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"Dunkirk" and The Week in Reviews
[I review the new movie "Dunkirk" as well as DVDs "Patriots Day" and the new Netflix original film "To The Bone." The Book of the Week is "The Cake and the Rain," a memoir by songwriter Jimmy Webb. I also bring you up-to-date with "My 1001 Movies I Must See Before I Die Project" with "Stairway to Heaven" (orig. title "A Matter of Life and Death."]
During World War II, by the end of May, 1940, the Nazis had driven British and French forces to the beach near the French village of Dunkirk and 400,000 soldiers waited anxiously on the beach to be evacuated.
Since the United States didn't enter WW II until after Pearl Harbor in 1941, the Battle of Dunkirk is not a WW II event that most Americans are familiar with. But Britain and France had been fighting the Germans since 1939, so it is vividly remembered by the British, whose soldiers, after fighting in France, were driven back by the Nazis and were stranded on the beach like sitting ducks as enemy fighters pummeled them with bombs and bullets. And the event is vividly remembered and stunningly recreated here by director Christopher Nolan (he also wrote the screenplay), who has already stunned us with "Interstellar," The Black Knight Rises," "Inception," "Memento" and other wonderfully smart and beautifully conceived films.
The story of the Battle of Dunkirk is told in three parts: from the vantage point of Tommy (newcomer Fionn Whitehead), a young British infantryman on the beach with his comrades waiting to be rescued; from the vantage point of Mr. Dawson (Mark Rylance), his son, Peter (Tom Glynn-Carney) and Peter's friend, civilians taking their own boat from Dover across the English Channel to help evacuate the soldiers; and Farrier (Tom Hardy), one of the ace Spitfire pilots sent to protect the evacuation.
So there is the action on the beach as Commander Bolton (Kenneth Branagh) and Captain Winnant (James D'Arcy) try to evacuate the stranded soldiers and get them on board boats and ships while at the same time dodging German bullets and bombs; there is the human story of Dawson picking up a shell-shocked soldier (Cillian Murphy) who doesn't want to go back to Dunkirk; and there is the action in the air with Farrier and his cohorts dog-fighting with the German Luftwaffe as they try to protect the soldiers on the ground.
It all adds up to a terrifically tense bit of filmmaking and a remarkable story.
There are some filmmakers that no matter what the subject matter can make a riveting film. Director Christopher Nolan is just such a filmmaker. I am not a huge fan of war films but from the opening scene when some British soldiers are seen walking down the lane of a quaint and quiet French village, and suddenly leaflets rain down upon them, leaflets from the Germans telling them they are surrounded to the final scene when our pilot, Farrier, commits to a heroic act, I was riveted.
There are also some amazingly emotional scenes. As the British civilian boats arrive at Dunkirk to help evacuate the soldiers, Captain Winnant asks Commander Bolton what he sees. He replies "Home." I lost it.
Near the end of the film when Tommy reads Churchill's famous speech:
"...we shall fight on the beaches,
we shall fight on the landing grounds,
we shall fight in the fields and in the streets,
we shall fight in the hills;
we shall never surrender..."
I lost it!
And at the very end of the film, I completely lost it.
From the editing to the cinematography to the acting, this film was perfect and will stand as one of the great war movies. And it wasn't too long, either, which is often an ego thing with directors that I often don't appreciate. It plays for less than two hours and it is white knuckles all of the way. Nolan likes to play with time and this film is no exception but it's not confusing as that device can often be.
As you probably already know, Tom Hardy is my guy. He is one of our preeminent actors who can do anything. Here as the ace pilot Farrier, despite the fact that his face is covered by an oxygen mask for almost the entire film, you knew what his character was thinking. He acted with his eyes.
Rylance also proves why he deserved an Academy Award in 2016 with another stunning performance as Mr. Dawson, the epitome of British stoicism as he calmly drives his boat toward Dunkirk and danger. Speaking of which, after what the British went through in WW II and being able to come out the other side, and after seeing this, all I can say is "Don't mess with the British!" They are a tough lot.
Oh, and for you fans of One Direction, Harry Styles acquitted himself well.
Even if you don't usually like war films, this one is a must see.
Rosy the Reviewer says...you know that when I have a really good movie experience, I cry, right? Did I cry after seeing this? I bloody well did! Academy Award nomination FOR SURE!
***Some Movies You Might Missed***
(And Some You Will Be Glad You Did)!
Patriots Day (2016)
Dramatic reenactment of the 2013 Boston Marathon bombings and the hunt for the perpetrators.
Mark Wahlberg seems to have made a career out of these dramatizations of real life events, if "Lone Survivor" and "Deepwater Horizon" are any indication (both also collaborations with Director Peter Berg, who directed this). I would like to see him in a one-to-one drama without any action for a change. But he is particularly believable here as a fictional Bah-ston cop, Tommy Saunders, because, well, Wahlberg is a real-life Bostonian.
The movie starts by setting up the various characters - first responders and runners - some of whom will be heroes and some of whom will be victims. Then we meet our young terrorists, the Tsarnaev brothers (Themmo Melikidze, Alex Wolff), and all of these people are on a collision course that will lead to death and a manhunt. Though there is not a lot of suspense here, because we all know how it turned out, the intensity lies in the seemingly benign and safe atmosphere of a marathon race as everyone participates and goes about their day while we wait for the bombs to go off.
Likewise, there is suspense in the aftermath as the cops and the FBI try to find the perpetrators, and we discover things we didn't know about them and their bombs. The damage those pressure cooker bombs could inflict was horrifying. However, I take issue with how long the film lingered on the whole chaotic period of evacuating the victims. We get it already. How many bloody legs and feet do we need to see to get the fact that this was a horrible event?
There was also suspense in wanting to find out more about the Tsarnaev brothers. In fact, I hate to say it, but I cared less about Wahlberg's character and the other "good guys" (the story went back and forth between what was happening with the victims and the police) and more about the Tsarnaev brothers, not because I am sympathetic to their cause, but because they were more interesting characters, especially the younger brother, Dzhokhar (Wolff) who was a seemingly reluctant participant but wanted to do what his older brother, Tamerlin (Melikidze), expected when all he really wanted to do was hang with his college friends and smoke dope. And it was also fascinating to see how the police and FBI were able to ID the brothers so quickly.
The film plays like a dramatized documentary and uses real footage from the day. Remember "Unsolved Mysteries," where actors would act out the scenario? This film felt very much like that except with A-list actors. So as a recreation of events, the film was effective. but when it came to Wahlberg's character, not a real person but a composite, he just seemed extraneous because his role wasn't really established early on, and I was totally distracted by his eyebrows which just scream that he has had plastic surgery.
The strength of this movie and the most interesting part of the film was the aftermath and how the police zeroed in on the perpetrators in the crowd.
"He was the only one who walked the other way."
Also I learned things about the brothers that I didn't know: what they did after the bombings and what their plans were (to do more bombings in New York City). There is a scene where they carjack a car from Dun Meng (Jimmy O. Yang), a young Chinese man and they take him along with them. It was his escape from the Tsarnaev brothers that called attention to them and led to the final shootout. Had that not happened, they might have gotten away with it.
It's an all-star cast consisting of mostly cameos. Michelle Monaghan, who I really like, has little to do as Tommy's wife; J.K. Simmons is not his usual hard ass bombastic self, but again, I couldn't help but wonder what he was doing there and Kevin Bacon as one of the FBI guys also shows up with not much to do.
John Goodman plays Commissioner Ed Davis and, oh gee, I feel a rant coming on. Goodman seems to be one of those actors who started out in a TV sit-com, became a movie star and now takes himself VERY seriously. Here he is particularly overdoing it as he huffs and puffs around acting officious and Commissioner-like, sticking his chest out and trying to sound like he is from Bah-ston when in fact his Boston accent stinks. Sorry, John.
Directed by Peter Berg with a screenplay by Berg, Matt Cook and Joshua Zeturner, I liked this film much better than I thought I would but not because of the all-star cast. It was those two brothers who fascinated me as I tried to understand what drove them to do what they did. And just what did Tamerlin's wife know? That's something we may never find out.
The film ended on a flagrantly sentimental, and actually jarring note, which I could have done without, when Mark/Tommy gets to make a speech about how terrorists will never take away our way of life and the love we have for our families followed by a long montage showing the real life first responders, the victims and how they are doing, followed by the Boston Strong message. The ending was very much a memorial to the victims of the Boston Marathon bombings, I get that, but it took away from a lasting impression of the drama itself.
Rosy the Reviewer says...a mostly well-done docudrama with a strong message: good will always defeat bad. I hope that's true.
To the Bone (2017)
Lily Collins plays Ellen, a young woman struggling with anorexia.
The film starts with a disclaimer that some of the pictures may be challenging for some viewers, which as I watched the film I found very strange. Yes, the young people in the film are very, very skinny, and we see rib cages and backbones sticking out, but I don't remember seeing such a disclaimer in any films that showed violence and destruction - no such disclaimer for "Patriots Day," for example, which lingered so long on bloody limbs and faces (see review above), so I thought that was very odd.
Anyway, based on writer/director Marti Noxon's own experiences with anorexia, this Netflix original movie (now streaming on Netflix) follows Ellen as she unwillingly undergoes treatment for anorexia. She is not only begrudgingly seeking treatment, she is in fact, defiant about not changing.
She returns home to live with her stepmother, Susan (Carrie Preston) half-sister, Kelly (Liana Liberato), with whom she is close, and her father. It is established early on that Ellen does not approve of her stepmother whose well-meaning attention annoys Ellen, but Ellen eventually learns that her step-mother has her back. Ellen's father is clearly a big force in Ellen's life, but he is never home and in fact never appears in the film. Ellen's mother came out when Ellen was 13 and basically abandoned her and lives a new age type of life on a horse therapy farm with her girlfriend in Phoenix and feels she can't cope with Ellen. It's a self-preservation thing, I guess.
After Ellen is kicked out of the latest treatment facility for being defiant, Susan hits upon yet another treatment center run by a Dr. Beckham (Keanu Reeves) who is known to use unusual but effective treatment methods, so off Ellen goes, with her usual negativity, to another rehab place.
But we see early on that this place is different. It's a big house with only six other people, five girls and a young man. Luke (Alex Sharp), who was once a dancer but an injury has curtailed his career, is obsessed with Raymond Chandler and Jonathan Gold's restaurant reviews. The other girls all have their own issues and eating fetishes. One is pregnant, one is obsessed with peanut butter, another loves "My Little Pony." This potpourri of troubled youth gathered together reminded me of the Angelina Jolie film, "Girl, Interrupted."
And then we meet Dr. Beckham - Keanu Reeves. OK, sorry, it's rant time.
Keanu has always been a problem for me as an actor. I have never understood his appeal or how he became an A-lister. He is so phlegmatic most of the time as an actor that I feel like I need to take his pulse to see if he is really alive, but then when he wants to be really dramatic, he does a 360 and starts yelling. So that is his range. Barely alive with the occasional shouting thrown in. Nothing in between. And that's how he plays every character.
Anyway, rant over. Moving on.
This film depicts the world of the anorexic: the daily rituals to avoid eating, memorizing the exact calorie count for every food, talking about which foods come back up the easiest (ice cream) and starvation as a high to avoid feeling.
"It's not about thin enough - it's the numbing of the thing you don't want to feel."
But the film also depicts how the disease affects the entire family and the well-meaning but ineffective things people say when confronted with something they don't understand. Word to the wise. Saying "Be strong" to someone struggling with a disease is not helpful.
An Anne Sexton poem is a particularly moving centerpiece to a turning point in the film:
"Your courage was a small coal you kept swallowing..."
Collins, the daughter of singer/musician Phil Collins, who was part of the rock band Genesis, already has a Golden Globe nomination under her belt ("Rules Don't Apply") and is a lovely actress - those eyebrows - and she is the centerpiece of this film. She has stated in interviews that she too has struggled with an eating disorder so it was particularly challenging to have to lose weight for this film.
Despite what might seem like depressing subject matter, this is a wonderful film that brings understanding to a disease that is rarely shown in films. It's also a film that you won't be able to take your eyes off of, mostly because of Collins' extraordinary performance. As for Keanu, well he's always Keanu.
Rosy the Reviewer says....an uplifting message: No matter what we are struggling with, in the end we all have to find our own way. I teared up and you know what that means.
***My 1001 Movies I Must See Before I Die Project***
191 to go!
Have YOU seen this classic film?
Stairway to Heaven Orig. Title: "A Matter of Life and Death" (1946)
You've seen the films and read the stories of people having to plead their case to the devil to stay out of hell? Well, this time a man must plead his case to God to stay out of heaven.
There are several movies directed by Michael Powell in the "1001 Movies You Must See Before You Die" book and there is a reason. They are beautiful, lush films in Technicolor with dramatic scores, extreme closeups and lots of fantasy. They are the kinds of films I grew up with watching with my Dad so I have a particular soft spot in my heart for these kinds of films.
Michael Powell and Emeric Pressburger wrote, directed and produced films under the name "The Archers," and they made 24 films together from 1939 to 1972. They shared writing duties, but Powell did most of the directing while Pressburger mostly produced. Their most famous film is probably "The Red Shoes," but they were very influential during the 1940's and 1950's with such films as "The Life and Death of Colonel Blimp," "Black Narcissus" and others. Originally titled "A Matter of Life and Death," this film was renamed "Stairway to Heaven" for the American market.
The film opens with...
"This is the story of two worlds the one we know and another which exists only in the mind of a young airman whose life and imagination have been violently shaped by war -- any resemblance to any other world known or unknown is purely coincidental."
Here David Niven stars as Peter Carter, a WW II Royal Air Force aviator whose plane has been hit by enemy fire, and he is going down. His crew has bailed out or are dead, and he has no parachute. He makes contact with June (Kim Hunter), an American radio operator at the military base, and the two share a moment before he plans to jump from his plane. He spouts poetry and makes witty wise-cracks in the face of death which was a common part of showing bravery in these old WW II movies. Then he jumps.
Miraculously, though (and I mean that literally), he wakes up in the sea -- and he's alive! Turns out there was a hiccup in heaven, and the angel, who was supposed to go get him and escort him to heaven, couldn't see him because of the fog. Well, that's his story anyway, and Peter escaped his time of death. Now heaven has to scramble to figure out what to do about it, because, as we all know, in filmdom, God has a file on all of us, and when it's our time, it's our time.
According to this film, Heaven is also a clean black and white place where there's a Coke machine, lots of wise-cracking airmen and soldiers who are all issued their wings - literally, their angel wings.
In the meantime, wouldn't you know, Peter picks himself up and heads for the road and meets, guess? Why it's June, who is riding her bike home from work. Because of "that moment" they shared on the radio, they are instantly in love with each other. Like I said, I love these old movies.
Now heaven really has a problem. Because of those unaccounted for hours when Peter was supposed to be dead, the two have fallen in love and that can't be discounted. Peter now has made a commitment to June.
So God sends Conductor 71 (Marius Goring) to fetch Peter. He is a French fop of an angel who is the one who missed Peter and screwed this all up in the first place.
Remember I said Heaven is depicted in black and white? Well, the "real world" is in color, kind of like how they worked "The Wizard of Oz." So when Conductor 71 arrives on earth to find Peter, he remarks, "We were starved up there for Technicolor." Then to really make the point, there is a whole montage of beautiful colored flowers blooming all over the place as the camera discovers Peter and June making out. Again, I love these old movies.
The Frenchman finds Peter and tells him a mistake has been made. He was supposed to escort him to heaven when he died from jumping from his plane, but he missed him because of the fog but now he must go with him to heaven. However, Peter says, "No way!" He makes his case as to why he should stay alive, so a deal is struck. Peter must go on trial in heaven to plead his case as to why he should stay alive. He can choose any dead person to defend him. Unfortunately, the prosecutor is Abraham Farlan (Raymond Massey), an orator from the American Revolution and, naturally, he hates Brits.
While all of this is going on, everyone on earth thinks Peter has gone nuts because he keeps seeing visions - mostly that French guy. June takes Peter to see Doctor Reeves (Roger Livesey) and brain surgery is prescribed. But before the surgery can be performed, Dr. Reeves is killed. Now Reeves is in heaven, and Peter chooses him to defend him.
At the trial, Reeves argues that, through no fault of his own, Peter was given additional time on Earth and that, during that time, he has fallen in love and now has an earthly commitment that should take precedence over the afterlife's claim on him.
And let's just say, love triumphs. Doesn't it always in the old movies?
"Nothing is stronger than the law in the universe, but on Earth, nothing is stronger than love."
Then we hear that Peter's brain surgery was a success.
Mmm - so did all of that really happen? What was real and what was all in Peter's mind?
Relatively early in Niven's career as a leading man, he was already the charming wise-cracker we came to know and love. This was an early role for American actress Hunter, too, and she went on to be a staple on TV dramas such as "Playhouse 90" and "General Electric Theatre."
A very smart film, it's a comment on war, wartime love affairs and a satire on English and American relations, and to get the jokes and the allusions, it helps to know something about history and literature. The set design is stunning and the whole film is visually beautiful and stimulating.
It's interesting to note that these old films are often of their time when it comes to outdated speech and mores, but this was a British film, and I find the British and European films from the 30's, 40's and 50's to be much edgier and contemporary than what Hollywood was putting out, probably because of the Motion Picture Production Code which was implemented in the U.S. in 1934 and which was basically censorship in the guise of "moral guidelines" for films.
Why it's a Must See: "...intended as a propaganda film to ameliorate strained relations between Britain and America...The movie outstrips its original purpose...ending up a lasting tale of romance and human goodness that is both visually exciting and verbally amusing."
---"1001 Movies You Must See Before You Die"
Rosy the Reviewer says...a wonderfully fun film experience. They don't make 'em like that anymore!
The Cake and the Rain: A Memoir by Jimmy Webb (2017)
If you ever wanted to know what the hell the song "MacArthur Park" was about and why the cake was left out in the rain, then here's your chance!
Webb was a prolific songwriter starting in the 1960's. If you loved a ballad back then, he probably wrote it. His memoir includes 17 single-spaced pages of songs he wrote and it's only a partial list! He was also the only artist to win Grammys for music, lyrics and orchestration and was hugely successful and rich by the time he was 26. In addition to "MacArthur Park," he was responsible for "Wichita Lineman," "Up, Up and Away," "Galveston" and countless other hits.
However, his success was a double-edged sword. He became known for his middle of the road pop hits at a time when the Rolling Stones and other rock groups were the anthems of the young. It's a little hard to be cool with the young in-crowd when Frank Sinatra was singing your songs in Vegas and the young people were listening to the Grateful Dead at big concert venues. He also had a personal identity problem. He was a pot-smoking, cocaine-snorting, young hippie writing songs for the straight folks.
Born Jimmy (not James or Jim), Webb's father was a self-proclaimed preacher who moved his family around the country. Webb grew up poor so when his songwriting career took off and the money started pouring in, he took off, too, into the world of sex, drugs and fast cars and it almost killed him.
This memoir covers Webb's life from 1955 to the early 1970's, and since he has had a 50 year music career, I would guess there will be another book. But this one covers his rise to fame, his inability to deal very well with lots of money at a young age, his drug use, and his penchant for married women which could explain why many of his songs are so sad. But he must have had something, because he always seemed to get the beautiful ladies his heart desired, including an early love who became "Miss America." He also shares anecdotes about rubbing elbows with some of the great singers and musicians of the day (The Beatles, Harry Nilsson, Janis Joplin) and includes some great behind-the-scenes stories about recording "Galveston," "Up, Up, Up and Away," and others.
Webb also shares his singing career. What is it about actors who want to be directors, comedians who want to be dramatic actors and songwriters who want to sing, songwriters whose voices are not that great? Burt Bacharach comes to mind...and Jimmy Webb. He shares his attempts at a singing career and is a bit self deprecating about it, though I will say, Webb's ego always seems to shine through.
Speaking of his ego, most of the bad stuff he did or bad decisions he made, he blames on the devil who actually plays a real role in the book. Though I found his story interesting and Webb is a good writer, I have to say that the conversations he has with the devil in the book, basically blaming his "bad side" on the devil - "The devil made me do it" as Flip Wilson used to say - was really annoying. About page 50 I was wishing that old Beelzebub would go away but he never did.
This is the story of the rise and fall of one of our great American songwriters and since the book ends with a drug overdose, I am assuming there will be another installment that will show him rising from the ashes of addiction.
But for this one, just remember the Devil made him do it.
Rosy the Reviewer says...If you enjoy reading about the musicians and the antics that characterized the 60's and 70's, you will enjoy Webb's experiences. But don't call him Jim or James. It's Jimmy!
"Rosy's Test Kitchen #4:
Cooking Successes and Cooking Conundrums -
"Sweets!"
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Labels: 1001 Movies , book reviews , books , movie reviews , Movies , My 1001 Movies I Must See Before I Die Project , War Movies
I loved Dunkirk! I am a sucker for historical fiction, and this was a part of history I honestly knew nothing about. Just because the U.S. didn't get into Europe until much later doesn't mean that the civilized world wasn't already fighting the Nazi's with everything they had. When I read later that this is the event that caused Churchill to proclaim, "We will NEVER surrender!", I realized how important Dunkirk was. If things had gone differently, the war in Europe might have been lost before the U.S. even got there. Two things about this movie that I loved - one, the Rylance character was supposedly based on the true story of Thomas Lightoller, who had been vilified for his role as on officer on the Titanic. Here he redeems himself by saving people he was unable to save thirty years earlier. Second, at the end of the evacuation Commander Bolton tells the Captain that he is not leaving on a boat because he is "waiting for the French"...that got me. If only now we could think of our fellow nations, and remember that we're all in this together. Sigh...
Rosy Brewer August 22, 2017 at 7:13 PM
I so agree. I didn't know that about the Rylance character. I thought all of the officers went down with the Titanic so that would explain his zeal to help. Wasn't Rylance wonderful in that role? I also cried so much at the end when Churchill's words were read...and they never did surrender. But like you, wonder what would have happened if we hadn't gotten involved...and why we didn't sooner!
"Wind River" and The Week in Reviews
"The Glass Castle" and The Week in Reviews
"Detroit" and The Week in Reviews
Rosy's Test Kitchen #4: Cooking Successes and Cook...


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Package java.rmi.registry

Provides a class and two interfaces for the RMI registry.

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Interface
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Registry

Registry is a remote interface to a simple remote
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 remote object references bound with arbitrary string names.

RegistryHandler
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no replacement

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LocateRegistry

LocateRegistry is used to obtain a reference to a bootstrap
 remote object registry on a particular host (including the local host), or
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Package java.rmi.registry Description

Provides a class and two interfaces for the RMI registry.
A registry is a remote object that maps names to remote objects.  A
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text:
Many Alaskans Think Sarah Palin Now Belongs to Arizona -- Nice Try, Alaska
Matthew Hendley July 30, 2013 12:40PM
A whole lot of Alaskans seem to think that Sarah Palin is now an Arizonan, according to poll results.
Nice try, Alaskans.
Palin, who was rumored to be considering a Senate run in Arizona in 2012, when Jeff Flake became the eventual Republican nominee and was elected Senator.
She didn't end up running but currently is the top Republican pick in Alaska for that state's U.S. Senate election next year, according to Public Policy Polling.
However, a lot of the people who don't like Palin apparently deny that she's even Alaskan anymore.
According to the PPP survey of Alaskans, 47 percent say Palin's still Alaskan, and 46 percent say she's not. Those people who say she's not Alaskan apparently think Palin -- who's claimed to be a part-time Scottsdale resident for the past couple of years -- is now Arizonan.
The poll results show that 37 percent of the people thought it was "more appropriate for Palin to run for office in Arizona, rather than Alaska. Forty-one percent found it more appropriate that Mama Grizzly runs in Alaska.
(Note that just 25 percent of the people participating in the poll said they were "very liberal" or even "somewhat liberal.")
Overall, it was a good effort by Alaskans, but no thanks -- you can keep her.
Check out the poll results here.
Send feedback and tips to the author.
Follow Matthew Hendley on Twitter at @MatthewHendley.
Matthew Hendley
Contact: Matthew Hendley

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text:
Bossa Tropical é um álbum de estúdio da cantora Gal Costa, lançado em 2002 pela gravadora MZA Music. O álbum renovou o repertório da cantora, com composições de Titãs, Rita Lee e Beatles, a sonoridade prioriza mais a voz e o violão. O álbum vendeu mais de 60 mil cópias no Brasil.

Faixas

Álbuns de 2002
Álbuns de Gal Costa
Álbuns de música popular brasileira
Álbuns em língua portuguesa

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text:
Відомча заохочувальна відзнака Міністерства оборони України — нагрудний знак «За розвиток військового співробітництва» входила до діючої до 2012 року системи відзнак Міністерства оборони України.

Історія нагороди
 Відзнака встановлена наказом Міністра оборони України Юрія Єханурова від 21 травня 2008 року № 218.
 30 травня 2012 року Президент України В. Ф. Янукович видав Указ, яким затвердив нове положення про відомчі заохочувальні відзнаки; міністрам, керівникам центральних органів виконавчої влади, керівникам (командувачам) військових формувань, державних правоохоронних органів було доручено забезпечити в установленому порядку перегляд актів про встановлення відомчих заохочувальних відзнак, приведення таких актів у відповідність із вимогами цього Указу. Протягом 2012–2013 років Міністерством оборони України була розроблена нова система заохочувальних відзнак, що вже не містила нагрудного знака «За розвиток військового співробітництва».

Положення про відзнаку
 Відомчою заохочувальною відзнакою Міністерства оборони України — нагрудним знаком «За розвиток військового співробітництва» нагороджуються особи старшого та вищого офіцерського складу, працівники Збройних Сил України та інші особи за вагомий внесок у справу розвитку співробітництва у військовій сфері, підтримання миру та дружніх стосунків між збройними силами держав.
 Нагородження Відзнакою здійснюється наказом Міністра оборони України (по особовому складу).
 Подання до нагородження Відзнакою здійснюється в порядку, визначеному в Положенні про заохочувальні відзнаки Міністерства оборони України.
 Нагородженому вручаються Відзнака та посвідчення до неї.
 У разі втрати (псування) Відзнаки дублікат не видається.
 Відзнака і посвідчення до неї після смерті нагородженого залишаються в сім'ї померлого як пам'ять.

Опис відзнаки
 Відомча заохочувальна відзнака Міністерства оборони України — нагрудний знак «За розвиток військового співробітництва» виготовляється з жовтого металу і має форму багатопроменевої зірки з розбіжними променями, у центрі якої у круглому медальйоні вміщено емблему Збройних Сил України на тлі стилізованого зображення земної кулі. Медальйон обрамлено вінком з калинових, дубових, лаврових гілок і військових козацьких атрибутів. Вертикальні та горизонтальні промені зірки залито бордовою емаллю.
 Розмір відзнаки — 50×50 мм.
 Зворотний бік зірки — плоский, з написом «За розвиток військового співробітництва» і вигравіруваним порядковим номером.
 Усі зображення і написи рельєфні.
 Стрічка відзнаки шовкова муарова бордового кольору з поздовжніми смужками синього і жовтого, блакитного, чорного, зеленого та бордового кольорів з країв. Ширина синьої і жовтої смужок — по 3 мм кожна, блакитного, жовтого, чорного, зеленого та бордового кольорів — по 2 мм кожна.

Порядок носіння відзнаки
 Нагрудний знак «За розвиток військового співробітництва» носять з лівого боку грудей і розміщують після нагрудного знака «За зміцнення обороноздатності».

Примітки

Джерела
 Наказ Міністра оборони України від 21.05.2008 № 218 «Про встановлення відомчої заохочувальної відзнаки Міністерства оборони України — нагрудного знака "За розвиток військового співробітництва"»
 Відзнака Міністерства оборони України — нагрудний знак «За розвиток військового співробітництва» // Сайт Міністерства оборони України (збережено на archive.today)
 Віталій АГЕЄВ: «Створення власної системи нагород стало вагомим кроком у розвитку української державності» // Володимир КОРОТИЧ, Військо України, № 12 (102) 2008, стор. 22-25 (збережено на archive.org)

Відзнаки міністерства оборони України
Нагороди, засновані 2008
Засновані в Україні 2008


Happy 12-12 Everyone! This is numerologically a perfect day to relax and release into Universal flow, moving with synchronicity, appreciating and acting on serendipities, feeling connection with everyone and everything in the Universe. Our Solar System now also vibrates to the number 12, ever since the 12-12-12 shift. The 12-12 teaches us to go with the flow, not fight against it. It teaches us to attract that which we desire to us by magnetism and The Law of Attraction, not by force. The 12-12 lets us know that we are One with Universal flow and Universal consciousness. Look for the signs of the 12-12 in your life, perhaps on clocks, license plates, phone numbers, billboards etc. And when you see it, know that all is in divine cosmic order as you flow in synchrony with the rhythm of the Universe.
What Do You Choose: Fear or Love?
The wheel of the year turns yet again as we find ourselves deep within the holiday season. This can be a stressful, overwhelming or sad time for many of us, as we cope with time and money issues, spend more time with family members who seem to push all our buttons, or find ourselves missing those loved ones who are no longer with us in physical form. The dark, short days and long, cold nights may also give us pause, as we naturally wish to turn inward during a time when our social calendar requires us to turn outward. All of these things may set us up for physical, energetic, emotional, mental and spiritual overload. And a body under stress is a body primed for illness. Remember, illness is nature's way of making us slow down, of allowing us to recalibrate. Sometimes we can't avoid it. But often, we can do things to reduce our levels of stress and to help us stay in a healthy balance.
We often need to honor our body, mind and spirit and follow our instincts. Perhaps it may be best for us to miss that party or get-together if we are not feeling up to it. Perhaps we need to allow ourselves to say "no" to those things we really don't want to do. During this season, we seem to be pulled in many directions, often toward people and events for which we feel an obligation. When that is the case, we have three options. The first is to attend the event even when we don't want to do it. This sets us up for having a less than satisfying encounter, which adds to our stress level. The second option is to decline the invitation and to feel good and empowered about our choice, not guilty! The third choice is to reframe the way we approach an event. When we change our perspective and expectations about an "obligation" to seeing it as an opportunity for joyful interaction or a chance to shine our light, the obligation may become a joyful encounter--one that we actually look forward to! You see, it is all a matter of tweaking our perception of the event from fear to love. We don't have to reserve this manner of thinking to the holiday season. We can stand in our full power and do this all year long! What events that you are dreading or feeling obligated to attend are looming on your horizon? What if you choose to see things differently? What if you choose to be joyful and see this as an opportunity to be the living example to others, radiating your love and light, standing in your truth and power? Would that change things?
Also important is to honor our spirit. The natural rhythm of the Earth, to which we are integrally connected, is to slow down, look inward, reflect on what has come to pass, what is, and what may be. It is a time to evaluate our creations of the waning year and to dream of what we will create in the New Year that is just around the corner. Time for meditation, prayer, silent communion with nature and time well-spent with those we love best is what our souls long for at this dark time of year. Allow time for this! Our soul and spirit wants and needs this deep, reverent pause within all the commercial chaos that this season has become. Why don't we choose not to get caught up in the material whirlwind and start new traditions that make our hearts sing?
As we celebrate the return of the Light, whether it be with the Winter Solstice, Christmas, Hanukkah, or whatever beautiful tradition we hold dear, let's remember the true meaning of the holiday. Let's celebrate the Love and Light within our lives, within our families (either biological, social or spiritual), within each and every human on this planet. Let's remember those less fortunate and hold them with love and compassion, perhaps donating our time and energy so that they, too, may feel love and light this season. Each of us holds within our hearts the Light from Home. The more we radiate this light to others, the quicker and more profoundly this world will change for the better.
Take back your power, Lightworkers, Earth Angels! Stand in your full power! We always have a choice. Be the change you want to see. Be the Light you want to feel. Know that we are all exactly where we are meant to be. Be loving, be compassionate, be joyful. Celebrate the season in whatever way soothes and exhilarates your soul! We are all powerful creators with a choice. And no matter the details, the choice always comes down to this: do we choose fear or love?
Best wishes and bright blessings to all of you this holiday season and always!


Private Beach Access and Amazing Lake Views!
There is direct walking or driving access to a private beach. Access to the lake is unobstructed by other homes, with stunning views through pine trees of the lake and surrounding mountains.
This neighborhood is located approximately 1/2 mile off Highway 50, 1 ½ miles from Stateline casinos, shopping centers, and the Edgewood golf course. World-renowned Heavenly Valley Ski Resort is just 4 miles away. It is located next to the Round Hill and Zephyr Cove neighborhoods.


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

With the progressive downscaling of electron devices, whose size has reached
well into the nanoscale, the quantum simulation of charge transport has become
increasingly important. This study requires the numerical solution of
the quantum wave equation for the
charge carriers in the device.

In the case of periodic boundary conditions, the solution in the Fourier
domain represents a spontaneous alternative to a study in the direct space.
In a Fourier analysis, all the functions appearing in the equation
are replaced by their Fourier expansions, and the Fourier coefficients of
the wave function become the unknowns of the problem,
which in direct space approaches are instead the values of the wave function
at the points of a discretization grid.

The main advantage of this technique is the correct
treatment of the derivatives. In a reciprocal space analysis, each of
the complex exponential functions appearing in the Fourier expansions can
be derived exactly, without any approximation. On the contrary, in a standard
finite difference solution, the
derivatives are replaced with finite difference approximations involving a
certain number of samples of the function. This approximation introduces a
distortion in the dispersion relation, which severely limits the accuracy of
high-order solutions. The discretization error decreases if a finer grid is
used, at the expense of an increase of the computational effort.
Moreover, in some cases,
such as the solution of the Dirac equation for massless fermions,
the discretization of the equation on a direct space grid gives rise to
problems such as the appearance of spurious solutions, or fermion doubling,
unless proper, non-standard discretization techniques are
applied,~\cite{stacey1982,bender1983,susskind1977} while these issues
do not appear using a Fourier solution technique. The solution in the
reciprocal space is especially convenient when a continuum, envelope function
description of the device is adopted and a slowly varying potential
(containing only a limited number of Fourier components) is considered.

Here, we illustrate how the use of a particular set of periodic basis
functions, obtained from the periodic repetition of sinc (cardinal sine)
functions, makes it possible to obtain an exact description of the derivatives
in the direct space and a solution approach equivalent to that in the
reciprocal space. This approach allows solving the differential problem
in a very efficient way in the
direct space, without the need to switch to the reciprocal space and
vice versa.

After a general presentation (in Sec.~\ref{method}) of the method that we
propose, in Secs.~\ref{functionder} and \ref{productsec} we describe how
each term of the differential equation is treated, and in Sec.~\ref{fourier},
we demonstrate the equivalence of the sinc-based method to a Fourier
approach.

Finally, in Sec.~\ref{dirac}, we apply this method to the solution of the
the wave equation in an armchair graphene nanoribbon (described using an
envelope function approach) with a potential varying only in the transverse
direction.
Graphene represents an interesting material that, since its isolation in
2004~\cite{geim}, has been the focus of a vast theoretical and experimental
research activity~\cite{rise,status,castroneto,experimental,mikhailov,
connolly,iwceconn,acsnano,iwceroche}. Among the many applications that have
been proposed for this material, the possibility to use graphene
nanoribbons for the implementation of nanoelectronic devices~\cite{raza,
schwierz,persp} is particularly intriguing.
The effective wave equation for the envelope functions, which in common
semiconductors corresponds to the effective mass Schr\"odinger
equation~\cite{yu,mitin} (see, for example,
Refs.~\cite{epl,prl,prb,fnl,cavita}),
in monolayer graphene is represented by the Dirac equation~\cite{ando,kp}.
Its numerical solution in graphene ribbons with an arbitrary potential
landscape has been studied in a few recent papers~\cite{tworzydlo2008,
hernandez2012,snyman2008}. The solution of the Dirac equation
within a slice with a potential varying only in the transverse direction
can be exploited to perform a transport analysis for an armchair
nanoribbon with a generic potential, if we subdivide the ribbon into a series
of slices in each of which the potential is approximately constant in the
longitudinal direction, and then we apply a recursive technique to obtain the
transmission through the overall device~\cite{icnf2013}.

\section{Mathematical method}
\label{method}

For the sampling theorem, if a function $z(x)$ is band limited with band
$B\le 1/(2\,\Delta)$ (Nyquist criterion), it can be perfectly
reconstructed~\cite{haykin} from its samples taken with a sampling
interval $\Delta$
\begin{equation}
z(x)=\sum_{n=-\infty}^{+\infty}z(n\,\Delta)\,
{\rm sinc}\left(\frac{x-n\Delta}{\Delta}\right)\,,
\end{equation}
where the sinc function is defined as ${\rm sinc}(x)\equiv\sin(\pi x)/(\pi x)$.

Moreover, if $z(x)$ is periodic with period $L$ and we take $N$ samples within
a period (and thus $L=N\,\Delta$), we have that
\begin{eqnarray}
z(x)&=&\sum_{\ell=0}^{N-1}\sum_{\eta=-\infty}^{+\infty}
z((\ell+\eta N)\Delta)\,\,
{\rm sinc}\left(\frac{x-(\ell+\eta N)\Delta}{\Delta}\right)\nonumber\\
&=&\sum_{\ell=0}^{N-1}z(\ell\,\Delta)\sum_{\eta=-\infty}^{+\infty}
{\rm sinc}\left(\frac{x-(\ell+\eta N)\Delta}{\Delta}\right)\nonumber\\
&=&\sum_{\ell=0}^{N-1}z(\ell\,\Delta)g_{\ell,\Delta}(x)\,,
\end{eqnarray}
where we have exploited the periodicity of $z(x)$ and we have defined the
function (periodic with period $L=N\,\Delta$)
\begin{equation}
g_{\ell,\Delta}(x)\equiv \sum_{\eta=-\infty}^{+\infty}
{\rm sinc}\left(\frac{x-(\ell+\eta N)\Delta}{\Delta}\right)
\end{equation}
(in Fig.~\ref{g}, we represent 4 periods of the function $g_{\ell,\Delta}(x)$
for $N=15$ and $\ell=5$).

\begin{figure}
\begin{center}
\includegraphics[width=8cm]{g.eps}
\caption{Four periods of the function $g_{\ell,\Delta}(x)$
for $N=15$ and $\ell=5$.}
\label{g}
\end{center}
\end{figure}

If we define the scalar product between two functions $\psi_1(x)$ and
$\psi_2(x)$ as
\begin{equation}
\langle \psi_1(x) | \psi_2(x) \rangle=
\frac{1}{\Delta}\int_0^L\psi_1^*(x)\psi_2(x) d\,x \,,
\label{product}
\end{equation}
we can prove (see Appendix~\ref{orthonormality}) that the set of functions
$g_{\ell,\Delta}(x)$ is orthonormal,
i.e. $\langle g_{j,\Delta}(x) | g_{\ell,\Delta}(x) \rangle=\delta_{j,\ell}$
(where $\delta$ is the Kronecker delta).

Here, we consider a wave equation defined on a domain $[0,L]$
and with periodic boundary conditions. The solutions of this differential
equation (i.e., the wave functions), extended by periodicity with period
$L$, $\varphi(x)$ (with $\varphi: \mathbb{R} \to \mathbb{C}$)
are elements of the infinite-dimensional Hilbert space (with scalar product
(\ref{product})) $L^2[0,L]$ of the $L$-periodic functions with finite
norm and are continuous up to the $(\tau-1)$th derivative (if the potential
is bounded, $\tau$ being the order of the differential equation).
In order to solve the problem numerically, here we reformulate it
onto the finite-size subspace generated by the $N$ basis functions $g$ that
we have just defined. This allows to preserve the smoothness of the
wave functions and, in particular, to numerically approximate the $N$
slowest-varying solutions, the ones we are mainly interested in.
In detail, we approximate each of the terms appearing in the wave equation
as a linear combination of the functions $g_{\ell,\Delta}(x)$, with coefficients
given (in their turn) by linear combinations of the samples of the (unknown)
wave function $\varphi(x)$ at the points $n\,\Delta$ (with $n=0,\ldots,N-1$).
In order to easily arrive at closed-form analytical expressions,
in our calculations, we assume $N$ to be an odd number, i.e., $N=2\,D+1$
(with $D$ an integer). Then, we project the wave equation onto the functions
$g_{j,\Delta}(x)$, with $j=0,\ldots,N-1$. In this way, exploiting the
orthonormality of the $g$ functions, we recast the wave equation into
a system of $N$ linear equations in the $N$ unknowns $\varphi(n\,\Delta)$
with $n=0,\ldots,N-1$ (corresponding to the $N \times N$ matrix
representation~\cite{datta} of the wave equation on the chosen set of
basis functions). In Sec.~\ref{fourier}, we will prove that this method
is equivalent to a Fourier-Galerkin analysis with cut-off
at the spatial frequency $D/L$ (and it is thus characterized by the same good
convergence properties that have been proved for the Fourier-Galerkin approach,
if the involved functions are sufficiently well-behaved: see, for
example, Ref.~\cite{canuto}).

In our discussion, we will consider the following terms
of the wave equation: a term proportional
to the unknown wave function $\varphi(x)$, a term proportional to a derivative
of the wave function, and a term proportional to the product between the wave
function $\varphi(x)$ and
a function $f(x)$ (with $f: \mathbb{R} \to \mathbb{C}$, in general)
which in $[0,L]$ represents (or is a function of) the potential
energy and is then extended by periodicity with period $L$.
Other possible terms
appearing in the wave equation will undergo a similar treatment.

\section{Treatment of the wave function and of its derivatives}
\label{functionder}

We assume the unknown periodic wave function $\varphi(x)$ to be a
band limited function with a maximum band $B=D/L$, such that, sampling it
with a step $\Delta$, the Nyquist criterion $B \le 1/(2\,\Delta)$ is satisfied.

Therefore, the term proportional to the unknown wave function is easily
rewritten in the desired form by applying the sampling theorem with a
step $\Delta$
\begin{equation}
\varphi(x)=\sum_{\mu=0}^{N-1}\varphi(\mu\,\Delta)g_{\mu,\Delta}(x)\,.
\label{exp-rho}
\end{equation}

Concerning the term proportional to the derivative of the wave function
$\varphi(x)$, if $\varphi(x)$ is band limited with a band equal to $B=D/L$,
also
its derivatives have the same property, and thus we can apply the sampling
theorem to them, too, obtaining (for the generic $s$-th derivative):
\begin{equation}
\frac{d^s\,\varphi(x)}{d\,x^s}=\sum_{\mu=0}^{N-1}
\left[\frac{d^s\,\varphi(x)}{d\,x^s}\right]_{x=\mu\,\Delta}g_{\mu,\Delta}(x)\,.
\end{equation}
We now have to express the values of the derivatives in the points
$\mu\,\Delta$ as a function of the samples of $\varphi(x)$.
This can be easily achieved by deriving Eq.~(\ref{exp-rho}). We have that
\begin{eqnarray}
\displaystyle \left[\frac{d^s\,\varphi(x)}{d\,x^s}\right]_{x=\mu\,\Delta}&=&
\left[\frac{d^s}{d\,x^s}
\left(\sum_{\ell=0}^{N-1}
\varphi(\ell\,\Delta)g_{\ell,\Delta}(x)\right)\right]_{x=\mu\,\Delta}\nonumber\\
&=&\displaystyle\sum_{\ell=0}^{N-1}\varphi(\ell\,\Delta)
\sum_{\eta=-\infty}^{+\infty}
\left[\frac{d^s}{d\,x^s}
{\rm sinc}\left(\frac{x-(\ell+\eta N)\Delta}
{\Delta}\right)\right]_{x=\mu\,\Delta}\nonumber\\
&=&\sum_{\ell=0}^{N-1}\varphi(\ell\,\Delta)
\alpha_{\mu \ell}^s \,,
\label{alpha}
\end{eqnarray}
where we have defined as $\alpha_{\mu \ell}^s$ the coefficients with which we
have to combine the samples of the unknown wave function $\varphi(x)$ in order
to determine the samples of its $s$th derivative.

In particular, let us develop the calculation for $s=1$ (first derivative) and
for $s=2$ (second derivative).

For $s=1$, we have that
\begin{equation}
\frac{d}{d\,x}
{\rm sinc}\left(\frac{x-(\ell+\eta N)\Delta}{\Delta}\right)
=\frac{\cos\left(\pi\,\frac{\displaystyle x-(\ell+\eta N)\Delta}
{\displaystyle \Delta}\right)}
{x-(\ell+\eta N)\Delta}-
\frac{\sin\left(\pi\,\frac{\displaystyle x-(\ell+\eta N)\Delta}
{\displaystyle \Delta}\right)}
{\frac{\displaystyle \pi}{\displaystyle \Delta}
\left(x-(\ell+\eta N)\Delta\right)^2}
\end{equation}
and thus
\begin{equation}
\left[\frac{d}{d\,x}
{\rm sinc}\left(\frac{x-(\ell+\eta N)\Delta}
{\Delta}\right)\right]_{x=\mu\,\Delta}=\begin{cases}
\displaystyle 0 & \text{if $\mu-\ell-\eta N=0$}\\[10pt]
\displaystyle \frac{(-1)^{\mu-\ell-\eta N}}{\Delta (\mu-\ell-\eta N)}
& \text{if $\mu-\ell-\eta N \ne 0$.}
\end{cases}
\end{equation}
If we develop the calculation assuming $N$ odd, we have that
\begin{equation}
\alpha_{\mu \ell}^1=\sum_{\eta=-\infty}^{+\infty}\left[\frac{d}{d\,x}
{\rm sinc}\left(\frac{x-(\ell+\eta N)\Delta}
{\Delta}\right)\right]_{x=\mu\,\Delta}=\begin{cases}
\displaystyle 0 & \text{if $\mu=\ell$}\\[10pt]
\displaystyle \frac{(-1)^{\mu-\ell}\,\pi}
{N\,\Delta \sin\left(\pi \,\frac{\displaystyle \mu-\ell}
{\displaystyle N}\right)}
& \text{if $\mu \ne \ell$.}
\end{cases}
\label{alpha1}
\end{equation}

Analogously, for $s=2$, we have that
\begin{eqnarray}
&\displaystyle \frac{d^2}{d\,x^2}
{\rm sinc}\left(\frac{x-(\ell+\eta N)\Delta}{\Delta}\right)
=-\frac{\pi}{\Delta}
\frac{\sin\left(\pi\,\frac{\displaystyle x-(\ell+\eta N)\Delta}
{\displaystyle \Delta}\right)}
{x-(\ell+\eta N)\Delta}&\nonumber\\
&\displaystyle
-2\frac{\cos\left(\pi\,\frac{\displaystyle x-(\ell+\eta N)\Delta}
{\displaystyle \Delta}\right)}
{\left(x-(\ell+\eta N)\Delta\right)^2}
+\frac{2\Delta}{\pi}
\frac{\sin\left(\pi\,\frac{\displaystyle x-(\ell+\eta N)\Delta}
{\displaystyle \Delta}\right)}
{\left(x-(\ell+\eta N)\Delta\right)^3}&\nonumber\\
\end{eqnarray}
and thus
\begin{equation}
\left[\frac{d^2}{d\,x^2}
{\rm sinc}\left(\frac{x-(\ell+\eta N)\Delta}
{\Delta}\right)\right]_{x=\mu\,\Delta}=\begin{cases}
\displaystyle -\frac{\pi^2}{3\Delta^2} & \text{if $\mu-\ell-\eta N=0$}\\[10pt]
\displaystyle -2\frac{\displaystyle (-1)^{\mu-\ell-\eta N}}
{\displaystyle \Delta^2 (\mu-\ell-\eta N)^2}
& \text{if $\mu-\ell-\eta N \ne 0$.}
\end{cases}
\end{equation}
Assuming $N$ odd, we obtain that
\begin{eqnarray}
\alpha_{\mu \ell}^2
&=&\displaystyle \sum_{\eta=-\infty}^{+\infty}\left[\frac{d^2}{d\,x^2}
{\rm sinc}\left(\frac{x-(\ell+\eta N)\Delta}
{\Delta}\right)\right]_{x=\mu\,\Delta}\nonumber\\[10pt]
&=&\displaystyle \begin{cases}
\displaystyle \frac{\pi^2}{3N^2\Delta^2}(1-N^2) & \text{if $\mu=\ell$}\\[10pt]
\displaystyle -\frac{2\pi^2}{N^2\Delta^2}(-1)^{\mu-\ell}
\frac{\cos\left(\frac{\displaystyle (\mu-\ell)\pi}{\displaystyle N}\right)}
{\sin^2\left(\frac{\displaystyle (\mu-\ell)\pi}{\displaystyle N}\right)}
& \text{if $\mu \ne \ell$.}
\end{cases}
\label{alpha2}
\end{eqnarray}

Notably, these expressions coincide with those of the so-called SLAC
derivative technique~\cite{drell}, which were
obtained~\cite{foerster}
switching to the reciprocal space, operating in that domain and then
transforming back (analogously to what we do in Sec.~\ref{fourier},
where we compare our method with the Fourier one).

\section{Treatment of the product term}
\label{productsec}

The last term that we consider in our analysis is the one proportional to the
product between the function $f(x)$ (the potential energy or a function of it)
and the wave function $\varphi(x)$.

To a first approximation (simplified sinc-based approach), we can proceed as
if the product term were band limited with band $B=D/L$ and thus directly
express it as a linear combination of the functions $g_{\ell,\Delta}(x)$, with
coefficients given by the samples of the product term at the points $n\,\Delta$
(with $n=0,\ldots,N-1$). Operating in this way, in the final system of
equations this term gives only a diagonal contribution consisting of its
samples at the points $n\,\Delta$ (analogously to Ref.~\cite{foerster}).
However, since we have assumed a band $B$ for $\varphi(x)$, the same
assumption is in general not verified for the product term (unless a constant
potential function is considered); and for high-order solutions,
this introduces a discrepancy with respect to the solutions from a
Fourier analysis.

In the following, we discuss a better approximation (advanced sinc-based
approach) that makes this direct-space method equivalent to the
Fourier one.

We introduce an odd (for analytical convenience) positive integer
number $M$, which accounts for the fact that in general the exploitation of
more samples of the potential than just those at the positions where we want
to evaluate the wave function $\varphi$ can be useful.
In the calculation we include the samples of the potential
function $f(x)$ at the $N\,M$ ($=2\,Q+1$, with $Q$ a positive integer)
points taken at intervals multiple of $\Delta'=L/(N\,M)=\Delta/M$.
We replace the function $f(x)$ with the function
\begin{eqnarray}
\tilde f(x)&=&
\sum_{m=0}^{MN-1}f\left(m\,\Delta'\right)
\sum_{\eta=-\infty}^{+\infty}
{\rm sinc}\left(\frac{\displaystyle x-(m+\eta MN)\Delta'}
{\displaystyle \Delta'}\right)\nonumber\\
&=&\sum_{m=0}^{MN-1}f\left(m\,\Delta'\right)
g_{m,\Delta'}(x)
\label{tildef}
\end{eqnarray}
(with a maximum band $B'=Q/L$), obtained by reconstruction from its $N\,M$
samples taken at intervals $\Delta'$.

Assuming for the wave function the expression (\ref{exp-rho}), we can
therefore write the product term as:
\begin{equation}
h(x)=\tilde f(x)\varphi(x)=
\left[\sum_{m=0}^{MN-1}f\left(m\,\Delta'\right) g_{m,\Delta'}(x)\right]
\left[\sum_{\ell=0}^{N-1}\varphi(\ell\,\Delta)g_{\ell,\Delta}(x)\right]\,.
\label{h}
\end{equation}

In order to obtain the final matrix representation of our wave equation,
we have to compute only the projections of $h(x)$ on the functions
$g_{\mu,\Delta}(x)$ (with $\mu=0,\ldots,N-1$), which corresponds to
approximating $h(x)$ with
\begin{equation}
h_0(x)=\sum_{\mu=0}^{N-1}\langle g_{\mu,\Delta}(x)|h(x)\rangle\, g_{\mu,\Delta}(x)
=\sum_{\mu=0}^{N-1}\left[\frac{1}{\Delta}\int_{0}^{L}g_{\mu,\Delta}^*(\chi)
h(\chi) d\,\chi\right]g_{\mu,\Delta}(x)\,.
\label{h0}
\end{equation}
These projections are equal (exploiting Eq.~\eqref{h}) to
\begin{eqnarray}
&& \displaystyle \frac{1}{\Delta}\int_{0}^{L}
g_{\mu,\Delta}^*(\chi)h(\chi) d\,\chi\nonumber\\
&& \displaystyle
=\sum_{\ell=0}^{N-1}\Bigg\{ \varphi(\ell\,\Delta)
\sum_{m=0}^{MN-1} \bigg[ f\left(m\,\Delta'\right)
\frac{1}{\Delta}\int_{0}^{L}
g_{\mu,\Delta}^*(\chi)g_{m,\Delta'}(\chi)g_{\ell,\Delta}(\chi)d\,\chi \bigg]\Bigg\}
\nonumber\\
&& \displaystyle
=\sum_{\ell=0}^{N-1}\Bigg\{ \varphi(\ell\,\Delta)
\frac{1}{M N^2}\sum_{m=0}^{MN-1}f\left(m\,\Delta'\right)
\nu_{\mu \ell m} \Bigg\}\nonumber\\
&& \displaystyle
=\sum_{\ell=0}^{N-1}\varphi(\ell\,\Delta)\beta_{\mu \ell}\,,
\label{pro}
\end{eqnarray}
where the coefficients $\beta_{\mu \ell}$ are given by
\begin{equation}
\beta_{\mu \ell} \equiv
\frac{1}{M N^2}\sum_{m=0}^{MN-1}f\left(m\,\Delta'\right)\nu_{\mu \ell m}\,.
\label{beta}
\end{equation}
With some analytical calculations  (a possible procedure will be briefly
described in the second part of Appendix~\ref{productnotes}) it is possible
to find a closed form for
\begin{equation}
\nu_{\mu \ell m} \equiv M N^2\frac{1}{\Delta}\int_{0}^{L}
g_{\mu,\Delta}^*(\chi)g_{m,\Delta'}(\chi)g_{\ell,\Delta}(\chi)d\,\chi\,.
\label{nu}
\end{equation}
In detail, if $M\ne 1$ we have that
\begin{equation}
\nu_{\mu \ell m}=\lambda_1(m,\mu)\,\lambda_1(m,\ell)
\label{mne1}
\end{equation}
with
\begin{equation}
\lambda_1(m,\mu)=\begin{cases}
\displaystyle N & \text{if $\mu M=m$}\\[10pt]
\displaystyle
\frac{\displaystyle\sin\left(\pi\left(\mu-\frac{m}{M}\right)\right)}
{\displaystyle\sin\left(\frac{\pi}{N}\left(\mu-\frac{m}{M}\right)\right)}
 & \text{if $\mu M \ne m$.}
\end{cases}
\end{equation}
Instead, if $M=1$ (and thus $\Delta'=\Delta$) we have that
\begin{equation}
\nu_{\mu \ell m}=\begin{cases}
\displaystyle \lambda_2(m,\mu) & \text{if $\ell =m$}\\[10pt]
\displaystyle
\frac{\displaystyle (-1)^{\ell-m}\,
\left(\lambda_3(\ell,\mu)-\lambda_3(m,\mu)\right)}
{\displaystyle
\sin\left(\frac{\pi}{N}\,(\ell-m)\right)}& \text{if $\ell \ne m$}
\end{cases}
\label{meq1}
\end{equation}
with
\begin{equation}
\lambda_2(m,\mu)=\begin{cases}
\displaystyle 3D^2+3D+1 & \text{if $m=\mu$}\\[10pt]
\displaystyle
\frac{\displaystyle \sin^2\left(\frac{\pi}{N}\,D\,(\mu-m)\right)}
{\displaystyle \sin^2\left(\frac{\pi}{N}\,(\mu-m)\right)}
 & \text{if $m \ne \mu$}
\end{cases}
\end{equation}
and
\begin{equation}
\lambda_3(m,\mu)=\begin{cases}
\displaystyle 0 & \text{if $m=\mu$}\\[10pt]
\displaystyle
\frac{\displaystyle \cos\left(\frac{\pi}{N}\,(\mu-m)\right)-(-1)^{\mu-m}}
{\displaystyle 2\sin\left(\frac{\pi}{N}\,(\mu-m)\right)}
 & \text{if $m \ne \mu$.}
\end{cases}
\end{equation}

\section{Correspondence with the Fourier analysis}
\label{fourier}

The differential problem we are
interested in can alternatively be solved using a classical solution method in
the transformed domain. We can replace the wave function
$\varphi(x)$ and the potential energy function $f(x)$ with their truncated
Fourier expansions. In detail, we can consider only the lowest (in modulus)
$N$ spatial frequencies of $\varphi(x)$ (i.e., the spatial frequencies
$\ell/L$ with $|\ell| \le D$, where $N=2\,D+1$) and we can numerically compute
the lowest-order $M\,N$ Fourier coefficients of $f(x)$ by means of a discrete
Fourier transform (DFT) of its $M\,N$ samples within the period $L$.
Then, we can project the resulting equation onto the $N$ basis functions
$\exp(i2\pi j x/L)$ with $|j| \le D$, thereby
recasting the problem into an $N\times N$ system of linear equations where
the Fourier coefficients of the wave function are the unknowns.
In this way, we approximate the infinite-dimensional problem to a finite-size
matrix problem, disregarding the Fourier components corresponding to
frequencies greater in modulus than $D/L$, both for the unknown wave function
and for all the terms appearing in the equation.

We will now show that the advanced sinc-based approach
described in the previous sections is equivalent to this
Fourier-based solution technique.

As we have seen,
a periodic function $\tilde z(x)$ with maximum band $D/L$ can be expressed
in terms of its $N=2\,D+1$ samples at the points $n\,\Delta$ within the
period $L$ as follows:
\begin{equation}
\tilde z(x)=\sum_{\ell=0}^{N-1}\tilde z(\ell\,\Delta)g_{\ell,\Delta}(x)\,.
\end{equation}
Therefore, we can compute its Fourier series coefficients $\tilde Z_p$
noting that the Fourier coefficients $[G_{\ell,\Delta}]_p$ of
the function $g_{\ell,\Delta}(x)$ are given by (exploiting the
definition of $g_{\ell,\Delta}(x)$ as a periodic repetition of sinc
functions~\cite{haykin})
\begin{equation}
\left[G_{\ell,\Delta}\right]_p=\frac{1}{L}\mathcal{F}
\left[{\rm sinc}\left(
\frac{x-\ell\Delta}{\Delta}\right)
\right]_{f=\frac{\scriptstyle p}{\scriptstyle L}}
=\frac{1}{N}e^{-i2\pi p \frac{\scriptstyle \ell}{\scriptstyle N}}
\label{trasfg}
\end{equation}
for $|p| \le D$, and 0 otherwise (with $\mathcal{F}$ the Fourier
transform) and thus
\begin{equation}
\tilde Z_p=
\sum_{\ell=0}^{N-1}\tilde z(\ell\,\Delta)\left[G_{\ell,\Delta}\right]_p=
\frac{1}{N}\sum_{\ell=0}^{N-1}\tilde z(\ell\,\Delta)
e^{-i2\pi p \frac{\scriptstyle \ell}{\scriptstyle N}}
\label{coeff}
\end{equation}
for $|p| \le D$, and 0 otherwise.

It is apparent that Eq.~\eqref{coeff} corresponds to the DFT of $\tilde z(x)$.

Therefore the DFT of a function $z(x)$ computed from its $N=2\,D+1$ samples
coincides with the exact Fourier series of the function $\tilde z(x)$ with
maximum band $D/L$ that has the same samples as $z(x)$ in the period $L$.

In particular, this consideration can be applied
to the potential function $f(x)$: calculating its DFT on the $M\,N$
samples (as we do in the Fourier analysis)
corresponds exactly to substituting $f(x)$ with the function
$\tilde f(x)$ of Eq.~(\ref{tildef}) (as we do in our advanced sinc-based
approach).

Moreover, as we have seen, when operating in the frequency domain,
we disregard the frequency components outside the interval $[-D/L,D/L]$
for all the terms of the differential problem.
Let us discuss, for each term, the equivalent of this frequency cut-off
in the direct domain.

Limiting the Fourier components of the unknown wave function
$\varphi(x)$ to frequencies that have a modulus less than $D/L$ corresponds
exactly to the assumption, we have made while operating in the direct space,
that $\varphi(x)$ is band limited with band $D/L$ (which has allowed us to
use the sampling theorem with $N=2\,D+1$ samples).

The same consideration is valid for the derivatives of $\varphi(x)$.
The expressions that we have found for the derivatives can actually
be obtained also from a reciprocal space analysis. Indeed, if
$\varphi(x)$ is periodic and we assume it to be band limited with band
$B=D/L$ with Fourier coefficients $a_p$, we can write it and its
derivatives in the form
\begin{equation}
\varphi(x)=\sum_{p=-D}^{D} a_p
e^{i2\pi p \frac{\scriptstyle x}{\scriptstyle L}}\,,
\end{equation}
\begin{equation}
\frac{d^s \varphi(x)}{d x^s}=
\sum_{p=-D}^{D} \left(i 2 \pi \frac{p}{L}\right)^s
a_p e^{i2\pi p \frac{\scriptstyle x}{\scriptstyle L}}\,.
\label{derfreq}
\end{equation}
The function $\exp(i2\pi p x/L)$ with $|p|\le D$
can be seen as a band limited function with band $B=D/L$ and thus
expressed in terms of its $N$ samples in the period $L$:
\begin{equation}
e^{i2\pi p \frac{\scriptstyle x}{\scriptstyle L}}=
\sum_{\mu=0}^{N-1}e^{i2\pi p \frac{\scriptstyle \mu}{\scriptstyle N}}
g_{\mu,\Delta}(x)\,.
\end{equation}
Substituting this expression into Eq.~(\ref{derfreq}) we obtain
\begin{equation}
\frac{d^s \varphi(x)}{d x^s}=
\sum_{\mu=0}^{N-1}\left[\sum_{p=-D}^{D} \left(i 2 \pi \frac{p}{L}\right)^s
a_p e^{i2\pi p \frac{\scriptstyle \mu}{\scriptstyle N}}\right]g_{\mu,\Delta}(x)\,.
\end{equation}
The value of the derivative for $x=\mu\,\Delta$ is the quantity between
square brackets.
Therefore (if we express $a_p$ using Eq.~(\ref{coeff})), we have that
\begin{eqnarray}
&&\left[\frac{d^s \varphi(x)}{d x^s}\right]_{x=\mu\,\Delta}\nonumber\\
&&=\sum_{p=-D}^{D} \left(i 2 \pi \frac{p}{L}\right)^s
\left(\frac{1}{N}\sum_{\ell=0}^{N-1}\varphi(\ell\,\Delta)
e^{-i2\pi p \frac{\scriptstyle \ell}{\scriptstyle N}}\right)
e^{i2\pi p \frac{\scriptstyle \mu}{\scriptstyle N}}\nonumber\\
&&=\sum_{\ell=0}^{N-1}\varphi(\ell\,\Delta)
\left[\left(\frac{i 2 \pi}{(2D+1)\Delta}\right)^s
\frac{1}{2D+1}
\sum_{p=-D}^{D}p^s
e^{i2\pi p \frac{\scriptstyle \mu-\ell}{\scriptstyle 2D+1}}\right]\nonumber\\
&&=\sum_{\ell=0}^{N-1}\varphi(\ell\,\Delta)\alpha_{\mu \ell}^s\,.
\label{alphaf}
\end{eqnarray}
It is easy to verify that the expression for $\alpha_{\mu \ell}^s$ in
Eq.~(\ref{alphaf}) yields exactly the same closed-form results that can be
found operating in the direct space [for example, for $s=1$ and $s=2$,
those reported in Eqs.~(\ref{alpha1}) and (\ref{alpha2})].
Indeed, the
expression in Eq.~(\ref{alphaf}) can be easily obtained also by substituting
the Fourier expansion (\ref{ser1}) of $g_{\ell,\Delta}(x)$ into the definition
$\alpha_{\mu \ell}^s=[d^s\,g_{\ell,\Delta}(x)/d\,x^s]_{x=\mu\,\Delta}$
of Eq.~(\ref{alpha}).

Finally, let us consider the effect of the frequency cut-off
on the product term $h(x)$.
Neglecting the frequency components of this function outside the interval
$[-D/L,D/L]$ corresponds to limiting its band to the range
$[-1/(2\Delta),1/(2\Delta)]$, because $1/(2\Delta)=(2D+1)/(2L)=(D/L)+[1/(2L)]$
but the product term, being periodic with period $L$, contains only frequency
components multiple of $1/L$.
If $H(f)$ is the Fourier transform of the function $h(x)$
($f$ being the spatial frequency), performing this frequency cut-off
is equivalent to considering, in the Fourier domain, a spectrum
$H_0(f)={\rm rect}(\Delta f) H(f)$, where ${\rm rect}(\Delta f)$ is a function equal
to 1 between $-1/(2\Delta)$ and $1/(2\Delta)$, and to 0 outside this interval.
The inverse Fourier transform of $H_0(f)$ is equal to
\begin{equation}
h_0(x)=\frac{1}{\Delta}\int_{-\infty}^{+\infty}
{\rm sinc}\left(\frac{x-\chi}{\Delta}\right)h(\chi) d\,\chi\,.
\end{equation}
Since the periodic function $h_0(x)$ is band limited with band
$B\le 1/(2\Delta)$, we can express it in terms of its samples taken with
sampling interval $\Delta$
\begin{eqnarray}
h_0(x)&=&\sum_{\ell=0}^{N-1}\left[
\frac{1}{\Delta}\int_{-\infty}^{+\infty}
{\rm sinc}\left(\frac{\chi-\ell\Delta}{\Delta}\right)h(\chi) d\,\chi\right]
g_{\ell,\Delta}(x)\nonumber\\
&=&\sum_{\ell=0}^{N-1}\left[
\frac{1}{\Delta}\int_{0}^{L}
g_{\ell,\Delta}^*(\chi)h(\chi) d\,\chi\right]
g_{\ell,\Delta}(x)
\end{eqnarray}
[where we have exploited the fact that the sinc function is even and
Eq.~(\ref{changedom})].
This exactly corresponds to what we have done in the direct space [see
Eq.~(\ref{h0})]. Indeed, we can verify that the expression of the
product term obtained operating in the reciprocal domain is equivalent to
that achieved in the direct space (see Appendix~\ref{productnotes}).

Therefore, with the approach we have described, the solutions in the direct
and in the reciprocal space are equivalent and correspond to linear systems
of the same size.

In order to further clarify this equivalence,
let us notice that the periodic and band limited function $\varphi(x)$ can be
equivalently expressed, using both the Fourier expansion and the
sampling theorem, in the following two ways:
\begin{equation}
\varphi(x)=\sum_{p=-D}^{D}(\sqrt{N}\,a_p)
\frac{e^{i2\pi p \frac{\scriptstyle x}{\scriptstyle L}}}{\sqrt{N}}=
\sum_{\ell=0}^{2D}\varphi(\ell\,\Delta)g_{\ell,\Delta}(x)\,.
\end{equation}
Two different sets of orthonormal basis functions have been used in this
equation: the functions $\exp(i2\pi p x/L)/\sqrt{N}$ and the
functions $g_{\ell,\Delta}(x)$ [with this choice, the exponentials have been
properly normalized with respect to the scalar
product (\ref{product})]. Performing the solution in the reciprocal domain
corresponds to using the former basis set, while performing the solution in
the direct space corresponds to using the latter basis set.\\
It is possible to switch from one basis to the other
noticing that each of the exponential functions that appear in the Fourier
expansion (being band limited with band $B\le 1/(2\Delta)$) can
be expressed in terms of its samples taken with sampling interval $\Delta$
in this way
\begin{equation}
\frac{e^{i2\pi p \frac{\scriptstyle x}{\scriptstyle L}}}{\sqrt{N}}=
\sum_{\ell=0}^{2D}
\frac{e^{i2\pi p \frac{\scriptstyle \ell}{\scriptstyle N}}}{\sqrt{N}}
g_{\ell,\Delta}(x)=
\sum_{\ell=0}^{2D}T_{\ell p}g_{\ell,\Delta}(x)\,.
\end{equation}
The matrix $T$, made up of the $T_{\ell p}$ elements, can be used to switch
from the Fourier basis to the direct-space one.\\
On the other hand, using Eq.~\eqref{trasfg}, we can write the Fourier
expansion of $g_{\ell,\Delta}(x)$ as
\begin{equation}
g_{\ell,\Delta}(x)=\sum_{p=-D}^{D}\left[G_{\ell,\Delta}\right]_p
e^{i2\pi p \frac{\scriptstyle x}{\scriptstyle L}}=
\sum_{p=-D}^{D}\sqrt{N} \left[G_{\ell,\Delta}\right]_p
\frac{e^{i2\pi p \frac{\scriptstyle x}{\scriptstyle L}}}{\sqrt{N}}
\end{equation}
and thus the matrix $\Theta=T^{\dagger}$, made up of the elements
$\Theta_{p \ell}=\sqrt{N} [G_{\ell,\Delta}]_p=
\exp(-i2\pi p \ell/N)/\sqrt{N}=T_{\ell p}^*\,$, operates
the change from the direct-space basis to the Fourier one.

In particular, if the matrix of the direct space linear system is $M_d$ and
the matrix of the reciprocal-space linear system is $M_r$, we have that
$M_d=T\,M_r\,T^{-1}=T\,M_r\,T^{\dagger}$.

\section{Application to the solution of the Dirac equation in an armchair
graphene nanoribbon}
\label{dirac}

Transport in graphene is described, within an envelope function approach,
by the Dirac equation~\cite{kp}. When considering graphene nanoribbons,
at the effective edges of the ribbon (i.e., at the lattice sites just
outside the ribbon), we have to enforce Dirichlet boundary conditions,
which couple the envelope functions corresponding to the two inequivalent
Dirac points $\vec K$ and $\vec K'$.
In particular, here we take into consideration the solution of the Dirac
equation in the case in which the potential in the ribbon depends only
on the transverse co-ordinate, i.e., in which the potential does not vary in the
longitudinal direction. In this case, the four envelope functions
$F_{S \vec P}(\vec r)$ (corresponding to the two Dirac points
$\vec P=\vec K,\vec K'$ and to the two sublattices $S=A,B$) can be written
as the product of a confined component in the transverse $y$ direction
$\Phi_{S \vec P}(y)$ and of a propagating wave in the longitudinal
$x$ direction (with longitudinal wave vector $\kappa_x$):
$F_{S \vec P}(\vec r)=\Phi_{S \vec P}(y)\,\exp(i \kappa_x x)$.

As shown in Refs.~\cite{pt,iwce2010}, the problem can be reformulated
into a differential problem with periodic boundary conditions on a doubled
domain ($2\tilde W$ instead of $\tilde W$, which represents the effective
width of the ribbon, i.e., the distance between the two effective edges
where the Dirichlet boundary conditions had to be enforced in the original
problem).
This new problem can be expressed in the following form:
\begin{equation}
\begin{cases}
\displaystyle
\sigma_z \left(\frac{d\,\vec\varphi(y)}{d\,y}+i\tilde K \vec\varphi(y)\right)
+\sigma_x f(y)\vec\varphi(y)=-\kappa_x \vec\varphi(y)\\[10pt]
\displaystyle
\vec\varphi(2\tilde W)=\vec\varphi(0)\,,
\end{cases}
\end{equation}
where the $\sigma$'s are the Pauli matrices and $\vec\varphi(y)$ is a
two-component function directly related to the four envelope functions
of graphene in the following way:
\begin{equation}\label{varphi}
\vec\varphi(y)=
\begin{cases}
\displaystyle
e^{-i \tilde K y}
\left[\begin{array}{c}
\Phi_{A \vec K}(y)\\
\Phi_{B \vec K}(y)
\end{array}\right] &
\text{if $y \in [0,\tilde W]$}\\[10pt]
\displaystyle e^{i \tilde K (2\tilde W-y)}\,i\,
\left[\begin{array}{c}
\Phi_{A \vec K'}(2\tilde W-y)\\
\Phi_{B \vec K'}(2\tilde W-y)
\end{array}\right] &
\text{if $y \in [\tilde W,2\tilde W]$.}
\end{cases}
\end{equation}
Moreover, $K=4\pi/(3a)$ (where $a$ is the graphene lattice constant) is the
modulus of the transversal co-ordinate of the Dirac points and
$\tilde K=K-\,{\rm round}\,(K /(\pi/\tilde W))\cdot\pi/\tilde W$ is a quantity
such that $\exp(i 2 \tilde K \tilde W)=\exp(i 2 K \tilde W)$ and that
$|\tilde K|<\pi/\tilde W$. We consider $\tilde K$ instead of $K$, in such
a way that slowly varying $\vec\varphi(y)$ functions correspond to
slowly varying transverse components of the envelope functions, i.e., those
in which we are most interested. Additionally,
$f(y)=(U(\tilde W-|\tilde W-y|)-E)/\gamma$ is the
potential term, where $U(y)$ is the potential energy, $E$ is the Fermi energy,
$\gamma=\hbar v_F$, $\hbar$ is the reduced Planck constant, and $v_F$ is the
Fermi velocity in  graphene. In this formulation, the problem has to be solved
over the domain $[0,2\,\tilde W]$ and the boundary condition for
$\vec\varphi(y)$ is periodic.

The periodic boundary condition makes it possible to solve the problem in the
Fourier domain or equivalently in the basis of $g$ functions.
Working in the basis of $g$ functions, we have rewritten the equation
as follows:
\begin{eqnarray}
&&\displaystyle \sigma_z \left[\sum_{\mu=0}^{2D}
\left(\sum_{\ell=0}^{2D}\vec\varphi(\ell\,\Delta) \alpha_{\mu \ell}^1\right)
g_{\mu,\Delta}(x)
+i\tilde K \sum_{\mu=0}^{2D}\vec\varphi(\mu\,\Delta)g_{\mu,\Delta}(x)
\right]\nonumber\\
&&\displaystyle +\sigma_x
\sum_{\mu=0}^{2D}\left(\sum_{\ell=0}^{2D}\vec\varphi(\ell\,\Delta)
\beta_{\mu \ell}\right) g_{\mu,\Delta}(x)=-\kappa_x
\sum_{\mu=0}^{2D}\vec\varphi(\mu\,\Delta)g_{\mu,\Delta}(x).\nonumber\\
\end{eqnarray}
Projecting this equation (using the scalar product (\ref{product}))
onto the generic function $g_{j,\Delta}(x)$, we obtain
\begin{equation}
\sigma_z \left[
\left(\sum_{\ell=0}^{2D}\vec\varphi(\ell\,\Delta) \alpha_{j \ell}^1\right)
+i\tilde K \vec\varphi(j\,\Delta)\right]+\sigma_x
\left(\sum_{\ell=0}^{2D}\vec\varphi(\ell\,\Delta)\beta_{j \ell}\right)
=-\kappa_x
\vec\varphi(j\,\Delta)
\end{equation}
for $j=0,\ldots,2D$, or equivalently
\begin{equation}
\sum_{\ell=0}^{2D}
\left[\sigma_z (\alpha_{j \ell}^1+i\tilde K \delta_{j,\ell})+\sigma_x
\beta_{j \ell}\right]\vec\varphi(\ell\,\Delta)
=-\kappa_x \vec\varphi(j\,\Delta)
\label{system}
\end{equation}
for $j=0,\ldots,2D$.
These $N=2\,D+1$ equations can be written in matrix form as an eigenproblem
with eigenvalues $-\kappa_x$ and eigenvectors containing the values
$\vec\varphi(j\,\Delta)$ of the unknown function $\vec\varphi(y)$ at the
$N$ points of the considered grid.

In order to adopt the first approximate approach for the product term
described at the beginning of Sec.~\ref{productsec}, we
just have to replace $\beta_{j \ell}$ with $f(j\,\Delta)\delta_{j,\ell}$
in Eq.~(\ref{system}).

\begin{figure}
\begin{center}
\includegraphics[width=8cm]{potential.eps}
\caption{Potential energy considered in the armchair ribbon.}
\label{potential}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=8cm]{eig.eps}
\caption{Position on the Gauss plane of all the longitudinal wave
vectors $\kappa_x$ obtained with the Fourier analysis and with our advanced
sinc-based approach.}
\label{eig}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=8cm]{phia.eps}
\caption{Envelope function $\Phi_{A \vec K} (y)$
corresponding to the eigenvalue with the largest real part, computed with
a Fourier analysis and with our advanced sinc-based approach.}
\label{phia}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=8cm]{eig1.eps}
\caption{Position on the Gauss plane of all the longitudinal wave
vectors $\kappa_x$ obtained with the simplified sinc-based approach.
The highest order wave vectors strongly differ from
those obtained with a Fourier analysis (see Fig.~\ref{eig}).}
\label{eig1}
\end{center}
\end{figure}

For example, we have considered a potential $U(y)$ (represented in
Fig.~\ref{potential}) equal to the sum of two Gaussian functions:
\begin{equation}
U(y)=0.15~{\rm eV}\cdot
e^{-\frac{\scriptstyle (y-300\>{\rm nm})^2}{\scriptstyle 5000\>{\rm nm}^2}}+
0.15~{\rm eV}\cdot
e^{-\frac{\scriptstyle (y-500\>{\rm nm})^2}{\scriptstyle 5000\>{\rm nm}^2}}
\end{equation}
in a 8131 dimer line ($\sim$1~$\mu$m) wide armchair nanoribbon,
considering an electron injection energy $E=0.1$~eV.
In Figs.~\ref{eig} and \ref{phia}, we show the results that we have obtained
considering $N=99$ and $M=21$ with the advanced approach in the direct
space, and we compare them with those achieved by means of a Fourier analysis.
In particular, in Fig.~\ref{eig}, we show the position on the Gauss plane
of all the obtained longitudinal wave vectors $\kappa_x$; and in
Fig.~\ref{phia}, we represent the behavior of the envelope function
$\Phi_{A \vec K}(y)$ corresponding to the eigenvalue with the largest real part.
As expected, the two methods yield exactly the same results.

Instead, in Fig.~\ref{eig1}, we show the values of the
longitudinal wave vectors $\kappa_x$ obtained with the simplified
treatment of the product term. As we can see from the highest order
wave vectors, this method is not equivalent to the Fourier one, even
though it yields quite good results for the low-order modes.

\section{Conclusion}
\label{conclusion}

We have proposed a method, based on a set of basis functions resulting from
the periodic repetition of sinc functions, for the numerical solution, in
the direct space, of differential problems
and, in particular, of quantum wave
equations, with periodic boundary conditions.

This approach is equivalent to a Fourier analysis and
(once we have chosen a given discretization) requires the
solution of a linear system with the same matrix size. It allows performing
a very efficient solution in the direct space, without
the need to Fourier transform the input functions and finally to inverse
transform the solutions.

We have applied this method to efficiently solve the
Dirac equation in an armchair graphene nanoribbon in the direct
space, verifying its equivalence to a reciprocal space approach.

We believe that this treatment also helps in clarifying the exact origin
of the differences in the convergence behavior observed between
direct space and reciprocal space approaches.

redpajama_set_name: RedPajamaArXiv


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You are here: Home > WHI Study News
Developments from the WHI Therapy Trials
What are the WHI Therapy Trials?
The Women's Health Initiative (WHI) was a major 15-year research program to address the most common causes of death, disability and poor quality of life in postmenopausal women— cardiovascular disease, cancer, and osteoporosis.
From the WHI home page (last updated Jan 2010):
The WHI was launched in 1991 and consisted of a set of clinical trials and an observational study, which together involved 161,808 generally healthy postmenopausal women. The clinical trials were designed to test the effects of postmenopausal hormone therapy, diet modification, and calcium and vitamin D supplements on heart disease, fractures, and breast and colorectal cancer.
The hormone trial had two studies: the estrogen-plus-progestin study of women with a uterus and the estrogen-alone study of women without a uterus. (Women with a uterus were given progestin in combination with estrogen, a practice known to prevent endometrial cancer.) In both hormone therapy studies, women were randomly assigned to either the hormone medication being studied or to placebo. Those studies have now ended. The women in these studies are now participating in a follow-up phase, which will last until 2010.
Estrogen-Alone Arm of Study Stopped
On March 2, 2004, the National Institutes of Health (NIH) announced that it had stopped the estrogen-alone arm of the WHI study after careful consideration of preliminary data and an average follow-up of nearly 7 years. Premarin was the form of estrogen used in the trials. While Premarin alone did not increase the risk of breast cancer or other cancers, it did slightly increase the risk of stroke and did not reduce risk of heart disease as previously thought.
There were two WHI postmenopausal hormone therapy trials:
the estrogen plus progestin study (stopped in 2002) of women with a uterus
the estrogen-alone study of women without a uterus (stopped March 2, 2004)
(Women with a uterus were given progestin in combination with estrogen, a practice known to prevent endometrial cancer.) In both hormone therapy trials, women were randomly assigned to either the hormone medication being studied or to placebo. Women in both trials have been asked to stop taking their study medication and to stay in the follow-up phase of the trial.
statements and findings from the NIH
NHLBI Statement, March 2, 2004: NIH Asks Participants in Women's Health Initiative Estrogen-Alone Study to Stop Study Pills, Begin Follow-up Phase
Sample letter to Estrogen-Alone Participants
NHLBI Advisory for Physicians on the WHI trial of Conjugated Equine Estrogens versus Placebo
Questions and Answers About the WHI Postmenopausal Hormone Therapy Trials
The Estrogen-Alone Study
The Estrogen-Plus-Progestin Study
Oral Contraceptives and Cardiovascular Disease
Project AWARE's Comments
We feel concern that the discontinuation of this study will send the message "all HRT and all estrogen supplementation is bad for women."
The estrogen used in the discontinued study was Premarin. No other type of estrogen was used. Premarin is synthesized from the urine of pregnant mares, and is not bio-identical to human estrogen.
There are transdermal and other non-oral forms of bio-identical estrogen available to women, and there is much evidence that shows these other forms are safer, providing a more "natural" effect on the body than any oral formulation, including bio-identical oral estrogens.
We encourage you to investigate the resources regarding HRT in this website. Learn which transdermal and other non-oral forms of bio-identical estrogen you might consider. Read about alternatives to the kind of HRT used in WHI trials so that you can consult your physician armed with information, and to find the right solution for you. For further guidance and more specific information, you can contact any compounding pharmacy.
The estrogen plus progestin arm of the WHI trials was stopped in 2002 when data showed that after 5 years the balance of benefits from combined hormone replacement therapy did not compensate for an increased risk of cardiovascular events.
We believe it is important to note that the progestin used was a molecularly altered synthetic, not natural, or bio-identical progesterone. An alternative to total rejection of HRT would be the initiation of HRT trials involving "natural", or bio-identical hormones.
We invite you to explore our website for more comprehensive information on bio-identical hormones.
Page uploaded March 2004, Updated August 2010


Beekeeping is on the wane in Romania.
Romania's honey production this year is expected to drop to half the yearly average of 20 - 22 thousand tons in recent years, due to the bad weather, which has caused the death of a significant number of bees. Beekeepers say though that there will be no shortage of this product on the Romanian market and neither are prices going to soar, because there are still reserves and the Romanians aren't great honey consumers. Here is Ioan Fetea, chair of the Romanian Beekeepers Association with more on the issue.
Ioan Fetea has also mentioned other issues currently facing beekeepers in Romania.
According to data released by the Ministry of Agriculture and Rural development, in 2015 Romania exported 10,863 tons of honey valued at 41.48 million euros, while imports stood at only 2450 tons, amounting to 5.55 million euros. In 2015 Romania reported 40 thousand beekeepers and 1.47 million bee families.
Romanian beekeepers are to receive subsidies of 3.4 euros per bee family in the following period of time, an amount meant to make up for some of the losses they incurred this year due to the bad weather conditions.
Over 2017 – 2019 Romania will benefit from a budget of 21.5 million euros part of the National Beekeeping Programme, a 7.2% increase as compared to the money allotted for the 2014 – 2016 period. The National Beekeeping Programme for 2017 – 2019 includes measures such as the purchase of medicine, bee families, queens or beekeeping equipment.
Statistics show the Romanians are consuming a yearly average of 450 – 500 grams of honey per person, consumption being on the rise, as compared to 10 years ago. However, in comparison with 2 kilos in Germany and 1.5 kilos in Belgium and the Netherlands, Romania is still at the bottom of the European standings in terms of honey consumption.


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Modulating innate immunity improves Hepatitis C virus infection and replication in stem cell-derived hepatocytes
Zhou, X., Sun, P., Lucendo-Villarin, B., Angus, A. G.N., Szkolnicka, D., Cameron, K., Farnworth, S. L., Patel, A. H. and Hay, D. C. (2014) Modulating innate immunity improves Hepatitis C virus infection and replication in stem cell-derived hepatocytes. Stem Cell Reports, 3(1), pp. 204-214. (doi: 10.1016/j.stemcr.2014.04.018) (PMID:25068132) (PMCID:PMC4110790)
97247.pdf - Published Version
In this study, human embryonic stem cell-derived hepatocytes (hESC-Heps) were investigated for their ability to support hepatitis C virus (HCV) infection and replication. hESC-Heps were capable of supporting the full viral life cycle, including the release of infectious virions. Although supportive, hESC-Hep viral infection levels were not as great as those observed in Huh7 cells. We reasoned that innate immune responses in hESC-Heps may lead to the low level of infection and replication. Upon further investigation, we identified a strong type III interferon response in hESC-Heps that was triggered by HCV. Interestingly, specific inhibition of the JAK/STAT signaling pathway led to an increase in HCV infection and replication in hESC-Heps. Of note, the interferon response was not evident in Huh7 cells. In summary, we have established a robust cell-based system that allows the in-depth study of virus-host interactions in vitro.
Angus, Dr Allan and Patel, Professor Arvind and Zhou, Dr Xiaoling
Zhou, X., Sun, P., Lucendo-Villarin, B., Angus, A. G.N., Szkolnicka, D., Cameron, K., Farnworth, S. L., Patel, A. H., and Hay, D. C.
Stem Cell Reports
First published in Stem Cell Reports 3(1):204-214
Funder and Project Information
Award No
Funder's Name
Funder Ref
Lead Dept
63348 1 Development of Novel in Vitro Models for Hepatitus B virus and Hepatitus C virus Infection and Replication Arvind Patel Medical Research Council (MRC) MR/K008757/1 III - CENTRE FOR VIRUS RESEARCH
65649 1 Basis of the host range and tissue tropism for hepatitis C virus Arvind Patel Medical Research Council (MRC) MC_UU_12014/2 MVLS III - CENTRE FOR VIRUS RESEARCH


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The late Fadzlan Yahya: DAP's great orator
On 8 July, 2020
If we were to look back at Malaysia's political history, one figure stood out as one of the best orators during his prime; the late Fadzlan Yahya.
Known as DAP's great orator, Fadzlan Yahya was the State Legislative Assembly Representative (ADUN) for Pasir Berdamar, Perak from 1982 to 1986. Impressively, he won the Pasir Berdamar seat during GE7 with a majority of 10,845 votes in 1986.
In the subsequent years, he was elected as the ADUN for Lahat, Perak in 1990.
The prominent DAP leader held many positions with the party during his political career, including DAPSY Chief and subsequently, the Vice Chairman of the party.
As an elected representative, Fadzlan was famous for his rousing speeches which many likened to that of the first Indonesian President, Sukarno.
In an interview with Roketkini back in 2016, Fadzlan was asked whether he had any regrets in joining DAP. He responded:
"I have zero regrets. It is the path that I have chosen for myself after careful deliberation. Coincidentally during the time, this decision was also supported by my mother-in-law."
"She asked me to do what my heart told me and to not listen to naysayers. My loving wife also gave full trust and support behind my decision then."
"I am really blessed to have such an understanding wife who will always be behind me even when some vilified me. Especially if we were to lose an election, 'they' will always make attacks and ridiculed us."
In the same interview, Fadzlan also shared his thoughts on DAP's then state of affairs. He commented that DAP has evolved since its inception.
Back then, DAP was more focused on issues relevant to the Malaysian-Chinese community that it created a 'wall'.
"But now everything is different. DAP delves and fights for national issues that are relevant and accepted by all of the races in Malaysia."
Sadly however, Fadzlan lamented how DAP became the victim of slander and misinformation. The mainstream media since his time have maliciously associated DAP with communism even when it is clear that the party is vehemently anti-communists.
Unfortunately, most of the Malay community have fallen for this 'character assassination' to the point that they believed this lie to be a fact when there is no merit to it whatsoever.
However, Fadzlan urged for a 'personal touch' to combat this whereby the party needs to approach those who have fallen for the negative narrative and change their perspective.
"DAP have had a lot of prominent and charismatic Malay leaders. This is a healthy development in line with the Malaysian Malaysia ambition. Furthermore, this is important in order to defeat the DAP image that is 'Chinese-chauvinist' portrayed by the party's political enemies."
"Now, they have been an open acceptance of DAP among the Malay community. This effort needs to be sustained for a more balanced DAP membership. Hence, a Malaysian Malaysia concept can be put forward."
The charismatic leader Fadzlan Yahya sadly passed away on 12 September 2016 in Ipoh, Perak. DAP's great orator will forever live on in the spirit and struggles of the party.
Tags : DAP, democratic action party, fadzlan, fadzlan yahya, great orator, malaysia, Perak, politics, speech, speeches, the great speaker
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Home What's Brewing? Mostly Cloudy, Microsoft Re-Org Could See Up to 3,000 Pink Slips
Mostly Cloudy, Microsoft Re-Org Could See Up to 3,000 Pink Slips
Microsoft is downsizing, or should we call it cloud-sizing? It's rain that comes down from the dark clouds in these climes, sometimes in a trickle or a mist, at other times in a downpour.
The company says it is reorganizing its workforce for the future of, what else, cloud computing.
In an effort to pare down employment rolls in some divisions while adding to the roster in more promising areas, up to 3,000 positions are said to be facing elimination.
While the word on both sides of the Lake is a bit gloomy, Microsoft has inferred that a majority of the employment cuts will occur outside of U.S. or P.S. shores.
Widespread rumors of layoffs in the thousands began percolating throughout the company last week. An internal memo to employees on Monday spelled out changes in Microsoft's consumer and commercial businesses and how it sells cloud services.
Talk about a silver lining, the commercial cloud run rate, an extrapolation of future financial results, hit $15.2 billion during the March quarter, up from $14 billion in the previous quarter. Meanwhile, revenue in the company's intelligent cloud group grew by 93 percent to $6.8 billion in the quarter.
Last year at this time, Microsoft cut 2,850 people from its smartphone and sales teams. For the entirety of the last fiscal year which ended June 30th, Microsoft cut 7,400 employees. Traditionally, the end of the FY is when layoffs have occurred.
As of March 31, Microsoft employed 121,567 worldwide, including 45,535 in Washington state.
The weather report for cloud computing puts Microsoft in the forecast but lagging behind Seattle frontrunner AWS. Amazon Web Services retained an impressive lead in analyst Gartner Inc.'s latest Magic Quadrant for Cloud Infrastructure as a Service report, with Microsoft Corp. and Google Inc. failing to make up much if any ground over their biggest rival.
It's no surprise that AWS continues to dominate the IaaS scene. The company has held a commanding lead ever since cloud computing became a thing a decade ago, and Gartner noted it's now the most commonly chosen provider for strategic, organizationwide adoption of cloud infrastructure services.
According to SiliconAngle.com, Microsoft Azure ranks second primarily due to its appeal among enterprise customers. Those customers tend to gravitate toward Azure primarily so they can extend their existing infrastructure-oriented relationship with Microsoft and their investments in its technologies.
Gartner also placed a high value on Azure's ability to integrate with Microsoft's application development tools, as well as its specialized PaaS capabilities that include Azure Data Lake, Azure Machine Learning and the Azure IoT Suite.
The Gartner Magic Quadrant
However, Gartner called out Microsoft's service experience as being "less enterprise-ready than expected."
As for the Google Cloud Platform, Gartner says it remains a good distance behind the top two, though it is still a good option for "cloud-native" companies. The company's strengths include its innovation engine and focus on portability.
The rest of the field drops precipitously after the three leaders. IBM Corp., which is often cited as being one of four so-called "cloud giants," was not reflected as favorably in Gartner's Magic Quadrant. Instead, IBM's SoftLayer offering finds itself deep down in the "Visionaries" box. "
Also struggling to catch up is Oracle Corp. The Gartner study concluded that Oracle's Gen 2 cloud offering is a "minimum viable product," with only a basic selection of cloud IaaS compute, storage and networking capabilities.
The last remaining "Visionary" in Gartner's list is a new entrant, Alibaba Cloud, operated by the Chinese web giant Alibaba Group Holding Ltd. Alibaba Cloud is the dark horse thanks to its financial muscle and its commanding lead in the Chinese IaaS market. For now, though, it remains considerably behind the likes of AWS and Microsoft because of its "limited track record" and the weakness of its international cloud platform, which doesn't possess the same capabilities or performance of its Chinese version. [24×7]
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Q: How to add the icons header tru mapping How can I add icons?
export interface Section {
    path: string;
    name: string;
    icon: string
}

This is my code that I will map in header
export function getSections() {
    const sections: Array<Section> = [
        {path: 'customers',
        name: 'Customers',
        icon: ''},
        {path: 'groups', name: 'Groups', icon: ''},
        {path: 'products', name: 'Products', icon: ''},
        {path: 'analytics', name: 'Analytics', icon: ''},
        {path: 'settings', name: 'Settings', icon: ''},
    ];
    return sections
}

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