2}$ and
therefore the
arbitrariness in defining $U_\nu$ arising from
the degeneracy of the solar pair does not affect the undelying generalized
$\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry. Equivalently, one finds
\cite{Hernandez:2013vya,Joshipura:2014qaa} that the quantity $I_\alpha \equiv
{\rm Im}(U_{\alpha 1}^* U_{\alpha 2})$ remains invariant under $U \to U
R_{12}(\theta_X) P_{\beta_2}$. These quantities can be written in the standard
parameterization of $U_{\rm PMNS}$ as
\beqa \label{invariant}
c_{12}s_{12}\sin\frac{\beta_1}{2}&=&\frac{1}{c_{13}^2} I_e~,\nonumber \\
c_{12}^2\sin \left(\delta_{\rm CP}-\frac{\beta_1}{2} \right)+s_{12}^2\sin
\left(\delta_{\rm CP}+\frac{\beta_1}{2} \right)&=&
\frac{1}{s_{23}c_{23}s_{13}}\left(I_\mu-\frac{s_{23}^2s_{13}^2-c_{23}^2}{c_{
13}^2}I_e\right),\end{eqnarray}
where $s_{ij}=\sin\theta_{ij}$ and
$c_{ij}=\cos\theta_{ij}$. Using the form of $U_{\rm PMNS}$ obtained in the
degenerate case above, one finds that $I_e =0$ and $I_\mu = -I_\tau = \pm
\frac{1}{2}\sin\theta_{13}$. Since these invariants are independent of
$\theta_X$, one can use the leading order values of $\theta_{12}$ to obtain
information on $\delta_{\rm CP}$. Theses are determined by the choice of $T_l$
and $T_\nu$. If $c_{12}s_{12}\neq 0$ at the leading order, then the above
equations predict $\beta_1=0$ and $\delta_{\rm CP}=\pm \frac{\pi}{2}$. On the
other hand if $c_{12}s_{12} = 0$ at the leading order than one gets
$\sin(\delta_{\rm CP} \pm \frac{\beta_1}{2})=\pm 1$. It is thus expected that
small perturbations will stabilize $\delta_{\rm CP}$ around the values obtained
in these two cases depending on the choice of the residual symmetries. Examples
of specific perturbations doing this have been considered in
\cite{Hernandez:2013vya}. Also general perturbations to the $U_{\rm PMNS}$
matrix obtained in case of the $A_5$ group were numerically analyzed in \cite{Joshipura:2014qaa} and
$\delta_{\rm CP}$ was found to be near $\pm \frac{\pi}{2}$ for the choices of
$T_l$ and $T_\nu$ made there. We shall give here an explicit model where one gets
the same result after perturbations.
\item The third column of $U$ is not
affected by arbitrariness in the choice of $\theta_{12}$ and the values of
$\theta_{13}$ is uniquely fixed by the choice of $T_\nu$ and $T_l$. We consider
leading order prediction of $\theta_{13}$ for DSG of $O(3)$ in the next section
concentrating mainly on $A_5$ . \end{itemize}

\section{Examples of generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry and $A_5$}
\label{example}
The groups $S_3,~D_N,~A_4,~S_4$ and $A_5$ are the only finite DSG of $O(3)$. Of
these only $A_4,~S_4$ and $A_5$ posses faithful three dimensional irreducible
representations. Any choice of residual symmetries within them consistent with
the previous discussion would lead to prediction Eq. (\ref{gen-pred}). The
mixing angle predictions for $A_5$ group have already been studied
\cite{Everett:2008et,Feruglio:2011qq,Cooper:2012bd,Ding:2011cm} in case of the
non-degenerate neutrinos. One gets either vanishing or large $\theta_{13}$ at the leading order in
this case. The same holds for the groups $A_4$ and $S_4$
even in case of the partially degenerate spectrum. The group $A_5$ provides only
non-trivial example which gives a non-zero $\theta_{13}$ close to its
experimental value if two of the neutrinos are degenerate. We discuss this case
explicitly and enumerate all the residual symmetries within $A_5$ giving
generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry.

The $A_5$ group has sixty elements which are generated using $E$, $F$ and $H$
where
\begin{equation} \label{def} H=1/2\left(\ba{ccc} -1&\mu_ -&\mu_+\\ \mu_-&\mu_+&-1\\
\mu_+&-1&\mu_-\\ \end{array} \right) ~;~~~E=\left(\ba{ccc} 0&1&0\\ 0&0&1\\ 1&0&0\\ \end{array}
\right) ~;~~~F=\left(\ba{ccc} 1&0&0\\ 0&-1&0\\ 0&0&-1\\ \end{array} \right)~,\end{equation}
with
$\mu_\pm=1/2(-1\pm \sqrt{5})$. We list all the sixty elements in terms of $E$,
$F$, $H$ defined above in the Appendix. Properties of $A_5$ group has been
studied earlier in \cite{Everett:2008et,Feruglio:2011qq,Cooper:2012bd} and
reference \cite{Ding:2011cm} also gives list of all elements using different
matrices. We have defined them in a way which makes the appearance of the
generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry for $A_5$ explicit.

We divide the sixty elements into four categories: (i) An identity element, (ii)
the 15 elements of order 2 to be collectively called $O_2$. The character $\chi$ of
these elements is $-1$, (iii) the 20 elements of order 3 to be called $O_3$, all
with $\chi=0$ and (iv) 24 elements of order 5 collectively called $O_5$. The 12
of these have $\chi=-\mu_+ $ and another 12 have $\chi=-\mu_-$. All these
elements and their diagonalizing matrices are listed in Table \ref{elements} in
Appendix. Following Eq. (\ref{ev-su3}), we find that all the elements in
category $O_3$ and $O_5$ have one real and two complex conjugate eigenvalues.
Thus there are 44 elements belonging to $O_3$ and $O_5$ which qualify to be the
residual symmetry $T_\nu$, $T_l$ of neutrinos and the charged leptons
respectively. The 15 elements in $O_2$ contain five distinct $Z_2\times Z_2$
subgroups which can be used as residual symmetry of $M_\nu$ in case of the
non-degenerate spectrum. For each of these five choices, there exists 44 $T_l$
giving generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } or $e$-$\tau$ or $e$-$\mu$ symmetry. The last two can be
converted to $U_{\rm PMNS}$ satisfying Eq. (\ref{gen-pred}) after proper
reordering in the columns of $T_l$. If any of the five $Z_2\times Z_2$ group is
used as residual symmetry of $M_\nu$ and any of 24 elements in class $O_5$ as
$T_l$ then one gets the following $|U_{\rm PMNS}|$:
\begin{equation} \label{nondeg}
|U_{\rm
PMNS}|= \left(
\begin{array}{ccc}
 0.8507 & 0.5257 & 0 \\
 0.3717 & 0.6015 & 0.7071 \\
 0.3717 & 0.6015 & 0.7071 \\
\end{array}
\right)~\end{equation}
or
matrix which
differs from above by reordering of row and columns. This matrix has the
property of golden ratio prediction \cite{Kajiyama:2007gx} for the solar mixing
angle $\sin^2\theta_{12}=0.276$. It however predicts $\sin^2 \theta_{13}=0$. This case provides a
good zeroeth order approximation and it has already been discussed in
\cite{Kajiyama:2007gx,Everett:2008et,Feruglio:2011qq,Ding:2011cm,Cooper:2012bd}.
If one chooses any of 20 elements in $O(3)$ as $T_l$ then one gets generalized
$\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry but the resulting form of $|U_{\rm PMNS}|$ differs significantly from the
observed one.

In case of the partially degenerate neutrino spectrum, one has the choice of 44 elements as residual
symmetries of $M_\nu$ and $M_l$ consistent with generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu }. The structure
of the PMNS matrix follows from the basic structure of $U_l$, $U_\nu$. In
particular, one gets from Eq. (\ref{gen-vl}) and Eq. (\ref{vnu})
$$
\sin^2 \theta_{13}=\left|\sum_{i} x_i u_i\right|^2~, $$
where $x_i$ ($u_i)$ denotes the
eigenvector of
$T_l$ ($T_\nu$) corresponding to the eigenvalue 1. This can be determined from
the structure of the elements $O_2$ and $O_5$ as given in the Appendix. All
possible values of $\theta_{13}$ obtained in this way are give by
$$\sin^2\theta_{13} = \{0.035,~0.111,~0.2,~0.556,~0.632\}~.$$ Similar exercise
in case of the $A_4$ and $S_4$ groups gives: \beqa \label{}
A_4~~&:&~~\sin^2\theta_{13} = 0.111~; \nonumber \\ S_4~~&:&~~\sin^2\theta_{13} =
\{ 0,~0.111,~0.333\}~.\end{eqnarray} The same results also follow from
\cite{Joshipura:2014qaa} in which an extensive analysis was performed on several
discrete subgroups of $SU(3)$ which can lead to the appropriate symmetries for
degenerate solar pair. The numerical results presented in Table I in
\cite{Joshipura:2014qaa} shows that among all the analyzed groups, the only group with prediction maximal
$\theta_{23}$ and $\delta_{\rm CP}$ for $0<\sin^2\theta_{13}<0.05$ is $A_5$ or
the group which contains it as a subgroup, for example $\Sigma(1080)$.

Of all the predicted values, $\sin^2\theta_{13}=0.035$ can be considered close
to experiments which can be brought within 3$\sigma$ limit of the experimental
value with relatively small corrections. This value is obtained if $T_l$ belongs
to $O_5$ and $T_\nu$ to $O(3)$ or vice versa. There exists more than one
structures of $|U_{\rm PMNS}|$ corresponding to the same value of
$s_{13}^2$. We
note here two qualitatively different cases.

If $T_l=T$ and $T_\nu= E^{-1}AE$ then one gets
$$|U_{\rm PMNS}|=\left(
\begin{array}{ccc} 0.8507 & 0.4911 & 0.1876 \\ 0.3717 & 0.616 & 0.6946 \\ 0.3717
& 0.616 & 0.6946 \\ \end{array} \right)$$
upto a rotation by an angle $\theta_{X}$ in the 12 plane, where $A$, $T$ are
defined in the Appendix. The
same $T_l$ but $T_\nu=AEA^{-1}$ gives instead $$|U_{\rm PMNS}|=\left(
\begin{array}{ccc} 0.9822 & 0 & 0.1876 \\ 0.1326 & 0.7071 & 0.6946 \\ 0.1326 &
0.7071 & 0.6946 \\ \end{array} \right)$$ In the first case, $c_{12}s_{12}$ is
non-zero at the leading order. Then invariants given in Eq. (\ref{invariant})
lead to $\beta_1=0$, $\delta_{\rm CP}=\pm \frac{\pi}{2}$. In the second case,
$c_{12}s_{12}=0$ and one gets $\sin(\delta_{\rm
CP}\pm \beta_1/2)=\pm 1$. The small perturbations are then required to fix
$\theta_{12}$ to its experimental value and to generate splittings in the solar
pair. Such perturbations would also fix $\delta_{\rm CP}$ close to the
values around this.
\section{An $A_5$ model} \label{model} We now provide explicit model in
which
the results of previous section can be realized. The model is very similar to
the one presented in \cite{Varzielas:2013hga}. Major difference being a
different vacuum alignment and the form of the charged lepton mass matrix. The
group $A_5$ has ${\bf 1}$, ${\bf 3_1}$, ${\bf 3_2}$, ${\bf 4}$ and
${\bf 5}$ dimensional irreps where ${\bf 3_1}$ and ${\bf 3_2}$ are
non-equivalent irreps. The model is supersymmetric with the
three generations of leptons $l_L$ and $l^c$ both transforming  as ${\bf 3_1}$ under
$A_5$
as in \cite{Varzielas:2013hga}. It follows from the product $${\bf 3_1}\times
{\bf 3_1}=({\bf 1}+{\bf 5})_{\rm sym.}+{\bf 3_1}_{\rm antisym.}$$ that symmetric
neutrino masses can arise from ${\bf 1}+{\bf 5}$ and the charged lepton masses
can arise from all three irreps. Accordingly, we introduce two flavons, a 5-plet
$\phi_\nu$ and a singlet $s_\nu$ to generate neutrino masses. The Higgs doublets
of the minimal supersymmetric standard model, $H_u$ and $H_d$, are singlet of
$A_5$. We introduce a weak triplet $\Delta$ as an $A_5$ singlet. The relevant
superpotential is: \begin{equation} \label{wnu} W_\nu= \frac{1}{2 \Lambda} l_L^T \Delta
l_L(h_{s\nu } s_\nu +h_{5\nu} \phi_\nu)~.\end{equation} The charged lepton masses are
generated by three additional flavons, a singlet $s_l$, a 5-plet $\phi_l$ and a
triplet $\chi_l$. The corresponding superpotential is \begin{equation} \label{wl} W_l=
\frac{1}{ \Lambda} l_L H_d l^c (h_{sl} s_l +h_{5l} \phi_l+h_{3l} \chi_l)~.\end{equation}

Among the various possible choices of the residual symmetries given in the
Appendix, we specialize to a particular choice with $T_l=E$ and $T_\nu=f_2 T
f_2$. A hermitian combination of the charged lepton mass matrix $M_lM_l^\dagger$
invariant under $T_l$ results if the vacuum expectation values (VEV)
$\vev{\chi_l}$ and $\vev{\phi_{l}}$ satisfy
\begin{equation} \label{lvac}
T_l(3)\vev{\chi_l}=\vev{\chi_l}~,~~~ T_l(5)\vev{\phi_l}=\vev{\phi_l}~,\end{equation} where
$T_l(3)$ $(T_l(5))$ denotes the matrices corresponding to the ${\bf 3_1}$ $({\bf
5})$ representation. The $T_l(3)=E$ and $T_l(5)$ is given
\cite{Varzielas:2013hga} by:
\begin{equation} T_l(5)=\left( \begin{array}{ccccc} 0 & 1 &
0 & 0 &
0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -\frac{1}{2} &
-\frac{\sqrt{3}}{2} \\ 0 & 0 & 0 & \frac{\sqrt{3}}{2} & -\frac{1}{2} \\
\end{array} \right)~. \end{equation} Denoting $\vev{\chi_l}=(\chi_1,\chi_2,\chi_3)^T$ and
$\vev{\phi_l}=(q_1,q_2,q_3,q_4,q_5)^T$, Eqs. (\ref{lvac}) are solved by \begin{equation}
\label{soll} v_1=v_2=v_3\equiv v_l~,~~ q_1=q_2=q_3\equiv q_l~~{\rm
and}~~q_4=q_5=0~.\end{equation}
Inserting this solution in the superpotential in Eq.
(\ref{wl}) leads to a charged lepton mass matrix
\begin{equation} \label{ml} M_l=\left(
\ba{ccc} m_0&m_1-m_2&m_1+m_2\\ m_1+m_2&m_0&m_1-m_2\\ m_1-m_2&m_1+m_2&m_0\\ \end{array}
\right)~. \end{equation}
The $M_lM_l^\dagger$ is diagonalized by the matrix $U_\omega$ which
also diagonalizes the corresponding symmetry generator $T_l(3)=E$. Explicitly,
\begin{equation} U_\omega^{\dagger} M_lM_l^\dagger
U_\omega={\rm Diag.} (m_e^2,m_\mu^2,m_\tau^2)~.\end{equation} with \beqa m_e^2&=& |m_0 + 2
m_1|^2~,\nonumber \\ m_\mu^2&=&|m_0-m_1- \sqrt{3} i ~m_2|^2~,\nonumber \\
m_\tau^2&=&|m_0-m_1 + \sqrt{3} i ~m_2|^2~\end{eqnarray}
Here $m_0$ can be taken real
without loss of generality. Note that the electron mass given above corresponds
to the eigenvector $(1,1,1)^T$ of $U_\omega$. This has to be identified as the
first column of $U_l$ in order to get the $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry as already mentioned. The
remaining two eigenvalues can be identified with muon and tau lepton masses and
can be interchanged. The contributions labeled by $m_0$, $m_2$, $m_1$ arise from
the VEVs of singlet, triplet and the 5-plet. The $M_l$ is symmetric in the
absence of triplet. In this case, $T_l$ invariance implies two degenerate
charged leptons. Thus a large triplet contribution $m_2$ is essential to split
the muon and tau lepton masses. Moreover, simultaneous presence of $m_0$ and $m_1$
is also required to suppress the electron mass. But given all the three
contributions, one can fit the charged lepton masses with appropriate choice of
parameters.

Neutrino masses follow analogously from Eq. (\ref{wnu}). In order to get
degeneracy, we impose the residual symmetry $T_{\nu}=f_2 T f_2$ and require that
\begin{equation} \label{nuvac} T_\nu(5)\vev{\phi_\nu}=\vev{\phi_\nu}~,\end{equation} where $T_{\nu}(5)$
can be shown to be\footnote{This is determined by noting that the presentations
$a,b,c$ introduced in \cite{Varzielas:2013hga} are given in terms of our
presentation as $f_3=a$, $E=b$, $ f_1= b^2 a b$ and $ H=b c$.}
\begin{equation}
T_\nu(5)=\left( \begin{array}{ccccc} -\frac{1}{2} & 0 & \frac{1}{2} & \frac{1}{2
\sqrt{2}} &\frac{\sqrt{3}}{2\sqrt{2}} \\ 0 & \frac{1}{2} & \frac{1}{2} &
-\frac{1}{\sqrt{2}} & 0 \\ \frac{1}{2} & -\frac{1}{2} & 0 & -\frac{1}{2
\sqrt{2}} & \frac{\sqrt{3}}{2\sqrt{2}} \\ -\frac{1}{2 \sqrt{2}} &
-\frac{1}{\sqrt{2}} & \frac{1}{2 \sqrt{2}} & -\frac{1}{4} & -\frac{\sqrt{3}}{4}
\\ -\frac{\sqrt{3}}{2\sqrt{2}} & 0 & -\frac{\sqrt{3}}{2\sqrt{2}} &
-\frac{\sqrt{3}}{4} & \frac{1}{4} \\ \end{array} \right)~.\end{equation}
Let
$\vev{\phi_\nu}=(p_1,p_2,p_3,p_4,p_5)^T$. A solution for Eq. (\ref{nuvac}) is
given by
\begin{equation} \label{ps} p_1=p_3=0~,~p_2=-\sqrt{2}
p_4~,~p_5=-\frac{p_4}{\sqrt{3}}~.\end{equation} Inserting these in the neutrino
superpotential leads to a neutrino mass matrix
\begin{equation}\label{mnu0} M_{0\nu}=\left(
\ba{ccc} m_{0\nu }-\frac{m_{1\nu}}{3}(\mu_+-\mu_-)&0&0\\ 0&m_{0 \nu
}-\frac{m_{1\nu}}{3}(\mu_- -1)&-m_{1\nu}\\ 0&-m_{1\nu}&m_{0 \nu
}-\frac{m_{1\nu}}{3}(1-\mu_+)\\ \end{array} \right)~.\end{equation}
As a consequence of the
residual symmetry, one gets two degenerate neutrinos with a mass
$m_{0\nu}-\frac{m_{1\nu}}{3}(\mu_+-\mu_-)$ and the third mass is given by
$m_{0\nu}+\frac{2 m_{1\nu}}{3}(\mu_+-\mu_-)$. The lower $2\times 2$ block of
$M_\nu$ is diagonalized by a rotation matrix with an angle $\theta$ given by:
$$\tan\theta=-\mu_-.$$ The full PMNS matrix at the leading order is thus given
by
\begin{equation} \label{upmns0} U_{0}\equiv U_\omega^{\dagger} R_{23}(\theta)=
\frac{1}{\sqrt{3}}\left( \ba{ccc} 1&c_\theta+s_\theta&c_\theta-s_\theta \\
1&c_\theta\omega^2+s_\theta\omega&c_\theta \omega-s_\theta\omega^2 \\
1&c_\theta\omega+s_\theta\omega^2&c_\theta \omega^2-s_\theta\omega \\ \end{array}
\right) ~,\end{equation}
where $c_\theta=\cos\theta,s_\theta=\sin\theta$. The generalized
$\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry is apparent from the
above. Moreover, \begin{equation}\label{s13model0}
s_{13}^2=\frac{1}{3}(c_\theta-s_\theta)^2=\frac{1}{3} \left(1+
\frac{2\mu_-}{1+\mu_-^2}\right)\approx 0.035~\end{equation} as would be expected from the
specific choice of the residual symmetry made in this example.

The above zeroth order prediction would get modified from the perturbations
which are required to split the degenerate states, fix the solar angle and to
change the zeroth order predictions for the mixing angles $\theta_{13}$ and
$\theta_{23}$. Effects of general perturbations were studied in \cite{Joshipura:2014qaa}
in the context of
$A_5$ symmetry with a slightly different choice of the residual symmetry which
also leads to the same zeroth order predictions as here. It was found that small
perturbations can cause significant changes in $\theta_{13}$ as required
experimentally and relatively small perturbations in the zeroth order values of
$\theta_{23}$ and the maximal CP violating phase. Moreover, all three neutrinos
are required to be quasidegenerate in order to reproduce all the mixing angles
correctly as long as perturbations are smaller than $\le 5\%$. The analysis in
\cite{Joshipura:2014qaa} was for the most general possible perturbations. In the
context of specific models, such perturbations can arise from the non-leading
higher order terms in the Yukawa superpotential which directly correct the
leptonic mass matrices and/or from the Higgs potential which may perturb the
Higgs vacuum expectation values from the symmetric choice. Let us consider
effect of a simple but interesting perturbation in the latter category. Assume
that the perturbations change one of the VEVs given in Eq. (\ref{ps}),
namely  $p_2\rightarrow p_2(1+\epsilon)$. Similar perturbations in the
VEV of other component would also arise in general but as we discuss here, this
perturbation alone has interesting consequences. The zeroth order mass matrix in
Eq. (\ref{mnu0}) now gets changed to
\begin{equation}\label{mnucorr} M_\nu=\left( \ba{ccc}
m_{\nu 0}-\frac{m_{1\nu}}{3}(\mu_+-\mu_-)&0&0\\ 0&m_{\nu
0}-\frac{m_{1\nu}}{3}(\mu_- -1)&-m_{1\nu}(1+\epsilon)\\
0&-m_{1\nu}(1+\epsilon)&m_{\nu 0}-\frac{m_{1\nu}}{3}(1-\mu_+)\\ \end{array} \right)~.\end{equation}
The above perturbed matrix is also diagonalized by a rotation in the 2-3
plane
but with a slightly different $\theta$ which is now given by
$$\tan\theta\approx
-\mu_- \left(1-\frac{\epsilon}{\sqrt{5}} \right)+{\cal O} (\epsilon^2)~.$$ This
changes the zeroth order prediction of the mixing angle $\theta_{13}$ and Eq.
(\ref{s13model0}) gets replaced by
\begin{equation} \label{news13}
s_{13}^2=\frac{1}{3}(c_\theta-s_\theta)^2=
\frac{\mu_+^2}{3(1+\mu_-^2)}-\frac{2\epsilon}{3\sqrt{5}}\frac{\mu_-^2}{
(1+\mu_-^2)^2}+{\cal O} (\epsilon^2)~.\end{equation}
Thus the appropriate choice of
perturbation can be used to get agreement with experiments. The other major
effect of $\epsilon$ is to split the degenerate pair and induce the solar scale:
$$\frac{\Delta m^2_{\rm sol}}{\Delta m^2_{\rm atm}} \equiv \frac{\Delta m^2_{21}}{\Delta
m^2_{31}}=\frac{4\epsilon}{5}\left(\frac{3\sqrt{5}m_{0\nu}-5 m_{1\nu}}{6\sqrt{5}
m_{0\nu}+5 m_{1\nu}}\right)+{\cal O} (\epsilon^2) $$

The overall effect of the perturbation is best appreciated by going to the
flavour basis with $M_lM_l^\dagger$ diagonal. In this basis
\beqa \label{mnuf}
M_{\nu f}&\equiv& U_\omega^{T} M_\nu U_\omega~\nonumber \\ &=&\left( \ba{ccc}
m_{0\nu}-\frac{2}{3} m_{1\nu} (1+\epsilon)&-\frac{m_{1\nu}}{3}(\mu_+
+\omega^2-\epsilon)&-\frac{m_{1\nu}}{3}(\mu_+ +\omega-\epsilon)\\
-\frac{m_{1\nu}}{3}(\mu_+ +\omega^2-\epsilon)&-\frac{m_{1\nu}}{3}(1-\mu_-
-\omega^2+2\epsilon)&m_{0\nu}+\frac{m_{1\nu}}{3}(1+\epsilon)\\
-\frac{m_{1\nu}}{3}(\mu_+
+\omega-\epsilon)&m_{0\nu}+\frac{m_{1\nu}}{3}(1+\epsilon)&-\frac{m_{1\nu}}{3}(1-\mu_-
-\omega+2\epsilon)\\ \end{array}\right) \end{eqnarray}
The interesting features of this matrix
are: \begin{itemize} \item Elements of $M_{\nu f}$ satisfy $$ \sum_{i}(M_{\nu
f})_{e i}=\sum_{i}(M_{\nu f})_{\mu i}=\sum_{i}(M_{\nu f})_{\tau i}~.$$ This
condition implies that one of the column vectors of $U_{\rm PMNS}$ has a
tri-maximal form as is the case with the zeroth order mixing matrix, Eq.
(\ref{upmns0}). Thus one gets the prediction
$\sin^2\theta_{12}\cos^2\theta_{13}=\frac{1}{3}$ if perturbation makes the state
with an eigenvector corresponding to the first column in Eq. (\ref{upmns0})
heavier compared to the second degenerate state. Perturbation in this case does
not change the zeroth order solar angle but it stabilizes it to that value by
splitting the degenerate states. \item If parameters $m_{0\nu}$, $m_{1\nu}$ and
$\epsilon$ are real then the $M_{\nu f}$ satisfies $(M_{\nu f})_{12}=(M_{\nu
f})_{13}^*$ and $(M_{\nu f})_{22}=(M_{\nu f})_{33}^*$. Thus $M_{\nu f}$
simultaneously enjoys the $Z_2\times Z_2$ symmetries corresponding to a
tri-maximal solar angle and generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } as envisaged and studied in
\cite{Gupta:2011ct}. In particular, one gets the maximal atmospheric angle and
the maximal CP violating phase as predictions even after perturbation.
\end{itemize}

As an example, we give a set of specific values of $\epsilon$, $m_{0\nu}$,
$m_{1\nu}$ determined numerically which fit the experimental values: $$
m_{0\nu}=0.025~{\rm eV},~~m_{1\nu}=0.016~{\rm eV},~~\epsilon= 0.228$$ This
gives the following
mixing matrix
$$|U_{\rm PMNS}|^2= \left( \begin{array}{ccc} 0.6421 & 0.3333 &
0.0246 \\ 0.179 & 0.3333 & 0.4877 \\ 0.179 & 0.3333 & 0.4877 \\ \end{array}
\right)~.$$
corresponding to
$$\sin^2\theta_{12}\cos^2\theta_{13}=\frac{1}{3}~,~\sin^2\theta_{23}=\frac{1}{2}~,~\sin^2\theta_{13}=0.0246~.$$
The $\delta_{\rm CP}$ gets stabilized to the value $-\pi/2$. The neutrino masses
giving correct $\Delta m^2_{\rm sol}$ and $\Delta m^2_{\rm atm}$ are determined by the above values of
parameters as
$$ m_{\nu_1}=0.0097~{\rm eV},~~m_{\nu_2}=0.0131~{\rm
eV},~m_{\nu_3}=0.0522~{\rm eV}.$$
The maximality of $\theta_{23}$ can be changed
by introducing small imaginary parts in parameters but the tri-maximal value of
$\theta_{12}$ remains unchanged. Small deviations can be introduced by
perturbing other component of the VEVs or by perturbing the charge lepton mass
matrix. Since general perturbations are already studied in
\cite{Joshipura:2014qaa}, we shall not pursue them further.

\section{Summary} The generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry of the leptonic mixing matrix is
known to predict maximal atmospheric mixing angle and maximal Dirac CP violation
in case of nonzero $\theta_{13}$. Both these predictions are consistent with the
current experimental observations within $3\sigma$ and their future precision
measurements will reveal weather such a symmetry is indeed realized in nature in
its exact form. It is therefore interesting to explore the symmetries of the
leptons which lead to generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry in the lepton mixing predicting
such special values of $\theta_{23}$ and $\delta_{\rm CP}$.

Assuming the Majorana neutrinos, we have shown in this paper that generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu }
symmetry naturally follows if the symmetry group $G_f$  of  leptons,
is a discrete subgroup of $O(3)$. It is required that the $G_f$ is broken
into
$Z_m$ with $m \ge 3$ as the residual symmetry of the charged lepton mass
matrix.
The residual symmetry of the Majorana neutrino mass matrix can be either (a)
$Z_2 \times Z_2 \in G_f$ or (b) $Z_n \in G_f$ with $n \ge 3$. The possibility
(a) leads to three non-degenerate neutrinos while one obtains two of the three
neutrinos degenerate in the case (b). The possible candidates of $G_f$ are only
$A_4$, $S_4$ and $A_5$ which can predict all the three mixing angles at the
leading order. Among these, only $A_5$ predicts $\theta_{13}$ very close to its
experimentally observed value in the case of two degenerate neutrinos which are
identified with the solar pair. The corrections to the leading order neutrino
mass matrix are needed to generate viable $\theta_{13}$, $\theta_{12}$ and the
solar mass difference. We have discussed in detail the group $A_5$ in the context of
generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry and provided an explicit model in which the leading
order predictions are realized. We have also discussed the perturbations which lead
to the realistic neutrino masses and mixing angles while maintaining the
predictions $\theta_{23}=\pi/4$ and $\delta_{\rm CP} = \pm \pi/2$.

Some example ansatz and symmetries of neutrino mass matrix leading to the
generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry have already been discussed in the literature. Our
findings of an emergence of generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry from the discrete subgroups
of $O(3)$ are more general and they accommodate some of the symmetries and
models
proposed in literature to obtain $\theta_{23}=\pi/4$ and $\delta_{\rm CP} = \pm
\pi/2$. In particular, we have shown that the generalized $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry in the lepton
mixing can follow without imposing $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry and/or CP on the neutrino mass
matrix. The $\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetry with CP conjugation is realized in our approach only
accidentally when an additional assumption is made on the free parameters.

\begin{acknowledgments} A.S.J. thanks the Department of Science and Technology,
Government of India for support under the J. C. Bose National Fellowship
programme, grant no. SR/S2/JCB-31/2010. K.M.P. thanks the Department of Physics
and Astronomy of the University of Padova for its support. He also acknowledges
partial support from the European Union FP7 ITN INVISIBLES (Marie Curie
Actions, PITN-GA-2011-289442). \end{acknowledgments}

\section{Appendix} We list all the sixty elements belonging to $A_5$ in terms of
their presentation matrices $E$, $F$ and $H$ defined in Eq. (\ref{def}). For
brevity, we have defined the following matrices which are used to
characterize
various elements.
$$
f_1=F~,~~f_2=Ef_1E^{-1}~,~~f_3=Ef_2E^{-1},~~T=f_1EH~,~~A=Hf_1~.$$
The elements are listed in Table \ref{elements}.
\begin{table}[!ht]
\label{generators} \begin{center}
\begin{math}
\begin{tabular}{ccc} \hline
\hline (Set, Order) & Set of elements & Diagonalizing matrix\\ \hline
\multirow{3}{*}{($O_2,~2)$}
&$f_a$ & $I$ \\
& $H$ & $U_H$ \\
& $ f_a Hf_a$ & $f_aU_H $\\
& $EHE^{-1}$ & $EU_H$\\
& $E^{-1}HE$ & $E^{-1}U_H$\\
& $Ef_a H f_aE^{-1}$ & $Ef_aU_H$ \\
& $E^{-1}f_aHf_aE$ & $E^{-1}f_aU_H$\\
\hline
\multirow{7}{*}{($O_3,~3$)}&$E, ~E^2=E^{-1}$ & $U_\omega$\\ & $f_a E
f_a,~f_aE^{-1}f_a$ & $f_a U_\omega$\\ & $A,~A^2=A^{-1}$ & $U_A$\\ & $EAE^{-1},
~EA^2E^{-1}$ & $EU_A$\\ & $E^{-1}AE, ~E^{-1}A^2E$ & $E^{-1}U_A$\\ &
$AEA^{-1},~AE^{-1}A^{-1}$ & $AU_\omega$\\ & $Af_{2,3}Ef_{2,3}A^{-1},
~Af_{2,3}E^{-1}f_{2,3}A^{-1}$ & $A f_{2,3} U_\omega$\\
\hline
\multirow{7}{*}{$(O_5,~5)$} & $
T^p$ & $U_T$ \\
& $f_2T^pf_2$ & $f_2 U_T$ \\
& $ET^pE^{-1}$ & $E U_T$ \\
& $E^{-1}T^p E$ & $E^{-1}U_T$ \\
& $Ef_2T^pf_2E^{-1}$ & $Ef_2U_T$\\ &
$E^{-1}f_2T^pf_2E$ & $E^{-1}f_2U_T$\\
\hline
\hline
\end{tabular}
\end{math} \end{center}
\caption{List of all the non-trivial elements of $A_5$. The last column gives
the list of diagonalizing matrices for the corresponding elements which are used as the residual symmetries of neutrino and/or charged
lepton mass matrices. The $T^p$ collectively denotes a list of four elements
$T^p=(T,T^2,T^3,T^4)$ while $a=1,2,3$~.} \label{elements} \end{table}
Here $U_\omega$ diagonalizes $E,E^2$ and  is  defined in Eq. (\ref{uw}). The unitary matrices
$U_A$, $U_T$ and $U_H$ respectively diagonalize $(A,A^2)$, $T^p$ and $H$ and are given by
\beqa\label{uAuT} U_A&=&\left( \ba{ccc}
\frac{i}{\sqrt{2}}&-\frac{i}{\sqrt{2}}&0\\
\frac{\mu_-}{\sqrt{6}}&\frac{\mu_-}{\sqrt{6}}&-\frac{\mu_+}{\sqrt{3}}\\
\frac{\mu_+}{\sqrt{6}}&\frac{\mu_+}{\sqrt{6}}&\frac{\mu_-}{\sqrt{3}}\\
\end{array}\right)~,~~U_T=\frac{1}{\sqrt{2}}\left( \ba{ccc} 1&1&0\\ x \mu_ -&x^*
\mu_-&-\frac{\sqrt{2}\mu_-}{(1+\mu_-^2)^{1/2}}\\ -x (\mu_- -1)&-x^* (\mu_-
-1)&-\frac{\sqrt{2}}{(1+\mu_-^2)^{1/2}}\\ \end{array}\right)~\nonumber \\
U_H &=&\left(
\ba{ccc}
0&-\frac{\sqrt{3}}{2}&\frac{1}{2}\\
\frac{\mu_+}{\sqrt{3}}&\frac{\mu_-}{2\sqrt{3}}&\frac{\mu_-}{2}\\
-\frac{\mu_-}{\sqrt{3}}&\frac{\mu_+}{2\sqrt{3}}&\frac{\mu_+}{2}\\
\end{array} \right) U_{12}~. \end{eqnarray}
Here $U_{12}$ denotes an arbitrary unitary rotation in the $12$ plane arising due to
degeneracy in two of the eigenvalues of $H$ and
$x=\frac{\lambda+1}{\lambda-1}$ with
$\lambda=\frac{1}{2}(\mu_-+i\sqrt{4-\mu_-^2})$. All the non-trivial elements of
$A_5$ given in the Table are expressed in the form $QPQ^{-1}$ with
$P=E,E^2,A,T^p,H$ and $Q=I,E,E^2,f_a,Ef_a,E^2f_a,Af_a$. This simplifies
diagonalization of
all the elements since $U_{QPQ^{-1}}=QU_P$. This allows one in principal to
calculate all possible $U_{\rm PMNS}$ in $A_5$ analytically in terms the
diagonalizing matrices of $E,A,T,H$. In particular, matrices diagonalizing $A$,
$T$, $E$ and therefore all elements in $O_5$, $O_3$ are seen to have
$\mu$-$\tau$~} \def\mnu{{\cal M}_{\nu } symmetric form given in Eq. (\ref{gen-vl}).

redpajama_set_name: RedPajamaArXiv


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Jul 1, 2008 . With today's recycling culture, sorting is surely set to increase. Not all companies can transport the waste to their own plants in order to separate it sometimes this work needs to be done on site. Mobile sorting machines are therefore a must, and one company that is leading the way in this field in waste.


Every active LottoSpring Syndicate player gets access to all our awesome marketing materials to help build their team.
Some of the most powerful tools we created for you are our special landing pages. They all have a different approach, purpose and target audience, but with the same objective to convert your prospects into active payers.
There's no need for any complicated instruction on how to use our landing pages because they are all automated. All you need to do is to share your personal link to any of our landing pages with your social circles and our system does the rest. Depending on the page your prospects land on we do our best to convert them into active Syndicate players by showing them a video, explaining LottoSpring Syndicates, sending them a sequence of educational emails or even giving them special offers. When your prospects join a Syndicate through your personal landing page they automatically join your team and earn you free rewards.
All you need to do is copy the generic URL and put "/?ref=your username" (username is the player's name you used when registered an account with LottoSpring) at the end of it.
Here's a list of landing pages you can share, their short description and example of referral link (username used in this example is bob).
LottoSpring's registration page has all the necessary features about our Syndicates, the benefits of becoming a LottoSpring member and a registration form to create a playing account to join the LottoSpring community.
Share your personal link with your social circles and when someone registers an account and buy a subscription, they will be in your direct referral in our system.
Our LottoSpring landing page is a great page to share with people you might think are interested in a fun and easy way to play lottery games and win every week. On this page, your prospects will see our short but informative LottoSpring video, which explains how the LottoSpring syndicate system works. This page also has a lot of other content explaining every aspect of LottoSpring's benefits, problems LottoSpring solves and the reasons why it's worth joining a LottoSpring syndicate.
When somebody you referred clicks the"Join / Play Now" button they're redirected to the registration page where they leave their name and email address. That means that even if they don't join LottoSpring straight away they will get a sequence of automated emails on your behalf with a special offer at the end.
And, if/when they register with LottoSpring and buy a subscription they'll join your team.
This landing page is designed for people who are interested in a Business Opportunity. Your prospects will see a generic Business Opportunity video, without getting any details of our business. It's ideal for driving cold, Business Opportunity traffic through online advertising or social media. When your prospects opt-in for more information they'll automatically be redirected to the full ProSpring Opportunity landing page where they get more information about the business.
Share this link to qualify people interested in a Business Opportunity and we will turn them into your business partners by sending them cleverly designed emails and giving them an irresistible offer.
This landing page is created for an audience who is ready to find out about our ProSpring and ProSpring VIP business opportunity. This page gives your prospects the full ProSpring opportunity presentation – both by video and information on the web page. It's fully automated, so as soon as someone fills in the form they start getting emails with useful information to convert them into your business partners.
When your prospects sign up and buy a subscription through your personal page they automatically join your team.
This is a page created for pure fun and enjoyment! Everybody can register with Giveaway Spring and enter our free sweepstake competitions to win a BMW Series 1 Sports Car and lots more amazing prizes. Plus, prospects who register here also get special deals from LottoSpring, which are only available for Giveaway Spring members.
Everyone who joins Giveaway Spring will start getting updates about upcoming competitions; newsletters with interesting news and emails from LottoSpring with special deals and last offers.
This is a great lander to share with all your friends to give then free entries to win prizes. When they do they will also get juicy promotions to join LottoSpring. It's easy to promote this page because even if they don't join LottoSpring they will still be entered into competitions to win great prizes, including a car. They literally have nothing to lose! However, if they do join LottoSpring using your link then they will automatically join your LottoSpring team.
If you have a registered account with LottoSpring you will find the list of all the landing pages you can share in your back office under the "Marketing Tools" tab. Each one has a name, short description and link address for sharing.
Log in to your playing account, copy your personal referral link to any landing page you want to share and send it to your friends using social media, any communication channel, email, Skype or messenger.
What can be easier than just copy/paste and send a link to your friends in order to start playing and winning for free?


Absolutely Unlimited Bandwidth sounds exhilarating, elating, and most of all an ideal deal you could ever find! But if this unlimited is really limitless?
When searching and comparing for a best web hosting service, the term UNLIMITED Bandwidth sounds music to ears, but it is very important for any website owner to be very vigilant while finding out different packages by any host provider and limitations of "unlimited" tag.
Unlimited bandwidth is one of the biggest marketing ploys since 'unlimited e-mails' used by web hosting companies to grasp attention and unfortunately many falls for it. You know that bandwidth is a commodity. The quality of that bandwidth, as well as the amount being provided; are both basic factors, needed to be considered while choosing 'bandwidth provider' to host your sites are both.
An average web page is 30-50K with an average of 50,000 – 70,000 pageviews per month. In Today's world, the average gigabyte of bandwidth on a shared hosting plan ranges from $1.50 to $20.00 (depending on where you're data center is located). If a "high bandwidth" site (i.e. 200 Gigs per month) were to sign up for an "Unlimited Bandwidth" plan, it would end up costing the web hosting company anywhere from $300.00 to $4000.00 to maintain these "high bandwidth" sites. This range far exceeds the average web hosting plan monthly fee (avg. is about $19.95). Apparently, that is not possible.
Ultimately, these high bandwidth websites running on unlimited bandwidth plans get disconnected without any prior notice and with no refunds, as the host conveniently declares the violation of Acceptable Use Policy or Terms of Service. In short, these plans are not really 'unlimited', but they're throttled by the capacity of the box and the link to the Internet.
Bottom line, if you find a host providing 'unlimited' bandwidth then consider it as a warning sign instead of getting super excited as you may not be getting a straight deal. Also, whenever you visit a site promoting "Unlimited Bandwidth" as one of the account features, be sure to visit the Acceptable Use Policy, or the Terms of Service. Read the fine text about the so-called "Unlimited" disclaimer in order to know the LIMIATION OF UNLIMITED!


"And now, how shall I cook you? Shall I make an omelet? No, it is better to fry you in a pan! Or shall I drink you? No, the best way is to fry you in the pan. You will taste better."
"Many, many thanks, indeed, Mr. Pinocchio, for having saved me the trouble of breaking my shell! Good-by and good luck to you and remember me to the family!"
With these words he spread out his wings and, darting to the open window, he flew away into space till he was out of sight.
"The Talking Cricket was right! If I had not run away from home and if Father were here now, I should not be dying of hunger. Oh, how horrible it is to be hungry!"
And as his stomach kept grumbling more than ever and he had nothing to quiet it with, he thought of going out for a walk to the near-by village, in the hope of finding some charitable person who might give him a bit of bread.


In one of my environments there is an executive decision that's enabled the announcing of meeting joiners and leavers. I find this very disruptive when folks drop in and out, not to mention the slight latency in the announcement.
You can disable this announcement within Outlook.


Ergonomic design, easy to plug.
USB 3.1 Type-C interface design allows for reversible plug in so users need not worry about plugging in the wrong way.
Aluminum casing minimizes electromagnetic(EM) interference.
No proprietary driver needed, even for Battery Charging Function.
Gereral support across al major OS and platforms that offer USB such as PC, MAC, Linux, etc.
Built-in HDMI / VGA interfaces makes it perfect for business trips.


text:
Double is a 1990 bilingual French / English double album by Canadian singer Roch Voisine as a follow up to the album Hélène. The first CD is all in French, the second all in English. The English tracks were also released as a separate album entitled Roch Voisine.

Album information
The album was recorded in Canada and France, and was mixed in France. Famous singer-songwriter and guitarist Francis Cabrel participated in one of the tracks of the second album.

In France, the album spawned three top three singles : "La berceuse du petit diable" (#3), "Darlin'" (#2), and later "La promesse" (#3) and "Avec tes yeux (Pretty Face)" (#3). The last two singles were released to support Voisine's live album Europe tour, but originally appears on this album.

Double debuted at #4 on 6 December 1990 on the SNEP Albums Chart and had a peak at number two for two weeks. It totaled 26 weeks in the top ten and 73 weeks in the top 50. In 1992, the album earned a 2 x Platinum disc for over 600,000 copies sold.

Track listing

Disc 1 : French
 "Prélude" (Beaulieu) — 1:51
 "Les Jardins de St-Martin (Princesse)" (Voisine) — 3:07
 "La promesse" (Lefebvre, Voisine) — 2:28
 "Elle a peur des hirondelles" (Campbell, Lefebvre, Lessard, Voisine) — 3:48
 "Tu ne sauras jamais" (Lefebvre, Voisine) — 3:59
 "Darlin'" (Ciciola, Decary, Izzo, Voisine) — 4:15
 "Au bout de l'île" (Lefebvre, Voisine) — 2:59
 "Pour toi" (Albert, Bourquin, Gasté) — 5:03
 "Blue Love Sue" (Decary, Voisine) — 3:02
 "La Berceuse du petit diable" (Little Devil's Lullaby) (Decary, Voisine) — 4:03
 "Bye-Bye" (Lefebvre, Voisine, Vollant) — 4:06

Disc 2 : English
 "On the Outside" (Campbell, Voisine) — 4:03
 "Waiting" (Voisine) — 3:28
 "Mountain Girl" (Lessard, Voisine) — 3:37
 "A Fishing Day" (Voisine) — 4:09
 "My Fairy Tale" (Campbell, Voisine) — 2:58
 "She Had a Dream" (Voisine) — 3:32
 "Until Death Do Us Part" (Francis Cabrel, Campbell, Voisine) — 3:17
 "Jamie's Girl" (Campbell, Voisine) — 2:25
 "Pretty Face" (Voisine) — 3:16
 "Helen" (English version) (Lessard, Voisine) — 3:43
 "All Wired Up" (Voisine) — 4:00

Personnel
 Musicians
 Backing vocals - Adessa, Sembele Assitan, Bobby Helms, Bruce L. Johnson, Carole Frédericks, Elise Duguay (tracks: 2.02), Georges Costa, James Campbell, Janiece Jamison, Kat Dyson, Kathleen Dyson, Kim Richardson, Norman Groulx, Richard Groulx, Roch Voisine
 Bass - Allan B. Abrahams, Bernard Paganotti, Denis Labrosse, Pierre Duchesne, Sylvain Bolduc
 Cello - Alain Aubut
 Drums - Denis Toupin, Dominique Messier, Peter Barbeau, Richard Provençal
 Guitar - Christian Leroux, Carl Katz, Christian Péloquin, Denis Forcier, Denis Lable, Francis Cabrel, James Campbell, Jean-Marie Benoit, Michael Pucci, Patrice Tison,  Rick Haworth, Roch Voisine, Réjean Bouchard, Réjean Lachance
 Keyboards - Aldo Nova, Luc Gilbert, Marc Beaulieu
 Percussion - Denis Benarrosh, Denis Toupin, Marc Chantereau, Paul Picard
 Piano - Marc Beaulieu, Scott Price
 Strings - Bernard Paganotti, Bertrand Lajudie, Marc Beaulieu, La Philharmonie de Paris, Le Quatuor de la Philharmonie de Paris
 Viola - Gérard Daigle
 Violin - Christian Prévost, Stéphane Allard

 Recording and photo
 Engineer - Philippe Laffont (strings, vocals and background vocals recording), Claude Pons (vocals and background vocals recording)
 Mixing - Manuel "Manu" Guiot, Serge "Bum-Bum" Pauchard
 Mixing assistant - Bruno Sourice
 Recording - Gaétan Pilon, Michael Delaney
 Recording assistant - Christophe Jauseau, Luc Pellerin, Stanislas BC
 Arranged by Aldo Nova, Bernard Paganotti, Francis Cabrel, Jean-Marie Benoit, Luc Gilbert, Marc Beaulieu, Roch Voisine, Scott Price
 Photo - Tony Frank
 Producer - André Di Cesare, Roch Voisine

References

External links
Roch Voisine Official site album page

1990 albums
Roch Voisine albums

redpajama_set_name: RedPajamaWikipedia