In November 2010 Marie Colvin spoke at a service in St Bride's church London, held to honour the 49 journalists, photographers and support staff killed in war zones since the turn of the century. She spoke about the risks she and her colleagues take to bring us the truth about what is happening to innocent victims of conflict around the world.
"Despite all the videos you see from the Ministry of Defence or the Pentagon, and all the sanitised language describing smart bombs and pinpoint strikes, the scene on the ground has remained remarkably the same for hundreds of years. Craters. Burned houses. Mutilated bodies. Women weeping for children and husbands. Men for their wives, mothers children."
Her dedication to this purpose was indefatigable, even after suffering the loss of one eye and severe post traumatic stress disorder.
"The real difficulty is having enough faith in humanity to believe that enough people, be they government, military or the man on the street, will care when your file reaches the printed page, the website or the TV screen."
This immensely brave and dedicated woman was risking her life, eventually losing it, and all she asked of us was that we take an interest and remember truthfully.
In one of her last reports Marie Colvin described casualties being brought into a makeshift clinic in an apartment, and estimated that 80 per cent of them were civilians, even describing how she had watched a baby die. This is the reality of war, not glory and honour but agony and grief.
The least we can do to honour Marie Colvin's memory is to disregard the propaganda we are fed by those who promote war, and try to do something, no matter how small, to bring a little more humanity to the world. Here http://www.paxchristi.org.uk/ or here http://stopwar.org.uk/ would be a good place to start. See also Ekklesia's report: "Reimagining Remembrance" which urges a more truthful recollection of the reality of war.


The Delta Kappa Gamma - Alpha XI, Sun City Scholarship is available to full - time junior and senior students at Arizona State University. You must have a minimum 3.0 GPA, demonstrate financial need, and be involved in campus activities to be eligible for this award. Preference is given to returning female students.


text:
Das Buch namens Qínjīng (chinesisch 禽经; "Vogelbuch") ist das früheste ornithologische Werk Chinas. Der Überlieferung nach wurde es von Shi Kuang (Shī Kuàng 师旷) in der Zeit der Frühlings- und Herbstannalen verfasst und von Zhang Hua (张华) (232–300) aus der Zeit der Westlichen Jin-Dynastie kommentiert. 1 Heft (juan).

Echtheit
Da darin Werke aus der Zeit nach Shi Kuang zitiert werden, sogar bis in die letzten Jahre der Südlichen Song-Dynastie hinein, ist leicht ersichtlich, dass sowohl die Verfasserangabe als auch die Angaben zum Kommentator (d. h. Zhang Hua aus der Zeit der Jin-Dynastie) falsch sind.

Datierung
Lu Dian (1042–1102) aus der Nördlichen Song-Dynastie hat es in seinem Tierbuch namens Piya zuerst zitiert; wahrscheinlich ist es ein unter einem anderen Namen veröffentlichtes Werk der Tang- oder Song-Zeit.

Inhalt
Das Buch berichtet über den (chinesischen mythischen) Phönix (fènghuáng) und mehrere Dutzend weitere bekannte Vogelarten, wie zum Beispiel Krähe, Fasan, Geier, Kranich, Kuckuck oder Rebhuhn. Im Mittelpunkt des Interesses steht die Ökologie, außerdem schenkt es der Anpassung des Körperbaus Aufmerksamkeit. Zum Beispiel heißt es darin: "Der Lebenszweck des Pelikans ist das Wasser, der des Spechtes das Holz." "Lebewesen fressende Vögel haben einen langen Schnabel, Körnerfresser einen kurzen." Nahezu sämtliches Wissen dieses Werkes ist durch Beobachtung erworben, es hat eine gewisse wissenschaftliche Bedeutung.

Es ist eine wichtige Quelle zur Geschichte der chinesischen Ess- und Trinkkultur.

Shi Kuang: der legendäre Musikmeister Kuang
Der angebliche Verfasser Shi Kuang ("Musikmeister Kuang") war ein berühmter Musiker des chinesischen Altertums, der am Hof von Herzog Ping (Jin Ping gong) im alten Staat Jin der Frühlings- und Herbstperiode tätig war und für sein gutes Gehör bekannt war. Der Legende nach wurde er ohne Augen geboren. Über ihn sind viele Legenden verbreitet; unter anderem ist von ihm im Buch Zhuangzi, und vor allem im reichen Legendenschatz des Zuozhuan des Öfteren die Rede. Es wird berichtet, dass er dem Herzog Ping den Angriff der Armee des Staates Qi auf den Staat Jin berichtete; aus den Vogelrufen wusste er, wann sich Qi zurückzog.

Weitere ornithologische Aufzeichnungen
Frühe ornithologische Aufzeichnungen sind noch das in der alten Bücherreihe Tang Song congshu enthaltene Werk namens Guihai qinzhi 1 juan "Aufzeichnungen über Vögel aus Guihai (d. h. aus dem fernen Süden) in einem Heft (juan)" Es stammt von Fan Chengda (1126–1193), dem Politiker und Dichter der Südlichen Song-Dynastie, der auch für seine Reiseberichte bekannt ist.

Wichtige Drucke (Auswahl)
Das Buch fand in vielen alten Bücherreihen Aufnahme (z. B. im Tang Song congshu oder im prestigeträchtigen Siku quanshu der Qianlong-Ära), woraus auf seine Popularität und seinen hohen Stellenwert geschlossen werden kann.

Moderne Ausgaben
Unter dem Titel Shi Kuang Qinjing wurde die Ausgabe der Bücherreihe namens Umfassende Sammlung von Congshu im Jahr 1991 im Verlag Zhonghua shuju 中华书局 in Peking nachgedruckt. Sie hat die ISBN 7-101-00894-1.

Literatur
 Cihai ("Meer der Wörter")
 Zhongguo pengren wenxian tiyao
 Zhongguo renming da cidian (lishi renwu juan) [Großes chinesisches Personenlexikon. Band: Historische Persönlichkeiten]; ISBN 7-5326-0033-5

Siehe auch
Liste von Quellen zur Geschichte der chinesischen Ess- und Trinkkultur

Weblinks
 General Index to the Congshu jicheng (chubian) (PDF-Datei; 1,3 MB)

Einzelnachweise

Literarisches Werk
Literatur (Chinesisch)
Sachliteratur (Ornithologie)
Chinesische Küche

redpajama_set_name: RedPajamaWikipedia


The drug lord, who has an unparalleled record of jailbreaks, is destined to spend 23 hours of each day in what amounts to a reinforced cage.
Come June, when El Chapo is sentenced after being found guilty in a New York court of unparalleled drug trafficking, he'll be making himself at home in one of America's "supermax" jails.
Most likely he'll be packed off to a prison in Florence, Colorado, dubbed the "Alcatraz of the Rockies".
Prisoners are locked up in a single-person reinforced concrete cell which measures a mere 7ft-by-12ft, which are specifically designed to create what one ex-prisoner described as a "high-tech version of hell, designed to shut down all sensory perception".
The bed is a poured concrete slab. A thin mattress is placed on top and there are blankets.
There is a combined toilet and a sink and drinking fountain. The shower has a timer to stop prisoners flooding their cell.
Furniture consists of an immovable concrete desk and stool.
Some prisoners are allowed a small black-and-white TV, but El-Chapo will be unable to catch up on Narcos on Netflix — it only shows carefully chosen educational and religious programmes.
Each cell has a slit-like 42in-tall while the four-inch-wide window is angled to prevent views of the sky.
Prisoners often go days even weeks "with only a few words spoken to them," an Amnesty International report found.
And to make sure inmates don't talk to each other cell walls are sound-proofed.
Meals are slipped through a flap in the door.
And should an inmate need a doctor, they must talk to them remotely through teleconferencing.
Former warden Robert Hood has described the supermax as "life after death", adding it's "far worse than death".
Commonly it is known it as "the prison of prisons" and human rights groups claim it is even "worse than Guantanamo".
When taken outside their cells for the daily one hour exercise, inmates wear leg irons, handcuffs and stomach chains.
This hideous existence is all a far cry from his billionaire life of luxury that was bankrolled on murder and the misery of drug addiction.
During his three month trial in Brooklyn it emerged that He had houses at every single resort in Mexico as well as ranches in every single state.
His Acapulco beach house featured a zoo with a "little train'' used to ride around and see lions, tigers and panthers.
There also was a yacht docked there called "Chapito''.
But in his new home he is going nowhere fast, despite his almost mythical reputation for breaking out.
Bribery is widely believed to have enabled that jailbreak, as well as a 2001 escape in which El Chapo was smuggled out of another top-security Mexican prison in a laundry basket.
"There had to be collusion from within," said Mike Vigil, a former US Drug Enforcement Administration agent who worked undercover in Mexico.
"There is no doubt corruption played a role in both of his spectacular escapes."


1. Fish ball noodle soup packed with fresh chili peppers on Monday. Did you know that chili contains endorphins? No wonder I love eating chili when I'm feeling down and starting to get a sore throat!
2. Pretty nails from O.P.I – this colour is 'Big Apple Red'. I love OPI nail polish, especially their interesting names! This is how they name their polishes. I love seeing my bright red nails when I type at work – it makes me happy (I know, I'm a girly-girl!!).
3. Breakfast at Panzerotti Cafe (on 42 Margaret Street in the CBD) this morning. They have delicious coffees and yummy breakfasts. I had poached eggs with bacon and Mari had a bacon and egg sandwich.
Happy Friday!!! Enjoy your weekend.


text: package "parallel"

redpajama_set_name: RedPajamaGithub


text:
Goldilocks & The Three Bears Singalong
Marketplace Theatre
Jolly Promotions
The much loved fairy-tale comes to the stage in an unforgettable live production of song and laughter. Goldilocks and the three bears, Ruff the dog, and Mr Bunny, perform live in this exciting and fun-filled musical production. The popular tale takes a few twists and turns along the way, with other new puppet characters and hopefully a happy ever after.
Kids will learn that music can be found everywhere as they sing and dance to all their favourite songs. Throughout the show there are interactive moments that children and parents alike can enjoy. Watch your kids take part in action songs like the 'Heads Shoulders Knees And Toes' and 'If You're Happy And You Know It' and 'I'm A Little Teapot', as well as many other all-time children's favourites. With different adventures from every character, the show will have parents and children alike laughing playing and singing.
Please note that there is a £1 per ticket booking fee for tickets purchased online.

redpajama_set_name: RedPajamaCommonCrawl


The exact same thing as last week. I didn't even move the blocks.
On the other hand, Josh's graduation quilt is finished! (OK, I still have to attach the label, but it's made!) Remind me to never use black backing. It shows everything. Hopefully later today I'll get a picture of it.
See what other are up to by checking out the links on Judy's blog.
Haha! I always tell myself every time I use black to remember never to use it again, but I LOVE it, and I keep doing it! It picks up every little thing, and it's so difficult to see on, but it sure does make a quilt sometimes!


Asus Tek Computer Inc. is one of the most famous brands in the PC and laptop industry, and it's no wonder you've come to check out our review of the ROG STRIX GL703VD.
This particular model, even though it's a gaming laptop in essence, is incredibly versatile and people from various lines of work, including music production, hold it in high regard and warmly recommend it. This review will help you find out what's so cool about it, why, and who should consider it.
The battery lifetime isn't exactly exemplary. Even though you'll be able to enjoy in your fun for quite a while, the recharge time is long.
Keyboard warms up rather quick if your GPU gets too overburdened.
Why should you consider Asus ROG STRIX GL703VD Laptop?
Among the plethora of things you could like about Asus' ROG Strix GL703VD lap, we've picked several prime benefits that most high-end laptops couldn't provide you with. These include a beautiful design, a set of premium-quality speakers, a wondrous vivid screen, the bright lit keyboard, and some heavy hardware which will forever change the way you game.
Let's start off with the design – Asus ROG Strix features a sleek, slim outlook. The brushed aluminium lid is pretty durable and looks expensive, but the outer shell is made of moderately sturdy plastic. Fear not, though, as the overall durability rating of ROG Strix is quite high, way above average in fact. The carbon-fibre finish puts on some flair on the simplistic outlook, although it doesn't affect its durability or performance.
Secondly, the keyboard is quite big, and it's quite awesome that the keys are lit with different colors – this help gamers organize their hotkeys more easily, programmers shouldn't struggle too much to find that one letter they're looking for after hours of coding and typing, and the same goes for music producers in general.
The screen of Asus ROG Strix is one of its most amazing features. It's big, fairly slim, and it features a resolution of 1920 by 1080 pixels. The 120Hz display works great for all sorts of purposes, although it's clear that Asus intended it for gaming. You can use the V-sync feature if your selected frame rate is higher than sixty frames per second to further improve your experience.
What's more, the saturation is plainly perfect – it's way above the average scale of sRGB standards with 92.8% coverage. Games, movies, animations, and colors in general look very rich and pronounced.
Some people report that they've experienced lags and frame-per-second issues. If that appears to be your case as well, there are several things you could do to resolve the issue.
Firstly, press the "Windows" button (found between control and alt) and type "Services". There you'll see a bunch of settings, and you're looking for "Superfetch" – once you've found it, stop it. Find the Superfetch properties before disabling the service in its entirety.
After that, switch back to the service panel and scroll down to the "Windows search" service. Just repeat what you did with the Superfetch service and you should be fine. Basically, these are not essential services and your laptop will work perfectly fine even if they're off.
Who is Asus ROG STRIX GL703VD Laptop ideal for?
Gamers have been the target group for Asus company for years, if not decades, and Asus' ROG Strix didn't change that. This laptop features some serious firepower, which is pretty surprising given that it's so small and lightweight – that makes it perfect for all kinds of gamers and people who simply can't get their hands off from their laptop during long, dull voyages.
Apart from that, music producers would welcome it, as it features a set of premium-quality built-in speakers on top the brutally strong spec rig – it has everything you need to run the latest audio edit software programs, it's user friendly, and we guarantee you'll be happy with it.
Asus ROG Strix GL793VD is a beautiful all-purpose laptop. It's exceptionally strong, it looks beautiful, and the features it comes outfitted with are quite exquisite. Although there are several things that could've been improved (such as quick heat buildup), it's clear that this laptop is among the finest you'd get for the buck – in fact, it's quite cheap for a high-end laptop, and it holds a superb value for the buck.


Find information about writing, inspiration for your next story, and the motivation to get it written.
We're excited to introduce another author who is making his debut after having used Novelize to write his first novel. Of course, Novelize is just one tool in this writer's arsenal. National Novel Writing Month gave him the push he needed to take writing seriously. Here, we got a chance to interview him about his recently released space opera The Blue Jewel and ask him about his writing process.
Kianna Alexander is living the dream. She's been a published author since 2009. Her latest accomplishment is her finished manuscript for the third book in her Sapphire Shores series. The first book was released in December of 2017, and book two just came out in September 2018. Kianna used Novelize to write book three, which is scheduled to be released in May 2019. This interview with her tells you how she did it and offers some tips for aspiring writers, too.


text:
Shanahan expects Leafs expect their young stars to take less money to stay together
A restricted free agent without much leverage other than sitting out, the 22-year-old Nylander and the Leafs have been unable to agree on a new contract as negotiations have dragged through the summer, into training camp and now the season.
By Joshua ClippertonThe Canadian Press
Wed., Oct. 3, 2018timer4 min. read
Maple Leafs president Brendan Shanahan says the team expects its star players to make financial sacrifices to be part of what's hoped will be a long championship window in Toronto.
And that begins with the still-unsigned William Nylander.
Speaking at an event before the Leafs opened the NHL regular season at Scotiabank Arena against the Montreal Canadiens on Wednesday, Shanahan reflected on his three Stanley Cup victories with the Detroit Red Wings as personal proof that taking a little less money can be worth it in the long run.
"When I get together with some of my old mates from the Cup years in Detroit ... we talk about winning together and growing together, and that's what we remember," Shanahan said. "We all found a way to fit with each other so that we could keep adding to the group.
"That's obviously what we're asking some of our young leaders to do."
Circus atmosphere grows as Leafs get ready for season opener
Puck about to drop on Leafs season .... still no Nylander
Wednesday NHL preview: Montreal Canadiens at Toronto Maple Leafs
"(Leafs general manager Kyle Dubas) and I have the same vision for what we want to continue to build here and make it sustainable," Shanahan said. "While these things are not easy to do, Kyle and I are on the exact same page as far as the things we need to do in order to achieve it.
"We're big fans of William, we want William to be a part of this, but obviously this is a process that he needs to go through."
The slick winger, who remains in Sweden after finishing third on the team in scoring last season (21 goals, 40 assists), has said he wants a long-term deal, with some reports pegging his demands at US$8 million per season.
The Leafs also have to get contracts done in the near future with Auston Matthews and Mitch Marner — star 21-year-old forwards both set to become restricted free agents next summer — so whatever Nylander gets will set the bar.
And with veteran centre John Tavares making $77-million over the next seven years after joining the Leafs in free agency, there's only a certain amount of money to go around.
"A lot of other voices, and understandably so," Shanahan said of contract negotiations. "(Playing in Toronto) is not for everyone and we're not for everyone, but we think the players that we currently have, while it's not going to be easy, we have great confidence that they have bought into being a part of this program and being a part of the Toronto Maple Leafs and representing Toronto in a way that they understand what's going to be most important.
"What I hope is they can look back on 20 years, 30 years down the road … and what's going to be most important to them is whether or not they maxed out as an individual and a team and have championships to look back on and remember fondly."
Shanahan, who won the Cup with Detroit in 1997, 1998 and 2002 in the pre-salary cap era, said he understands what Nylander is going through from a player's perspective.
"He wants to play," Shanahan said. "As a group we have a perspective of what we want to build here, how we need all these players. When I talk about looking back, we all did quite well.
"We all did quite well financially, we all did quite well professionally on the ice."
Leafs defenceman Morgan Rielly and centre Nazem Kadri have already taken less than what they might have signed for on the open mark to remain in Toronto, while Tavares took fewer dollars than what other teams offered.
"He wanted to come here and win hockey games, he wanted to be treated fairly, and he is," Shanahan said of Tavares. "And yes, that is what we hope for and expect from our players as we go forward. It's not for everyone, but for the ones that will play here, that's what we want from them, and I think that's what they want from each other.
"That's what they'll look back on and remember the most — how they interacted with their teammates and whether they were able to be a part of something like this."
On hand at an event unveiling a plaque honouring "the national historical significance" of the Leafs, Shanahan was also asked about the expectations in a city buzzing with dreams of ending a 51-year Cup drought.
"Our fans and our players have earned the right to be excited," he said. "They've earned the right to be in a position that is shared by other teams, that they have the ability to now go all the way and compete for a Stanley Cup.
"What's more important to them and to us in what we set upon to do four years ago was build this in a manner in which this was not a one-year window. You can't take for granted the opportunities in the years that you have in front of you, and you have to focus on the present, but it's been our goal ... to do this in a manner in which we can sustain this over a long period of time."
Brendan Shanahan, William Nylander


text:
Q: Bootstrap4 Jumbotron Text and Button elements not scaling to Fit in Smaller View Screen So I'm using Bootstrap 4's Jumbotron to style a landing page. Inside the jumbotron I have the standard headings and paragraphs with 2 call to action buttons. I have wrapped these elements in a div to position them correctly for a desktop screen. however when I shrink the window, the elements fall out of the jumbotron and mess up the look of the page.
Here is how it looks on desktop

And Here is how it looks on "mobile"

I hope you can see what I mean from the images. The background image of the jumbotron is perfectly responsive but the contents (text, buttons) are overflowing down the page.
Here is the relevant html

Welcome to Other Mother!

Scroll Down To See Whats On Offer

Click Below To Start Shopping or Make a Request Instantly

            Shop Now
            Contact OM

And Here is the relevant CSS
.jumbotron {
    position: relative;
    top: -170px;
    height: 100vh;
    background-image: url(images/homepg.jpg);
    background-size: 100% 100%;
    margin: 0 auto;
    color: white;
    z-index: -1;
}

.jumboContents {
    position: relative;
    top: 30%;
    left: 30%;
}

A: I suggest using the bootstrap grid classes on your divs to let them stack properly when shrunken screen. (ex: div class="col-md-4 col-sm-1")

redpajama_set_name: RedPajamaStackExchange


text:
ActiveRecord::Schema.define(version: 20151216153024) do

  create_table "film_actors", force: :cascade do |t|
    t.string   "actor",      limit: 50
    t.datetime "created_at",            null: false
    t.datetime "updated_at",            null: false
  end

  add_index "film_actors", ["actor"], name: "index_film_actors_on_actor", unique: true, using: :btree

  create_table "film_actors_films", id: false, force: :cascade do |t|
    t.integer "film_id",       limit: 4
    t.integer "film_actor_id", limit: 4
  end

  add_index "film_actors_films", ["film_id", "film_actor_id"], name: "index_film_actors_films_on_film_id_and_film_actor_id", using: :btree

  create_table "film_areas", force: :cascade do |t|
    t.string   "area",       limit: 15
    t.integer  "film_id",    limit: 4
    t.datetime "created_at",            null: false
    t.datetime "updated_at",            null: false
  end

  add_index "film_areas", ["area"], name: "index_film_areas_on_area", unique: true, using: :btree

  create_table "film_categories", force: :cascade do |t|
    t.string   "category",   limit: 10
    t.datetime "created_at",            null: false
    t.datetime "updated_at",            null: false
  end

  add_index "film_categories", ["category"], name: "index_film_categories_on_category", unique: true, using: :btree

  create_table "film_categories_films", id: false, force: :cascade do |t|
    t.integer "film_id",          limit: 4
    t.integer "film_category_id", limit: 4
  end

  add_index "film_categories_films", ["film_id", "film_category_id"], name: "index_film_categories_films_on_film_id_and_film_category_id", using: :btree

  create_table "film_directors", force: :cascade do |t|
    t.string   "director",   limit: 50
    t.datetime "created_at",            null: false
    t.datetime "updated_at",            null: false
  end

  add_index "film_directors", ["director"], name: "index_film_directors_on_director", unique: true, using: :btree

  create_table "film_directors_films", id: false, force: :cascade do |t|
    t.integer "film_id",          limit: 4
    t.integer "film_director_id", limit: 4
  end

  add_index "film_directors_films", ["film_id", "film_director_id"], name: "index_film_directors_films_on_film_id_and_film_director_id", using: :btree

  create_table "film_downloads", force: :cascade do |t|
    t.string   "download_url",         limit: 500
    t.string   "download_contact_url", limit: 500
    t.string   "expand",               limit: 50
    t.integer  "film_id",              limit: 4
    t.datetime "created_at",                       null: false
    t.datetime "updated_at",                       null: false
    t.string   "download_other_url",   limit: 500
    t.integer  "source",               limit: 4
  end

  create_table "film_introductions", force: :cascade do |t|
    t.string   "introduction", limit: 2000
    t.integer  "film_id",      limit: 4
    t.datetime "created_at",                null: false
    t.datetime "updated_at",                null: false
  end

  create_table "film_logos", force: :cascade do |t|
    t.string   "logo_url",   limit: 500
    t.integer  "film_id",    limit: 4
    t.datetime "created_at",             null: false
    t.datetime "updated_at",             null: false
  end

  create_table "film_titles", force: :cascade do |t|
    t.string   "chinese_name", limit: 255
    t.string   "english_name", limit: 255
    t.boolean  "status",       limit: 1
    t.integer  "film_id",      limit: 4
    t.datetime "created_at",               null: false
    t.datetime "updated_at",               null: false
  end

  add_index "film_titles", ["chinese_name", "english_name"], name: "index_film_titles_on_chinese_name_and_english_name", unique: true, using: :btree

  create_table "films", force: :cascade do |t|
    t.integer  "film_title_id",        limit: 4
    t.string   "chinese_name",         limit: 255
    t.string   "english_name",         limit: 255
    t.integer  "film_logo_id",         limit: 4
    t.integer  "film_area_id",         limit: 4
    t.integer  "film_introduction_id", limit: 4
    t.integer  "film_language",        limit: 4
    t.string   "make_date",            limit: 50
    t.integer  "show_type",            limit: 4
    t.integer  "click_rate",           limit: 4
    t.integer  "grade",                limit: 4
    t.string   "duration",             limit: 20
    t.datetime "created_at",                       null: false
    t.datetime "updated_at",                       null: false
  end

  add_index "films", ["chinese_name", "english_name"], name: "index_films_on_chinese_name_and_english_name", unique: true, using: :btree

end


text:
Home / Culture / Human Forms in Nature. Ernst Haeckel's Trip to South Asia and Its Aftermath
Human Forms in Nature. Ernst Haeckel's Trip to South Asia and Its Aftermath
An early promoter and populariser of Darwin's evolutionary theory, the German biologist and artist Ernst Haeckel was a hugely influential figure of the late 19th century.
October 31, 2020 by Public Domain Review Leave a Comment
This article, Human Forms in Nature. Ernst Haeckel's Trip to South Asia and Its Aftermath was originally published in The Public Domain Review under a Creative Commons Attribution-ShareAlike 3.0. If you wish to reuse it please see: https://publicdomainreview.org/legal/
Studio portrait of Ernst Haeckel in his tropical explorer's gear, date unknown — Source.
Ernst Haeckel and his sixteen steamer trunks arrived at the harbor of Colombo "in the glorious light of a cloudless tropical morning" on November 21, 1881, after a four-week journey via Trieste, Suez, Aden, and Bombay. The zoologist had high hopes: during the coming four months in Ceylon — now Sri Lanka — he planned to get to know flora and fauna in "that highest and most marvellous variety of form" which existed on the island. Haeckel's enthusiasm was understandable: he was literally entering the promised land of his naturalist dreams.
Admittedly, Ceylon at that time was not defined by dream beaches, a devastating tsunami, or decades of conflict between the country's Sinhalese and the Tamil inhabitants — all topics we commonly associate with Sri Lanka today. In Haeckel's era, the teardrop-shaped island nation off India's southeastern coast embodied the concept of "the tropics". It was a place of paradisiacal plenty bursting with things that were unfamiliar and often brilliantly colored: animals, plants, fruit, spices, and even gemstones. The realm of riches seemed like a dream world from another time. It was also a land of extremes, where the heat and humidity provided fertile conditions for a host of dangers — little-understood diseases that could quickly turn life-threatening, plagues of insects, predatory animals stalking through the thick vegetation, sharks swimming offshore, and torrential downpours.
As he later recalled in his book A Visit to Ceylon, Haeckel planned to trade "the restraint of our artificial social life" for an existence "in the midst of the simple children of nature . . . forming some conception of that visionary primeval paradise into which the human race was born." After all, he believed that humanity had originated near Ceylon on the hypothetical continent of "Lemuria" in the Indian Ocean. This conviction certainly influenced his decision to travel to Ceylon — along with the fact that his role models, Alexander von Humboldt and Charles Darwin, never made it to that corner of the world.
Chromolithograph by W. Koehler, after Ernst Haeckel's 1882 painting "World's End", printed in Haeckel's Wanderbilder (1905) — Source.
For Haeckel, at the time forty-seven years old and the most important champion of Darwin's theory of evolution on the European continent, the journey to Ceylon represented the fulfillment of a long-held but initially vague desire that took concrete shape over many years. "As an eight-year-old boy, I loved no book more than Robinson Crusoe," he once recalled, "and a lonely life in nature on a distant island played a large part in many of my childhood fantasies." It's clear that he caught his tropical fever early, and that it intensified over decades.
But his journey was also motivated by immediate scientific concerns. Five years earlier, Britain's Challenger expedition had ended. Despite its global aspirations, this trailblazing contribution to marine research neglected the Indian Ocean. Haeckel did not take part in the voyage, but he was later commissioned to study and classify the Radiata specimens — such as jellyfish and starfish — that had been collected. Haeckel wanted to add several chapters to the resulting report. In the end, he had a range of reasons for traveling to Ceylon: a longing for adventure he had nourished since childhood, scientific ambition, and — when we keep the illustrations he would later create in mind — artistic ambition as well.
Chromolithograph by W. Koehler, after Ernst Haeckel's 1882 painting depicting a jungle scene on the Kelany-Ganga (Kelani River) in Ceylon (present-day Sri Lanka), printed in Haeckel's Wanderbilder (1905) — Source.
Once Haeckel reached Ceylon, however, it was the island's rich gardening culture that first drew his attention. "Often I have fancied myself in some beautiful wild spot with tall trees on all sides, wreathed and overgrown with creepers", he wrote. "But a hut shrouded under the branches of a bread-fruit-tree, a dog or a pig trotting out of the brushwood, children at play and hiding under the caladium leaves, have betrayed the fact that I was in a native (Sinhalese) garden." The ecology of these practical gardens, known as gewattas, has only been investigated closely in recent decades. The seemingly chaotic mixture of trees, bushes, vines, and herbaceous underbrush is actually a symbiotic system in which each element has a place and fulfills a purpose. Cultivated using low inputs of labor and fertilizers, they are amazingly productive, especially when the full range of their bounty is considered: from firewood to food and spices.
Sri Lanka is home to a surprisingly diverse array of environments: from dense tropical rainforests to coral reefs, from tidelands to savannahs, from mangrove forests to sand dunes. Haeckel set to work familiarizing himself with almost everything they contained. He was especially impressed by the palms, those "princes among plants". In the hills whose lower stretches were covered with meadows and rice fields, he discovered thick brushland where the trees "have grown up without any kind of order, and in such wild confusion — so tangled with creepers and climbers, with parasitic ferns, orchids and other hangers-on, every gap closed with a compact network of bush and brake — that it is quite impossible to unravel the knot and distinguish the closely matted stems." According to Haeckel, simply taking a few steps into this tangled growth was a dangerous undertaking. He recalled being attacked by mosquitos, bitten by ants, and plagued by all the nettles and thorns deployed by the plants to "bar the way into their mysterious labyrinth". It was here that Haeckel first glimpsed the magnificent Gloriosa superba, "the poisonous climbing lily of Ceylon, with its golden-red crown". Large black apes and swarms of green parrots surrounded him, and he killed "an enormous lizard above six feet long (the singular Hydrosaurus salvator)" — a creature resembling a crocodile — with a shot to the head. He also encountered the talipot tree or "century palm", with its tall trunk like a slender marble column. The talipot blooms just once, usually between the ages of fifty and eighty, when a pyramidal bush composed of millions of yellowish-white flowers emerges from its crown.
Chromolithograph by W. Koehler, after Ernst Haeckel's 1882 painting depicting screwpine on the south coast of Ceylon, printed in Haeckel's Wanderbilder (1905) — Source.
When Haeckel spent four days at the botanical garden of Peradeniya — located near the center of the island, on the outskirts of the former royal city of Kandy — he recognized the place as "an admirable natural forcing-house" and considered it the heart of the horticultural paradise that was Ceylon. But while the garden was and is a remarkable place, Haeckel's most surprising experience on the island took place outside its confines. Setting up a zoological laboratory in Weligama, on the island's southern tip, Haeckel set off to explore the nearby waters. Some of these excursions were short trips to cull specimens from the ocean's surface, but at times he also sailed out for up to four hours in a traditional boat. Once he arrived at his destination — a point forty or fifty nautical miles from the shore (south, towards "Lemuria") — he busily collected jellyfish and larvae. His supply of specimen bottles was quickly exhausted. But despite the wealth of sea life he found, his study of the specimens strengthened his conviction that the life forms existing in the earth's various oceans resembled one another much more closely than the land creatures from continent to continent.
Haeckel set aside the last month of his stay to explore the island's high-altitude regions. His goal was to visit "the wildest parts of the primaeval forest" beyond the reach of human civilization. One spring morning he climbed through thick forests and past rushing streams and waterfalls to the top of the eight-thousand-meter Pidurutalagala, the highest point on the island. There were surprises in store, including five-foot-long sky-blue earthworms, the brilliantly colored woodcock, and a type of ash-gray monkey. On the Horton Plains, a primarily jungle-covered area that is now a national park, fresh piles of dung signaled that wild elephants were nearby. During the days that followed, Haeckel and his companions traveled through the tangled brush along so-called elephant trails. Unfortunately — or perhaps fortunately — they never actually encountered an elephant herd.
Chromolithograph by W. Koehler, after Ernst Haeckel's 1882 painting of Nillu bushes and hanging bamboo in the highlands of Sri Lanka, printed in Haeckel's Wanderbilder (1905) — Source.
Although plants and animals were the main objects of Haeckel's interest during his journey, descriptions of the local people recur throughout A Visit to Ceylon, creating a kind of leitmotif. Even though his gaze upon nature was perhaps never purely apolitical, it is here in his reflections on the human population that we encounter a side to Haeckel much more challenging, if not to say disturbing, to today's readers. During a stopover in India, Haeckel was already taking note of the naked brown bodies "of the members of the poorest class", who were clad in nothing but "a loin-cloth or apron-like swimming-drawers". "Thus the naked figures of the Hindoos", he wrote, "give a strange and strong impression of their being real savages, though, in point of fact, they are descended from the same 'Mediterranean', or Aryan, stock as the various races of Europe."
In Ceylon he became aware of the complex mixture of the population. In addition to the Buddhist Sinhalese and the Hindu Tamils — the largest groups — the island was home to Indo-Arabs, Malays, Chinese, Africans, and Englishmen, along with descendants of the Portuguese and Dutch settlers, who were known as "Burgers", and the indigenous people, the Veddas. He mentioned the latter briefly — and almost in passing:
The Veddahs, or Vellahs, are commonly regarded as being the dispersed remnants of this race (hindoo); a few wild hordes still linger in the remotest parts of the interior and in the most primitive state. But, according to others, the Veddahs are, on the contrary, debased and degenerate descendants of the Cinghalese, outcasts that have reverted to savagery, like the Rodiyas.
Some of these ethnic groups lived together creating what Haeckel saw as "very interesting problems to the anthropologist who should try to classify them." He claimed that he detected a "singularly slender and feminine character of the limbs" in the older Sinhalese men, which set them apart from the much shorter Tamils. In general, Haeckel resisted classifying the Tamils among the "lower races of man", although his justification for this generosity of spirit was purely an aesthetic one. His hikes through the tea plantations on the high plateaus provided plenty of opportunities to admire what he perceived as the Tamils' unspoiled physical perfection. According to Haeckel, a sculptor would do far better to
study the true beauty and proportion of the human form among these naturally developed models, than in the life-schools of European academies, where some model, found with difficulty among the degenerate sons of civilization and forced into some unwonted attitude, is but a poor substitute for the genuine child of nature.
Photograph of a Tamil girl, featured in the third edition of Haeckel's Indische Reisebriefe(1895) — Source: author's own scan.
On the other hand, Haeckel dismissively claimed that the "finely cut mouth" and "very dark, inspired-looking eyes" of the typical Tamil were deceptive, "promising much more than the brain within fulfills." Wherever he was able to give his own preferences free rein, he indulged his weakness for classical Greek clichés. For example, he gave one of his four servants in Weligama the name Socrates. The lower-caste boy who opened coconuts for him each morning and fanned him with palm branches was dubbed Ganymede — after the youth who had the honor of serving the Olympian gods. We may ask ourselves why Haeckel did not use the boy's real name, Gamameda. Wouldn't it have been a matter of respect? Could he not see his young servant for who he "really" was? Or does today's reader attribute too much weight to this question? That said, Haeckel can be credited with having formed closer relations with natives than with the European colonials — though this may in part have just been due to the fact that there were not many of the latter around.
Haeckel returned to Europe in March 1882, taking thirty crates with him. He later claimed that his triumphant arrival back in Europe could "only be compared with that of a soldier returning after a victorious campaign." Although this first voyage to tropical lands did not lead to any research breakthroughs as such, his contact with its flora and people surely inspired Haeckel in the two main branches of activity for which he is known today.
On the one hand, in the botanical and zoological bounty of Ceylon we can see the seed for the work by which he is perhaps best remembered, the splendid Art Forms in Nature, which he began publishing in 1899. The book's complex, ornamental lithographs and halftone prints bridge art and science and continue to exert a particular fascination. Glowing with color, they brought the exotic and the remote to libraries around the world and were the culmination of decades of work for Haeckel. On the other hand is Haeckel's work relating to race and eugenics, a side to his output certainly less palatable to a modern audience.
Buy as a Print
Plate 28, Discomedusae, from Haeckel's hugely popular Art Forms in Nature (1904) — Source. (Buy as a print)
Indeed, no matter how colorful and lively the descriptions of Haeckel's travels may be, a shadow lies over A Visit to Ceylon. Haeckel's legacy is not simply an appreciation for the beauty of nature. In light of the views that he later propagated — either in his writings or at his popular lectures — his travelogue seems tame and nearly innocent. But the complex constellation of attitudes that he developed over time and embodied does not fit easily into current categories of left and right and deserves a closer look. In political terms he was what may be called a liberal nationalist. At the same time, he made isolated anti-Semitic statements. Even more disturbingly, he was also a proponent of racial inequality and believed that the races he understood as "inferior" were dying out in response to their contact with the more "civilized" races.
The hardening of his views on the hierarchy of races that took place during the last two decades of the nineteenth century must be understood in this broader context. It was a period when the sciences were undergoing intense development and specialization. Haeckel was not working in a vacuum: racial anthropology was a popular area of inquiry and had a number of champions. The tremendous economic success of certain societies due to industrialization and colonialism provided scientists with a seemingly convincing argument for claiming the superiority of the "favoured races in the struggle for life" (Darwin) and cementing it in the public consciousness. At the same time, these circumstances do not absolve Haeckel of his personal responsibility. He was a man of his time, but he also bears a measure of blame. After all, he was far more than just a cog in the machine. As an extremely popular author — his The History of Creation was translated into all the major European languages — he played an active role in helping concepts regarding the inferiority of certain "races" to spread and have an impact.
In his view, "cultural value" determines the value of human life itself: he saw an unbridgeable chasm between the "thinking soul of the cultured man and the unthinking brutish soul of the savage, natural man" — a soul he compared to that of a dog. In his lecture on the origins of humankind at the Fourth International Zoological Congress, held in Cambridge in 1898, he discussed the Veddas as "Ceylon's dwarf-like indigenous people", who in his view were one step beyond humanoid apes. He even claimed that the distance between advanced European peoples and "savages" was greater than that between the latter and apes. His views later found an enthusiastic following among scientists who expounded racial theories to support National Socialist ideology.
Illustration of the "Tree of Evolution", from Haeckel's Anthropogenie (1874) — Source.
Equally troublesome is his stance on eugenics, which is light years removed from any ethically defensible standard today. His beliefs in this area transgress the purely descriptive limits of evolutionary theory to embrace the philosophy of the social Darwinists. Yet he did not merely echo their literal, limiting interpretation of the notion of "survival of the fittest" but took this chain of reasoning a fatal step further. Haeckel repeatedly cited the ancient Spartans in his defense of the killing of disabled infants, whom he considered to be of little or no value.
Eugenics and euthanasia were embraced by many secular social reformers at the time, including some socialists. However, most of these thinkers advocated reproductive control for those they considered "inferior", not their murder. But Haeckel did support killing, and he had a clear idea about where to draw the line between who was "worthy of living" and who was not. After his death, others followed who were no longer content with mere intellectual games. Adolf Hitler later claimed that it was actually more humane to kill disabled individuals than to allow them to live. To kill the disabled was seen as morally good, even though it overturned Judeo-Christian attitudes which protected the disabled. Haeckel's works were widely read in the early twentieth century, and many eugenicists, racists, and anti-Semites absorbed the ideas they contained.
Close to two decades after his trip, in 1900, Haeckel set off for the tropics once again. Art Forms in Nature had just been published. This time, Ceylon was just a stop along a much longer route. Haeckel's real destination was Southeast Asia: Java and Sumatra. It was unavoidable that this voyage lacked the excitement of his first trip, and it was no longer motivated by scientific ambitions. His observations in the resulting account of his travels read like the dutiful report of a researcher who, while he still had some capacity for wonder, was very set in his ways. His thoughts gravitated to humanoid apes and ape-like humans, leaving him little time for strolls through the coral gardens.
If you believe in the work we are doing here at The Good Men Project and want to join our calls on a regular basis, please join us as a Premium Member, today.
All Premium Members get to view The Good Men Project with NO ADS.
Need more info? A complete list of benefits is here.
Talk to you soon.
Photo credit: iStockphoto.com
Filed Under: Culture Tagged With: nature
About Public Domain Review
Founded in 2011, The Public Domain Review is an online journal and not-for-profit project dedicated to the exploration of curious and compelling works from the history of art, literature, and ideas.
In particular, as our name suggests, the focus is on works which have now fallen into the public domain, that vast commons of out-of-copyright material that everyone is free to enjoy, share, and build upon without restriction. Our aim is to promote and celebrate the public domain in all its abundance and variety, and help our readers explore its rich terrain – like a small exhibition gallery at the entrance to an immense network of archives and storage rooms that lie beyond.
With a focus on the surprising, the strange, and the beautiful, we hope to provide an ever-growing cabinet of curiosities for the digital age, a kind of hyperlinked Wunderkammer – an archive of content which truly celebrates the breadth and diversity of our shared cultural commons and the minds that have made it.
How To Stop Caring What Other People Think of YouDo you really care what other people think of you?
You Can End Fear. and Be Free.In praise of the practical wisdom of J. Krishnamurti.
Apply the Technique to Yourself: No ConflictI look within and remember to have compassion.
The Case of the Nail Polish ThiefI wish I could say there was at least some sort of lesson learned here.
I Am Truly Sorry I 'Borrowed' Your HusbandI didn't understand the dynamics of your marriage, I hope one day you will.
How To Begin Again for the First Time
Make-up Only To Break-Up?


text:
\section{Introduction}
\label{sec:intro}
Analog network coding (ANC) extends to multihop wireless networks the idea of linear network coding \cite{103liYeungCai}, where an intermediate node sends out a linear combination of its incoming packets. In a wireless network, signals transmitted simultaneously by multiple sources add in the air. Each node receives a \textit{noisy sum} of these signals, \textit{i.e.} a linear combination of the received signals and noise. A communication scheme wherein each relay node merely amplifies and forwards this noisy sum is referred to as analog network coding \cite{107kattiGollakottaKatabi, 110maricGoldsmithMedard}.

The rates achievable with ANC in layered relay networks is analyzed in \cite{110maricGoldsmithMedard, 111liuCai}. In \cite{110maricGoldsmithMedard}, the achievable rate is computed under two assumptions: (A) each relay node scales the received signal to the maximum extent subject to its transmit power constraint, (B) the nodes in all $L$ layers operate in the high-SNR regime, where $\min_{k \in l} P_{R,k} \ge 1/\delta, l = 1, \ldots, L$ for $\delta \ge 0$, where $P_{R,k}$ is the received signal power at the $k^\textrm{th}$ node. It is shown that the rate achieved under these two assumptions approaches network capacity as the source power increases. The authors in \cite{111liuCai} extend this result to the scenarios where the nodes in at most one layer do not satisfy these assumptions and show that achievable rates in such scenarios still approach the network capacity as the source power increases\footnote{However, it is assumed that the noises at the nodes in this particular layer are independent, resulting in the computed ANC rate overestimating the optimal ANC rate in general.}.

However, requiring each relay node to amplify its received signal to the upper bound of its transmit power constraint results in suboptimal end-to-end performance of analog network coding, as we show in \cite{111agnihotriJaggiChen, 112agnihotriJaggiChen}. Further, even in low-SNR regimes amplify-and-forward relaying can be capacity-achieving relay strategy in some scenarios, \cite{107gomadamJafar}.

In this paper we are concerned with analyzing the performance of analog network coding in general layered networks, without the above two assumptions on input signal scaling factors and received SNRs. However, such a characterization of the performance of analog network coding results in a computationally intractable problem in general \cite{111liuCai, 111agnihotriJaggiChen}.

In \cite{112agnihotriJaggiChen}, we establish that a globally optimal set of scaling factors for each node, {\it i.e.} a choice of relaying strategies that optimizes end-to-end throughput over all ANC strategies, can be computed in a layer-by-layer manner. This result allows us to computationally efficiently characterize exactly the optimal ANC rate in a large class of layered networks that cannot be addressed using existing approaches under the assumptions A and B. Further, for general layered relay networks, this result significantly reduces the computational complexity of computing a set of non-trivial achievable rates.

However, even layer-by-layer computation of a network-wide scaling vector that maximizes the end-to-end ANC rate for general layered networks is a computationally hard problem. In this paper, we propose a greedy scheme to bound from below the optimal rate achievable with analog network coding in general layered networks. The proposed scheme allows us to exactly compute the optimal ANC rate in a much wider class of layered networks than those that can be so addressed using existing approaches, including our approach in \cite{112agnihotriJaggiChen}. In particular, for the Gaussian $N$-relay diamond network \cite{111nazarogluOzgurFragouli}, the proposed scheme allows us to exactly compute the optimal rate achievable with analog network coding. To the best of our knowledge, this is the first characterization of the optimal ANC rate for Gaussian diamond network. Further, for general layered networks, our scheme allows for the computation of the optimal rates within a constant gap from the cut-set upper bound asymptotically in the source power.

\textit{Organization:} In Section~\ref{sec:sysModel} we introduce a general wireless layered relay network model and formulate the problem of maximum rate achievable with ANC in such a network. Section~\ref{sec:diamond} addresses the problem of maximum ANC rate achievable in a Gaussian $N$-relay diamond network and shows that a greedy scheme optimally solves this problem. In Section~\ref{sec:genNet} we first generalize the greedy scheme for Gaussian diamond networks to characterize the optimal performance of a specific subnetwork of the general layered network. We then construct a scheme to bound from below the optimal performance of ANC in general layered networks. Section~\ref{sec:exa} illustrates that the proposed scheme leads to the exact computation of the maximum ANC rate in a specific class of layered networks and tight characterization of the optimal rate in the general layered networks asymptotically in the source power. Section~\ref{sec:conclFW} concludes the paper.

\section{System Model}
\label{sec:sysModel}
Consider a $(L+2)$-layer wireless network with directed links. The source $s$ is at the layer `$0$', the destination $t$ is at the layer `$L+1$', and the relay nodes from the set $R$ are arranged in $L$ layers between them. The $l^\textrm{th}$ layer contains $n_l$ relay nodes, $\sum_{l=1}^L n_l = M$. An instance of such a network is given in Figure~\ref{fig:layrdNetExa}. Each node is assumed to have a single antenna and operate in full-duplex mode.

\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{./layrdNetExa}
\caption{Layered network with 3 relay layers between the source `s' and destination `t'. Each layer contains two relay nodes.}
\label{fig:layrdNetExa}
\end{figure}

At instant $n$, the channel output at node $i, i \in R \cup \{t\}$, is
\begin{equation}
\label{eqn:channelOut}
y_i[n] = \sum_{j \in {\mathcal N}(i)} h_{ji} x_j[n] + z_i[n], \quad - \infty < n < \infty,
\end{equation}
where $x_j[n]$ is the channel input of the node $j$ in the neighbor set ${\mathcal N}(i)$ of node $i$. In \eqref{eqn:channelOut}, $h_{ji}$ is a real number representing the channel gain along the link from node $j$ to node $i$. It is assumed to be fixed (for example, as in a single realization of a fading process) and known throughout the network. The source symbols $x_s[n], - \infty < n < \infty$, are i.i.d. Gaussian random variables with zero mean and variance $P_s$ that satisfy an average source power constraint, $x_s[n] \sim {\cal N}(0, P_s)$. Further, $\{z_i[n]\}$ is a sequence (in $n$) of i.i.d. Gaussian random variables with $z_i[n] \sim {\cal N}(0, \sigma^2)$. We also assume that $z_i$ are independent of the input signal and of each other. We assume that the $i^{\textrm{th}}$ relay's transmit power is constrained as:
\begin{equation}
\label{eqn:pwrConstraint}
E[x_i^2[n]] \le P_i, \quad - \infty < n < \infty
\end{equation}

In analog network coding each relay node amplifies and forwards the noisy signal sum received at its input. More precisely, a relay node $i$ at instant $n+1$ transmits the scaled version of $y_i[n]$, its input at time instant $n$, as follows
\begin{equation}
\label{eqn:AFdef}
x_i[n+1] = \beta_i y_i[n], \quad 0 \le \beta_i^2 \le \beta_{i,max}^2 = P_i/P_{R,i},
\end{equation}
where $P_{R,i}$ is the received power at the node $i$ and choice of the scaling factor $\beta_i$ satisfies the power constraint \eqref{eqn:pwrConstraint}.

One important characteristic of layered networks with unidirectional links is that all paths from the source to destination have same number of hops. Also, each path from the $i^\textrm{th}, i \in R$, relay node to the destination has the same length. Therefore, in a layered network with $L$ layers, all copies of a source signal traveling along different paths arrive at the destination with time delay $L$ and all copies of a noise symbol introduced at a node in $l^\textrm{th}$ layer arrive at the destination with time delay $L-i+1$. Therefore, the outputs of the source-destination channel are free of intersymbol interference. This simplifies the relation between input and output of the source-destination channel and allows us to omit the time-index while denoting the input and output signals.

Using \eqref{eqn:channelOut} and \eqref{eqn:AFdef}, the input-output channel between the source and destination can be written as
\begin{equation}
\label{eqn:sdchnl}
y_t =  \bigg[\sum_{(i_1, \ldots, i_{L}) \in K_{L}} \hspace{-0.25in} h_{si_1}\beta_{i_1}h_{i_1 i_2} \ldots \beta_{i_{L}}h_{i_{L} t}\bigg] x_s +  \sum_{l=1}^{L} \sum_{j=1}^{n_l}\bigg[\sum_{(i_l, \ldots, i_{L}) \in K_{lj,L}} \hspace{-0.25in} \beta_{lj} h_{lj, i_l} \ldots \beta_{L i_{L}}h_{L i_{L}, t}\bigg] z_{lj} + z_t, \nonumber
\end{equation}
where $K_L$, is the set of $L$-tuples of node indices corresponding to all paths from the source to the destination with path delay $L$. Similarly, $K_{lj,L-l+1}$, is the set of $L-l+1$-tuples of node indices corresponding to all paths from the $j^{\textrm{th}}$ relay of $l^\textrm{th}$ layer to the destination with path delay $L-l+1$.

We introduce \textit{modified} channel gains as follows. For all the paths between the source $s$ and the destination $t$:
\begin{equation}
\label{eqn:modChnlParams}
h_s = \sum_{(i_1, \ldots, i_{L}) \in K_{L}} h_{si_1}\beta_{i_1}h_{i_1 i_2} \ldots \beta_{i_{L}}h_{i_{L} t}
\end{equation}
For all the paths between the $j^{\textrm{th}}$ relay of $l^\textrm{th}$ layer to the destination $t$ with path delay $L-l+1$:
\begin{equation}
\label{eqn:modChnlParams2}
h_{lj} = \sum_{(i_1, \ldots, i_{L-l+1}) \in K_{lj,L-l+1}} \beta_{lj} h_{lj, i_1} \ldots \beta_{L i_{L}}h_{L i_{L}, t}
\end{equation}

In terms of these modified channel gains\footnote{Modified channel gains for even a possibly exponential number of paths as in \eqref{eqn:modChnlParams} and \eqref{eqn:modChnlParams2} can be efficiently computed using line-graphs \cite{103koetterMedard}. Further, the number of such modified channel gains scales polynomially in the size of the graph being considered.}, the source-destination channel in \eqref{eqn:sdchnl} can be written as:
\begin{equation}
\label{eqn:chnlmod}
y_t = h_s x_s + \sum_{l=1}^{L} \sum_{j=1}^{n_l} h_{lj} z_{lj} + z_t
\end{equation}

\textit{Problem Formulation:} For a given network-wide scaling vector $\bm{\beta}=(\beta_{li})_{1 \le l \le L, 1 \le i \le n_l}$, the achievable rate for the channel in \eqref{eqn:chnlmod} with i.i.d. Gaussian input is (\hspace{-0.001cm}\cite{110maricGoldsmithMedard, 111liuCai, 111agnihotriJaggiChen}):
\begin{equation}
\label{eqn:infoRateFin}
I(P_s, \bm{\beta}) = (1/2) \log\big(1 + SNR_t\big),
\end{equation}
where $SNR_t$, the signal-to-noise ratio at the destination $t$ is:
\begin{equation}
\label{eqn:snr}
SNR_t = \frac{P_s}{\sigma^2}\frac{h_s^2}{1 + \sum_{l=1}^{L} \sum_{j=1}^{n_l} h_{lj}^2}
\end{equation}

The maximum information-rate $I_{ANC}(P_s)$ achievable in a given layered network with i.i.d. Gaussian input is defined as the maximum of $I(P_s, \bm{\beta})$ over all feasible $\bm{\beta}$, subject to per relay transmit power constraint \eqref{eqn:AFdef}. In other words:
\begin{equation}
\label{eqn:maxAFrate}
I_{ANC}(P_s) \stackrel{def}{=} \max_{\bm{\beta}:0 \le \beta_{li}^2 \le \beta_{li, max}^2} I(P_s, \bm{\beta})
\end{equation}
It should be noted that $\beta_{li, max}$ (the maximum value of the scaling factor for $i^\textrm{th}$ node in the $l^\textrm{th}$ layer) depends on the scaling factors for the nodes in the previous $l-1$ layers.

Given the monotonicity of the $\log(\cdot)$ function, we have
\begin{equation}
\label{eqn:eqProb}
\bm{\beta}_{opt} = \argmax_{\bm{\beta}:0 \le \beta_{li}^2 \le \beta_{li, max}^2}  I(P_s, \bm{\beta}) = \argmax_{\bm{\beta}:0 \le \beta_{li}^2 \le \beta_{li, max}^2}  SNR_t
\end{equation}
Therefore in the rest of the paper, we concern ourselves mostly with maximizing the received SNRs.

In \cite{112agnihotriJaggiChen}, we discussed the computational complexity of exactly solving the problem \eqref{eqn:maxAFrate} or equivalently the problem \eqref{eqn:eqProb}. Further, we also introduced a key result \cite[Lemma 2]{112agnihotriJaggiChen} that reduces the computational complexity of the problem of computing $\bm{\beta}_{opt}$ by computing it layer-by-layer as a solution of a cascade of subproblems. This result allowed us to characterize the optimal end-to-end rate achievable with analog network coding in communication scenarios that cannot be so addressed using previous approaches. However, each of these subproblems itself is computationally hard for general network scenarios as it involves maximizing the ratio of \textit{posynomials} \cite{107boydkimVandenberghe, 105chiang}, which is known to be computationally intractable in general \cite{105chiang}. Therefore, in this paper, we introduce a greedy scheme that optimally solves these subproblems and consequently the problem \eqref{eqn:eqProb} for a large class of layered networks that cannot be addressed with current schemes. For general layered networks, the proposed scheme allows us to tightly bound from below the optimal ANC performance. However before discussing this scheme, we motivate it by computing the maximum rate of information transfer achievable with analog network coding over the diamond network with $N$ relay nodes.

\section{Diamond Network: The optimal rate achievable with analog network coding}
\label{sec:diamond}
Consider the diamond network of Figure~\ref{fig:diamond}. We can consider diamond network as a layered network with only one layer of relay nodes. Then using \eqref{eqn:modChnlParams}, \eqref{eqn:modChnlParams2}, and \eqref{eqn:snr}, we compute the SNR at the destination $t$ for any scaling vector $\bm{\beta} = (\beta_1, \ldots, \beta_N)$ as
\begin{equation}
\label{eqn:diamondSNR}
SNR_t = \frac{P_s}{\sigma^2} \frac{(\sum_{i=1}^N h_{si} \beta_i h_{it})^2}{1+\sum_{i=1}^N \beta_i^2 h_{it}^2}
\end{equation}

Using \eqref{eqn:maxAFrate}, the problem of computing the maximum ANC rate for this network thus can be formulated as
\begin{equation}
\label{eqn:diamondProb}
\max_{0 \le \bm{\beta}^2 \le \bm{\beta}_{max}^2} SNR_t,
\end{equation}
where $\bm{\beta}_{max} = (\beta_{1,max} \ldots, \beta_{N,max})$ with $\beta_{i,max}^2 = P_i/(h_{si}^2 P_s + \sigma^2), i \in {\mathcal N}, {\mathcal N} = \{1, \ldots, N\}$.

Equating the first-order partial derivatives of the objective function with respect to $\beta_i, i \in \mathcal{N}$, to zero, we get the following $N+1$ conditions for local extrema:
\begin{align}
&\sum_{i \in \mathcal{N}} h_{si} \beta_i h_{it} = 0 \label{eqn:b1bN} \\
&\beta_i = \frac{h_{si}/h_{it}}{\sum_{j \in \mathcal{N}\setminus\{i\}} h_{sj} \beta_j h_{jt}}\bigg(1+\sum_{j \in \mathcal{N}\setminus\{i\}} \beta_j^2 h_{it}^2\bigg) \label{eqn:bi_ito_rest}
\end{align}

\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{./diamond_N}
\caption{A diamond network with $N$ relay nodes.}
\label{fig:diamond}
\end{figure}

Let $SNR_{\beta_i \beta_j} = \frac{\partial^2 SNR_t}{\partial \beta_i \partial \beta_j}$ denote the second-order partial derivatives of $SNR_t$ with respect to $\beta_i$ and $\beta_j$, $i, j \in \mathcal{N}$ and $H(\bm{\beta})$ denote the determinant of $N \times N$ Hessian matrix.

First, consider the set of stationary points $S_{\bm{\beta}} = \{\bm{\beta}: \bm{\beta} \mbox{ satisfies } \eqref{eqn:b1bN}\}$. For all points in $S_{\bm{\beta}}$ we can prove that
\begin{align*}
SNR_{\beta_1 \beta_1} &> 0\\
H(\bm{\beta}) &= 0
\end{align*}
Therefore, the second partial derivative test to determine if the points in $S_{\bm{\beta}}$ are local minimum, maximum, or saddle points fails. However, we can establish that for every $\bm{\beta} \in S_{\bm{\beta}}$, the following set of conditions holds
\begin{align}
\pd{SNR_t}{\beta_i}\bigg|_{\bm{\beta}+\bm{\delta}} &< 0, \mbox{ if } \sum_{i \in \mathcal{N}} h_{si} h_{it} \delta_i < 0, \label{eqn:ineq1}\\
\pd{SNR_t}{\beta_i}\bigg|_{\bm{\beta}+\bm{\delta}} &> 0, \mbox{ if } \sum_{i \in \mathcal{N}} h_{si} h_{it} \delta_i > 0, \label{eqn:ineq2}\\
H(\bm{\beta}) &> 0, \mbox{ if } \sum_{i \in \mathcal{N}} h_{si} h_{it} \delta_i < 0, \label{eqn:ineq3}\\
H(\bm{\beta}) &> 0, \mbox{ if } \sum_{i \in \mathcal{N}} h_{si} h_{it} \delta_i > 0, \label{eqn:ineq4}
\end{align}
for all $\bm{\delta}=(\delta_1, \ldots, \delta_N) \rightarrow \bm{0}$. In other words, \eqref{eqn:ineq1} and \eqref{eqn:ineq2} imply that the slope of the function changes sign at $\sum_{i \in \mathcal{N}} h_{si} h_{it} \delta_i = 0$, and \eqref{eqn:ineq3} and \eqref{eqn:ineq4} imply that the convexity of the function, however, does not change at $\sum_{i \in \mathcal{N}} h_{si} h_{it} \delta_i = 0$. Therefore, together these imply that \eqref{eqn:b1bN} leads to a local minimum of the objective function.

Next, consider the set of points defined by \eqref{eqn:bi_ito_rest}. For all such points we can prove that
\begin{align*}
SNR_{\beta_1 \beta_1} &< 0\\
H(\bm{\beta}) &> 0
\end{align*}
Therefore, from the second partial derivative test the objective function attains its local maximum at the set of points characterized by \eqref{eqn:bi_ito_rest} above. However, no real solution of the simultaneous system of equations in \eqref{eqn:bi_ito_rest} exists. In other words, no solution of \eqref{eqn:diamondProb} exists where all relay nodes transmit strictly below their respective transmit power constraints. This is illustrated by the following example.

{\textit{\textbf{Example 1} (Three node Diamond Network):}} Consider the Gaussian diamond network of Figure~\ref{fig:diamond} with three relay nodes. For this network, \eqref{eqn:bi_ito_rest} results in:
\begin{align}
\beta_1 &= \frac{h_{s1}/h_{1t}}{h_{s2} \beta_2 h_{2t}+h_{s3} \beta_3 h_{3t}}\big(1+ \beta_2^2 h_{2t}^2 + \beta_3^2 h_{3t}^2\big) \label{eqn:beta1diamond3} \\
\beta_2 &= \frac{h_{s2}/h_{2t}}{h_{s1} \beta_1 h_{1t}+h_{s3} \beta_3 h_{3t}}\big(1+ \beta_1^2 h_{1t}^2 + \beta_3^2 h_{3t}^2\big) \label{eqn:beta2diamond3} \\
\beta_3 &= \frac{h_{s3}/h_{3t}}{h_{s1} \beta_1 h_{1t}+h_{s2} \beta_2 h_{2t}}\big(1+ \beta_1^2 h_{1t}^2 + \beta_2^2 h_{2t}^2\big) \label{eqn:beta3diamond3}
\end{align}

Substituting \eqref{eqn:beta2diamond3} in \eqref{eqn:beta3diamond3}, after a little algebraic manipulation we get:
\begin{equation*}
\label{eqn:beta3condition}
\{\beta_3 h_{3t} h_{s1} \beta_1 h_{1t} - h_{s3}(1+\beta_1^2 h_{1t}^2)\}\{(h_{s1} \beta_1 h_{1t}+h_{s3} \beta_3 h_{3t})^2 + h_{s2}^2(1 + \beta_1^2 h_{1t}^2 + \beta_3^2 h_{3t}^2)\}=0
\end{equation*}
Solving this gives three solutions for $\beta_3$, namely:
\begin{align}
\beta_{3,1} &= \frac{h_{03}/h_{3t}}{h_{s1} \beta_1 h_{1t}} (1+\beta_1^2 h_{1t}^2) \label{eqn:beta3sol1} \\
\beta_{3,2} &= \frac{- h_{s3} h_{s1} \beta_1 h_{1t} + i h_{s2} \sqrt{(h_{s2}^2 + h_{s3}^2)(1+\beta_1^2 h_{1t}^2) + (h_{s1} \beta_1 h_{1t})^2}}{h_{3t}(h_{s2}^2 + h_{s3}^2)}, i = \sqrt{-1} \label{eqn:beta3sol2} \\
\beta_{3,2} &= \frac{- h_{s3} h_{s1} \beta_1 h_{1t} - i h_{s2} \sqrt{(h_{s2}^2 + h_{s3}^2)(1+\beta_1^2 h_{1t}^2) + (h_{s1} \beta_1 h_{1t})^2}}{h_{3t}(h_{s2}^2 + h_{s3}^2)}, i = \sqrt{-1} \label{eqn:beta3sol3}
\end{align}
Substituting each of \eqref{eqn:beta3sol1}, \eqref{eqn:beta3sol2}, and \eqref{eqn:beta3sol3} in \eqref{eqn:beta2diamond3} results in three corresponding solutions for $\beta_2$, namely:
\begin{align*}
\beta_{2,1} &= \frac{h_{02}/h_{2t}}{h_{s1} \beta_1 h_{1t}} (1+\beta_1^2 h_{1t}^2) \\
\beta_{2,2} &= \frac{- h_{s2} h_{s1} \beta_1 h_{1t} + i h_{s3} \sqrt{(h_{s2}^2 + h_{s3}^2)(1+\beta_1^2 h_{1t}^2) + (h_{s1} \beta_1 h_{1t})^2}}{h_{2t}(h_{s2}^2 + h_{s3}^2)}, i = \sqrt{-1} \\
\beta_{2,3} &= \frac{- h_{s2} h_{s1} \beta_1 h_{1t} - i h_{s3} \sqrt{(h_{s2}^2 + h_{s3}^2)(1+\beta_1^2 h_{1t}^2) + (h_{s1} \beta_1 h_{1t})^2}}{h_{2t}(h_{s2}^2 + h_{s3}^2)}, i = \sqrt{-1}
\end{align*}
Therefore, we have three possible solutions for optimal $(\beta_2, \beta_3)$, \textit{i.e.} $(\beta_{2,1}, \beta_{3,1})$, $(\beta_{2,2}, \beta_{3,2})$, and $(\beta_{2,3}, \beta_{3,3})$.

Substituting $(\beta_{2,1}, \beta_{3,1})$ in \eqref{eqn:beta1diamond3} results in
\begin{equation*}
\beta_1^2 = -\frac{h_{s2}^2 + h_{s3}^2}{h_{1t}^2 (h_{s1}^2 + h_{s2}^2 + h_{s3}^2)},
\end{equation*}
which leads to complex valued solutions for optimal $\beta_1$.

Similarly, substituting $(\beta_{2,2}, \beta_{3,2})$, and $(\beta_{2,3}, \beta_{3,3})$ in \eqref{eqn:beta1diamond3} results in
\begin{equation*}
\beta_1^2 = -\frac{1 + h_{s2}^2 + h_{s3}^2}{h_{1t}^2 (h_{s2}^2 + h_{s3}^2)}
\end{equation*}
which also leads to complex valued solutions for optimal $\beta_1$.

This allows us to conclude that no real solution of the system of simultaneous equations in \eqref{eqn:beta1diamond3}-\eqref{eqn:beta3diamond3} exists.  {\hspace*{\fill}~\IEEEQEDclosed\par}

The above discussion implies that all points satisfying \eqref{eqn:b1bN} lead to the global minimum of the objective function in \eqref{eqn:diamondProb} and the global maximum of the objective function occurs at one of the $N$ hyperplanes (of dimension $N-1$) defined by $\beta_k = \beta_{k,max}, k \in \mathcal{N}$. Next we identify this hyperplane and characterize the corresponding optimal solution.

Consider the system of simultaneous equations in \eqref{eqn:bi_ito_rest} on the $(N-1)$-dimensional hyperplane defined by $\beta_k = \beta_{k,max}$.
\begin{equation}
\label{eqn:bi_ito_rest_on_kth_plane}
\beta_i = \frac{h_{si}}{h_{it}} \frac{1+\beta_{k,max}^2 h_{kt}^2+\sum_{j \in \mathcal{N}\setminus\{i,k\}} \beta_j^2 h_{jt}^2}{h_{sk} \beta_{k,max} h_{kt} + \sum_{j \in \mathcal{N}\setminus\{i,k\}} h_{sj} \beta_j h_{jt}}
\end{equation}
Note that the solution of the above system of equations is the set of scaling-factors for the nodes in the set of relay nodes $\mathcal{N} \setminus \{k\}$ that maximizes $SNR_t$ on hyperplane $\beta_k = \beta_{k,max}$. Solving the system of equations in \eqref{eqn:bi_ito_rest_on_kth_plane} results in the following set of optimal solutions for $\beta_i$ on hyperplane $\beta_k = \beta_{k,max}$:
\begin{equation}
\label{eqn:optBeta_on_kth_plane}
\beta_{i}^k = \frac{h_{si}}{h_{it}} \frac{1+\beta_{k,max}^2 h_{kt}^2}{h_{sk} \beta_{k,max} h_{kt}}, i \in \mathcal{N}\setminus \{k\}
\end{equation}

However, the optimal scaling factors in \eqref{eqn:optBeta_on_kth_plane} for $N-1$ nodes are computed without considering the upper bound $\beta_{i,max}$ on each $\beta_i, i \in \mathcal{N}\setminus\{k\}$. Therefore, taking into consideration the upper bound on the scaling factor for each node, the modified solution is computed as per the following lemma.

\begin{pavikl}
\label{lemma:optBeta}
The optimal scaling vector $\bm{\beta}_{opt}^k=(\beta_{1,opt}^k, \ldots, \beta_{N,opt}^k)$ for $N$ nodes on $\beta_k = \beta_{k,max}$ hyperplane such that each scaling factor satisfies the corresponding upper bound on its maximum value is given as
\begin{equation*}
\beta_{i,opt}^k = \begin{dcases}
                   \beta_{i,max}, i \in S^k \\
                     \frac{h_{si}}{h_{it}} \frac{1+\sum_{j \in S^k} \beta_{j,max}^2 h_{jt}^2}{\sum_{j \in S^k} h_{sj} \beta_{j,max} h_{jt}}, i \not\in S^k,
                  \end{dcases}
\end{equation*}
where $S^k$ is the set of nodes such that on hyperplane $\beta_k = \beta_{k,max}$, the optimal value of the scaling factor of a node is saturated to its corresponding upper bound, $S^k = \{k\} \cup \{i: \beta_{i}^k \ge \beta_{i,max}, i \in \mathcal{N}\setminus\{k\}\}$.
\end{pavikl}
\begin{IEEEproof}
Following the argument similar to the one used to prove the global extrema properties of \eqref{eqn:b1bN} and \eqref{eqn:bi_ito_rest}, we can prove that on the $\beta_k = \beta_{k,max}$ hyperplane, the $SNR_t$ achieves its global minimum at a hyperplane defined by
\begin{equation*}
\sum_{i \in \mathcal{N}\setminus\{k\}} h_{si} \beta_i h_{it} = 0
\end{equation*}
and its global maximum at the points defined by $\beta_i^k$ given in \eqref{eqn:optBeta_on_kth_plane}.

Let $M^k$ denotes the set of nodes for which $\beta_i^k$ computed in \eqref{eqn:optBeta_on_kth_plane} is greater than or equal to the corresponding upper bound $\beta_{i,max}$ on the maximum value of the scaling factor, \textit{i.e.} $M^k = \{i: \beta_{i}^k \ge \beta_{i,max}, i \in \mathcal{N}\setminus\{k\}\}$. For all such $\beta_i^k, i \in M^k$, after proving that $\pd{SNR_t}{\beta_i}\big|_{\beta_{i,max}} \ge 0$, we set $\beta_i^k = \beta_{i,max}$ and update $S^k$, the set of nodes such that on hyperplane $\beta_k = \beta_{k,max}$, the optimal value of each node is saturated to its corresponding upper bound; as follows: $S^k = S^k \cup M^k$. As $\beta_i^k$ computed in \eqref{eqn:optBeta_on_kth_plane} for a node $i \not\in S^k$ may no longer be optimal after the above re-assignment of $\beta_i^k, i \in M^k$, we need to solve the following simultaneous system of $N - |S^k| = N-|M^k|-1$ equations with $i \in \mathcal{N} \setminus S^k$:
\begin{equation}
\label{eqn:newSysOeqns}
\beta_i = \frac{h_{si}}{h_{it}} \frac{1+\sum\limits_{j \in S^k} \beta_{j,max}^2 h_{jt}^2+\sum\limits_{j \not\in S^k \cup \{i\}} \beta_{j}^2 h_{jt}^2}{\sum\limits_{j \in S^k} h_{sj} \beta_{j,max} h_{jt} + \sum\limits_{j \not\in S^k \cup \{i\}} h_{sj} \beta_{j} h_{jt}},
\end{equation}
Solving this system of equations results in
\begin{equation}
\label{eqn:recomputedBeta}
\beta_{i,opt}^k = \frac{h_{si}}{h_{it}} \frac{1+\sum_{j \in S^k} \beta_{j,max}^2 h_{jt}^2}{\sum_{j \in S^k} h_{sj} \beta_{j,max} h_{jt}}, i \not\in S^k
\end{equation}
Some of the recomputed scaling factors $\beta_{i,opt}^k, i \not\in S^k$ may violate the corresponding upper bound on their maximum value. All such nodes are added to set $S^k$, thus updating it. Then, the system of equations in \eqref{eqn:newSysOeqns} is solved again for this updated set $S^k$. This iterative process continues until none of the recomputed $\beta_i$ in \eqref{eqn:recomputedBeta} violates its corresponding upper bound. This iterative process is presented formally in terms of an algorithm: \textit{Algorithm 1}, given on the top of the next page.

Note that \textit{Algorithm 1} always halts with either $S^k = \mathcal{N} \setminus \{k\}$ or $S^k \subset \mathcal{N} \setminus \{k\}$ and $\beta_i^k < \beta_{i,max}, i \in \mathcal{N} \setminus S^k$.
\end{IEEEproof}

Using Lemma~\ref{lemma:optBeta}, for each of $N$ hyperplanes, defined as $\beta_k = \beta_{k,max}$, $k \in \mathcal{N}$, we can compute $\bm{\beta}_{opt}^k$, the set of scaling factors for all nodes at which $SNR_t$ attains its maximum on $\beta_k = \beta_{k,max}$ hyperplane. Then the hyperplane at which $SNR_t$ attains its global maximum is identified as follows:

\begin{pavikp}
\label{prop:optHyperplane}
The hyperplane at which $SNR_t$ attains its global maximum is defined as
\begin{equation*}
k^\star = \argmax_{k \in \mathcal{N}} SNR_t(\bm{\beta}_{opt}^k)
\end{equation*}
\end{pavikp}

Combining Proposition~\ref{prop:optHyperplane} and Lemma~\ref{lemma:optBeta}, we can characterize the scaling vector $\bm{\beta}_{opt}$ that solves the problem \eqref{eqn:diamondProb} as follows.

\newpage
\hspace{-1.0em}\hrulefill

\hspace{-1.0em}{\textbf{Algorithm 1}}

\vspace{-0.3cm}\hspace{-1.0em}\hrulefill
\begin{codebox}
\li Initialization:
\zi $S^k = \{k\}$, the set of nodes whose scaling factors are saturated to their respective upper-bounds
\zi on hyperplane $\beta_k = \beta_{k,max}$.
\zi $U^k = \mathcal{N}\setminus\{k\}$ , the set of nodes whose scaling factors are not saturated to their respective upper-bounds
\zi on hyperplane $\beta_k = \beta_{k,max}$.
\li Compute $\beta_{i}^k = {\displaystyle \frac{h_{si}}{h_{it}} \frac{1+\sum_{j \in S^k} \beta_{j,max}^2 h_{jt}^2}{\sum_{j \in S^k} h_{sj} \beta_{j,max} h_{jt}}}, i \in U^k$.
\li \While ($\exists \, \beta_{i}^k \ge \beta_{i,max}, i \in U^k$)
\li \hspace{-0.45in}  Compute $M^k = \{i: \beta_{i}^k \ge \beta_{i,max}, i \in U^k\}$.
\li \hspace{-0.45in}  $\beta_{i}^k = \beta_{i,max}, i \in M^k$.
\li \hspace{-0.45in}  $S^k = S^k \cup M^k$.
\li \hspace{-0.45in}  $U^k = U^k \setminus M^k$.
\li \hspace{-0.45in}  Compute $\beta_{i}^k = {\displaystyle \frac{h_{si}}{h_{it}} \frac{1+\sum_{j \in S^k} \beta_{j,max}^2 h_{jt}^2}{\sum_{j \in S^k} h_{sj} \beta_{j,max} h_{jt}}}, i \in U^k$.
    \End
\end{codebox}
\vspace{-0.2cm}\hrulefill

\begin{pavikt}
\label{thrm:optBeta4diamond}
A network-wide scaling vector $\bm{\beta}_{opt} = (\beta_{1}^{opt}, \ldots, \beta_{N}^{opt})$ that maximizes the $SNR_t$ for a diamond network with the relay nodes performing ANC is given as
\begin{equation*}
\beta_{i}^{opt} = \begin{dcases}
                    \beta_{i,max}, i = k^\star, k^\star = \argmax\limits_{j \in \mathcal{N}}SNR_t(\bm{\beta}_{opt}^j),\\
                    \beta_{i,max}, i \in S^{k^\star},\\
                    {\displaystyle \frac{h_{si}}{h_{it}} \frac{1+\sum\limits_{j \in S^{k^\star}} \beta_{j,max}^2 h_{jt}^2}{\sum\limits_{j \in S^{k^\star}} h_{sj} \beta_{j,max} h_{jt}}, i \not\in S^{k^\star}},
                  \end{dcases}
\end{equation*}
where $S^{k^\star} = \{k^\star\} \cup \{i: \beta_{i}^{k^\star} \ge \beta_{i,max}, i \in \mathcal{N}\setminus\{k^\star\}\}$.
\end{pavikt}

Based on our approach in this section to compute the optimal ANC rate in the Gaussian diamond networks, in the next section we introduce a greedy scheme to bound from below the maximum end-to-end rate achievable with analog network coding in the general layered networks.

\section{General layered networks: a greedy scheme to lower bound the maximum ANC rate}
\label{sec:genNet}
In a general layered network with $L$ layers of relay nodes, consider layer $l, 1 \le l \le L$, and a node in the next $l+1^\textrm{st}$ layer, denoted as $t_{l+1}$ or with a little abuse of notation as $t$. This scenario is depicted in Figure~\ref{fig:l2lplus1}. For this subnetwork, for any scaling vector $\bm{\beta}$ we have
\begin{equation}
\label{eqn:subnetSNR}
SNR_t = \frac{P_s(\sum_{i=1}^N s_i \beta_i h_{it})^2}{\mathbb{E}(z_t+\sum_{i=1}^N z_i \beta_i h_{it})^2}
\end{equation}

Using \eqref{eqn:maxAFrate} the problem of computing the maximum ANC rate for this subnetwork can be formulated as
\begin{equation}
\label{eqn:l2lplus1prob}
\max_{0 \le \bm{\beta} \le \bm{\beta}_{max}} SNR_t,
\end{equation}
where $\bm{\beta} = (\beta_1, \ldots, \beta_N)$ and $\bm{\beta}_{max} = (\beta_{1,max} \ldots, \beta_{N,max})$ with $\beta_{i,max}^2 = P_i/\mathbb{E}(s_i x_s + z_i)^2, i \in {\mathcal N}, {\mathcal N} = \{1, \ldots, N\}$.

Equating the first-order partial derivatives of the objective function with respect to $\beta_i, i \in \mathcal{N}$, to zero, we get the following $N+1$ conditions for local extrema:
\begin{align}
&\sum_{i \in \mathcal{N}} s_{i} \beta_i h_{it} = 0 \label{eqn:gen_b1bN} \\
&\beta_i = \frac{s_i + s_i \mathbb{E}\big(\sum\limits_{j \in \mathcal{N} \setminus \{i\}} z_j \beta_{j} h_{jt}\big)^2 -  \alpha_i \gamma_i}{h_{it}(\alpha_i \mathbb{E}z_i^2 / \sigma^2 - s_i \gamma_i)} \label{eqn:gen_bi_ito_rest}
\end{align}
where
\begin{align*}
\alpha_i &= \sum\limits_{j \in \mathcal{N} \setminus \{i\}} s_j \beta_{j} h_{jt}, \quad\qquad \mbox{(signal component at destination $t$ from all the nodes except node $i$)}\\
\gamma_i &= \sum\limits_{j \in \mathcal{N} \setminus \{i\}} \beta_{j} h_{jt} \mathbb{E}(z_i z_j) / \sigma^2, \quad \mbox{(noise component at destination $t$ from all the nodes except node $i$)}
\end{align*}

As we established the extremal properties of conditions \eqref{eqn:b1bN} and \eqref{eqn:bi_ito_rest} in Section~\ref{sec:diamond}, we can also prove that condition \eqref{eqn:gen_b1bN} leads to the global minimum of the objective function in \eqref{eqn:l2lplus1prob} and the global maximum of the objective function occurs at one of the $N$ hyperplanes defined by $\beta_k = \beta_{k,max}$. Following a sequence of arguments similar to those used to establish Theorem~\ref{thrm:optBeta4diamond} for diamond networks, we can characterize the scaling vector for the nodes in the $l^\textrm{th}$ layer that optimally solve the problem~\eqref{eqn:l2lplus1prob} for the subnetwork under consideration. Note that in this subnetwork, the noises at different nodes in a relay layer are correlated, unlike the noises at relay nodes in the diamond network in Figure~\ref{fig:diamond}. This explains the difference between the $SNR$ expression in \eqref{eqn:subnetSNR} and the one in \eqref{eqn:diamondSNR} for the diamond network, and results in more complex analysis in the present case.

\begin{pavikl}
\label{lemma:optBeta4l2lplus1}
A scaling vector $\bm{\beta}_{opt} = (\beta_{1}^{opt}, \ldots, \beta_{N}^{opt})$ that maximizes the $SNR_t$ for any subnetwork, as in Figure~\ref{fig:l2lplus1}, of the general layered network with the relay nodes in $l^\textrm{th}$ layer performing analog network coding is given as
\begin{equation*}
\beta_{i}^{opt} = \begin{dcases}
                    \beta_{i,max}, i = k^\star, k^\star = \argmax_{\{\bm{\beta}^j:j \in \mathcal{N}\}} SNR_t(\bm{\beta}^j),\\
                    \beta_{i,max}, i \in S^{k^\star},\\
                    \frac{s_i + s_i \mathbb{E}\big(\sum\limits_{j \in S^{k^\star}} z_j \beta_{j,max} h_{jt}\big)^2 - \alpha_j \gamma_j}{h_{it}(\alpha_j\mathbb{E}z_i^2 / \sigma^2 - s_i \gamma_j)}, i \not\in S^{k^\star},
                  \end{dcases}
\end{equation*}
where $\bm{\beta}^j=(\beta_1^j, \ldots, \beta_N^j)$ with
\begin{align*}
\beta_i^j &= \begin{dcases}
               \beta_{j,max}, i = j\\
               \frac{s_i + \{s_i \mathbb{E}z_j^2 / \sigma^2 - s_j \mathbb{E}(z_i z_j) / \sigma^2\}\beta_{j,max}^2 h_{jt}^2}{h_{it}\{s_j \mathbb{E}z_i^2 / \sigma^2 - s_i \mathbb{E}(z_i z_j) / \sigma^2\}\beta_{j,max} h_{jt}}, i \neq j,
             \end{dcases}\\
\alpha_j &= \sum_{j \in S^{k^\star}} s_j \beta_{j,max} h_{jt}, \qquad\qquad \mbox{(signal component at destination $t$ from the nodes in $S^{k^\star}$)}\\
\gamma_j &= \sum_{j \in S^{k^\star}} \beta_{j,max} h_{jt} \mathbb{E}(z_i z_j) / \sigma^2, \quad \mbox{(noise component at destination $t$ from the nodes in $S^{k^\star}$)}
\end{align*}
and $S^{k^\star} = \{k^\star\} \cup \{i: \beta_{i}^{k^\star} \ge \beta_{i,max}, i \in \mathcal{N}\setminus \{k^\star\}\}$.
\end{pavikl}

Note that Lemma~\ref{lemma:optBeta4l2lplus1} reduces to Theorem~\ref{thrm:optBeta4diamond} when the noise components at the relay nodes are uncorrelated.

\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{./l2lplus1}
\caption{A subnetwork of general layered network with $L$ relay layers, depicting $l^\textrm{th}$ layer with $N$ relay nodes and a node in the $l+1^\textrm{st}$ layer. The received signal component at node $i, 1 \le i \le N$, in the $l^\textrm{th}$ layer is denoted as $s_i x_s$, where $x_s$ is the source symbol and corresponding noise component is denoted as $z_i$.}
\label{fig:l2lplus1}
\end{figure}

Using Lemma~\ref{lemma:optBeta4l2lplus1}, we can compute $\bm{\beta}_{l,opt}^{l+1, j}$, the scaling vector for the nodes in the $l^\textrm{th}$ layer that maximizes the received SNR for node $j, 1 \le j \le n_{l+1}$, in the $l+1^\textrm{st}$ layer. Among these $n_{l+1}$ scaling vectors for the nodes in the $l^\textrm{th}$ layer, let $\bm{\beta}_{l}^{low}$ denote the one that solves the following problem
\begin{equation}
\label{eqn:bestlbeta}
\bm{\beta}_{l}^{low} = \argmax_{\substack{\bm{\beta}_{l,opt}^{l+1, j} \\ 1 \le j \le n_l}} \prod_{k=1}^{n_{l+1}} (1+SNR_k)
\end{equation}
The following corollary of Lemma~2 in \cite{112agnihotriJaggiChen} establishes that among $n_{l+1}$ such scaling vectors, the scaling vector characterized by $\bm{\beta}_{l}^{low}$ computes the tightest lower bound for the optimal value of the objective function in \eqref{eqn:bestlbeta} as well as \eqref{eqn:eqProb}.

\begin{pavikc}[\hspace{-0.001cm}\cite{112agnihotriJaggiChen}, Lemma 2]
\label{cor:betterBeta}
Consider two scaling vectors $\bm{\beta}_l$ and $\bm{\hat\beta}_l$ for the nodes in $l^\textrm{th}$ layer. If $\prod_{k=1}^{n_{l+1}} (1+SNR_k)\big|_{\bm{\beta}_l} > \prod_{k=1}^{n_{l+1}} (1+SNR_k)\big|_{\bm{\hat\beta}_l}$, then $SNR_t(\bm{\beta}_l) > SNR_t(\bm{\hat\beta}_l)$.
\end{pavikc}

Computing $\bm{\beta}_{l}^{low}$ as above for each layer $l, 1 \le l \le L$, in conjunction with Corollary~\ref{cor:betterBeta}, allows us to construct a network-wide scaling vector $\bm{\beta}_{low} = (\bm{\beta}_{1}^{low}, \ldots, \bm{\beta}_{L}^{low})$ to compute a lower bound\footnote{Clearly, choosing $\bm{\beta}_{l}^{low}$ as in \eqref{eqn:bestlbeta} for each layer $l$ may lead, in general, to some performance loss at each layer as $\bm{\beta}_{l}^{low}$ may not be the optimal vector of the scaling factors for the nodes in the layer $l$ that solves
\begin{equation*}
\argmax_{0 \le \bm{\beta}_{l} \le \bm{\beta}_{l,max}} \prod_{k=1}^{n_{l+1}} (1+SNR_k)
\end{equation*}
The cumulative effect of this performance loss at each layer is that the end-to-end ANC rate computed at $\bm{\beta}_{low}$ may not lead to the optimal solution of problem~\eqref{eqn:maxAFrate}. However, our results in the next section show that for a large class of layered networks there is no loss in the optimality and for other layered networks, the loss is \textit{small} asymptotically in the network parameters.} to the optimal solution of \eqref{eqn:maxAFrate}. Formally, for a given layered network, $\bm{\beta}_{low}$ is constructed as follows.
\begin{pavikp}
\label{prop:lowBetaGenNet}
Consider a layered relay network of $L+2$ layers, with the source $s$ in layer `$0$', the destination $t$ in layer `$L+1$', and $L$ layers of relay nodes between them. The $l^\textrm{th}$ layer contains $n_l$ nodes, $n_0 = n_{L+1} = 1$. A network-wide scaling vector $\bm{\beta}_{low}=(\bm{\beta}_{1}^{low}, \ldots, \bm{\beta}_{L}^{low})$ that provides a lower bound to the optimal solution of \eqref{eqn:maxAFrate} for this network, can be computed recursively for $1 \le l \le L$ as
\begin{equation*}
\bm{\beta}_{l}^{low} = \argmax_{\substack{\bm{\beta}_{l,opt}^{l+1, j} \\ 1 \le j \le n_l}} \prod_{k=1}^{n_{l+1}}(1+SNR_{l+1, k}(\bm{\beta}_{1}^{low}, \ldots, \bm{\beta}_{l-1}^{low}, \bm{\beta}_{l,opt}^{l+1, j}))
\end{equation*}
Here $\bm{\beta}_{l}^{low} = (\beta_{l 1}^{low}, \ldots, \beta_{l n_l}^{low})$ is the vector of scaling factors for the nodes in the $l^\textrm{th}$ layer, and $\bm{\beta}_{l,opt}^{l+1, j}$ (computed using Lemma~\ref{lemma:optBeta4l2lplus1}) is the scaling vector for the nodes in the $l^\textrm{th}$ layer that maximizes the received SNR for node $j, 1 \le j \le n_{l+1}$, in layer $l+1$.
\end{pavikp}

In the next section we analyze the performance of the greedy scheme of the Proposition~\ref{prop:lowBetaGenNet} in the context of both a special class of layered networks and the general layered networks.

\section{Illustration}
\label{sec:exa}
We first demonstrate that the greedy scheme of Proposition~\ref{prop:lowBetaGenNet} allows us to exactly compute the optimal ANC rate for a broad class of layered networks. Then, we give an example to show that for the general layered networks, the proposed scheme leads to the optimal rates within a constant gap from the cut-set upper bound asymptotically in the source power.

\textit{\textbf{Example 2} (A class of exactly solvable layered networks):} Let us consider a class of symmetric layered networks where the channel gains along all outgoing links from a node are equal. An instance of such a network is obtained from the network in Figure~\ref{fig:layrdNetSpl_Exa} when $h_{s1}=h_{s2}, h_{13}=h_{14}$, and $h_{23}=h_{24}$. An implication of this property of the channel gains is that the received SNRs at every node in a layer are equal: $SNR_{l, j}=SNR_l, 1 \le j \le n_l, 1 \le l \le L$. In this case, for each layer $l, 1 \le l \le L$, $\bm{\beta}_{l}^{low}$ computed in Proposition~\ref{prop:lowBetaGenNet} is equal to the optimal $\bm{\beta}_l^{opt}$ computed in \cite[Lemma 2]{112agnihotriJaggiChen}. Therefore, $\bm{\beta}_{low}$ is the optimal solution of problem \eqref{eqn:eqProb} for this class of networks.

\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{./layrdNetSpl_Exa}
\caption{General layered network with 2 relay layers between the source `s' and destination `t'. Each layer contains two relay nodes.}
\label{fig:layrdNetSpl_Exa}
\end{figure}

Consider an instance of the network in Figure~\ref{fig:layrdNetSpl_Exa} when $h_{s1}=h_{s2}, h_{13}=h_{14}$, and $h_{23}=h_{24}$. Such an instance belongs to the class of symmetric networks we are concerned with in this example. Using Proposition~\ref{prop:lowBetaGenNet}, the optimal solution of problem \eqref{eqn:eqProb} for this instance is:
\begin{align*}
&\bm{\beta}_{opt} = \bigg(\beta_{1,max}, \frac{1+\beta_{1,max}^2 h_1^2}{h_2 \beta_{1,max} h_1}, \beta_{3,max}, \beta_{4,max}\bigg), \mbox{ where}\\
&\beta_{1,max}^2 = \frac{P_1}{h_0^2 P_s + \sigma^2} \\
&\beta_{3,max}^2 = \frac{P_3}{S^2 P_s + Z^2 \sigma^2}, \beta_{4,max}^2 = \frac{P_4}{S^2 P_s + Z^2 \sigma^2} \\
&S = h_0(\beta_{1,opt} h_1 + \beta_{2,opt} h_2), Z^2 = 1 + \beta_{1,opt}^2 h_1^2 + \beta_{2,opt}^2 h_2^2
\end{align*}
and we assume that $P_1 h_1^2 > P_2 h_2^2$. {\hspace*{\fill}~\IEEEQEDclosed\par}

{\textit{\textbf{Example 3} (General layered networks):}}
Let us consider the layered network of Figure~\ref{fig:layrdNetSpl_Exa}. We compute a lower bound to the optimal ANC rate for this network using the greedy scheme in the Proposition~\ref{prop:lowBetaGenNet} and compare it with the MAC upper bound in Figure~\ref{fig:rateComparison}. Also, plotted in this figure is the ANC rate achievable when the scaling factors for all relay nodes are set to their respective upper-bounds. We observe that in this case the ANC rate achieved with the greedy scheme of Proposition~\ref{prop:lowBetaGenNet} approaches the capacity within one bit when $P_s > 100$.

\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{./rateComparison1t}
\caption{Comparison of the ANC rate achievable with the scheme in Proposition~\ref{prop:lowBetaGenNet} with the MAC upper bound for the layered network in Figure~\ref{fig:layrdNetSpl_Exa} with $P_1=P_2=P_3=P_4=10$, $h_{14}=h_{24}=2$ and all other channel gains are equal to $10$. Also plotted is the ANC rate when the scaling factors for all relay nodes are set to their respective upper-bounds.}
\label{fig:rateComparison}
\end{figure}

\section{Conclusion and Future Work}
\label{sec:conclFW}
We consider the problem of maximum rate achievable with analog network coding in general layered networks. Previously, this problem has been considered under certain assumptions on per node scaling factor and received SNR as without these assumptions the problem was presumed to be intractable. The key contribution of this work is a greedy scheme to exactly compute the optimal rates in a wider class of layered networks than those that can be addressed using prior approaches. In particular, using the proposed scheme for the Gaussian $N$-relay diamond network, to the best of our knowledge, we provide the first exact characterization of the optimal rate achievable with analog network coding. Further, for general layered networks, our scheme allows us to compute optimal rates at most a constant gap away from the cut-set upper bound asymptotically in the source power. In the future, we plan to extend this work to non-layered networks, and to construct the optimal distributed relay schemes.

redpajama_set_name: RedPajamaArXiv


text:
ASF Covid-19 Scheme
The Virtual School Team provide support and advice to education settings and professionals who are working with children who have previously been in care.
Adoptive families will be offered emergency support including online counselling and couples therapy as the Government expands the scope of the Adoption Support Fund to meet needs arising from the outbreak of coronavirus (COVID-19).
The Education Secretary has today (Friday 10 April) announced that up to £8 million will be available to pay for different types of therapeutic support for families whose adopted children may have already suffered trauma and be made more anxious owing to the uncertainty of the effects of the virus.
It comes as the Department for Education confirms it has spent more than £150 million on the Adoption Support Fund since it was launched in 2015, helping more than 45,000 adoptive and special guardianship order families across the country.
The £8 million fund can be used by local authorities and regional adoption agencies to pay for activities such as virtual peer to peer support, access to helplines, couples therapy and online counselling, given the social distancing measures that have been brought in by the Government to protect the NHS and save lives.
For more information, please click on the below link.
https://www.gov.uk/government/news/covid-19-adoption-support-fund-scheme-to-help-vulnerable-families


A landmark, in the history of CMS was the CMS Teaching Hospital inaugurated by the dynamic present Chairman & CEO Mr. Nagender K. Pampati. This has revolutionized the healthcare scenario in this region with a combination of clinical expertise and advanced diagnostic facilities. The College of Medical Sciences-Teaching Hospital is the focus of the clinical teaching activity of the College. A 1050 bedded facility with the largest ICU & OT Complex in the region with the state of the art equipments. The CMS teaching hospital is a tertiary care facility that provides the trainee with the opportunity to acquire knowledge and skills from the finest doctors in the country.
The present physical facilities at the College of Medical Sciences has been created to provide its students the best educational environment in this country, multi-storied spacious technologically advanced well ventilated Lecture Halls, Library, Laboratories, Seminar and Tutorial Rooms, Media Lab with Computer aided Learning Centre which are connected with modern audio visual equipments and CCTVs and all the OPDs, basic, clinical and post doctoral departments are air-conditioned.
The college has been coordinating and collaborating with various national teaching Institutions like the B.P.Koirala Institute of Health Sciences, Dharan, B.P.Koirala Memorial Cancer Hospital, Bharatpur Govt. hospital of Bharatpur. The college has also liaison with the Royal College of Physicians of Edinburgh and the Armed Forces Research Institute of Medical Sciences, (AFRIMS) Bangkok, Thailand.
The international Society for Medical Education Private Limited (ISME) and the Ministry of Education (MoE), Government of Nepal signed an agreement of the 8th day of August 1993 giving birth to the College of Medical Sciences-Nepal at Bharatpur, Chitwan (Dist.) and has undergone lot of modifications since than providing affordable advanced quality health care services and medical education. The College of Medical Sciences was established under the leadership of Mr. Nagendar K. Pampati, MBA, Chairman & CEO of ISME Pvt. Ltd.
The College of Medical Sciences will actively promote and preserve the highest values and ethics in education to produce medical and paramedical professionals of global standards, to create a centre of excellence in the medical education, the modern facilities and state of art equipments for excellent specialized health care in the region.
The goal of the College of Medical Sciences is to select capable students who can gain by attending academic activities to excel in both the art and science of medicine. It is emphasized that student is always of a heterogeneous group. We recognize that a diverse student body promotes an atmosphere of creativity, experimentation and create as conducive atmosphere for learning. The exposure to a variety of perspectives and experiences prepare students to care for patients from all walks of life and from every segment of society. Individuality, age, colour, gender, race, regional origin, religion, status, disability are not determinants of diversity and are not identified as unique characteristics during the admission process.
Joinus 4 Global Education assists in easy and direct admission in mbbs in Nepal for indian students, cheapest mbbs in Nepal for indian students, visa process and Admission in MBBS in Nepal for indian students and other related guidance. Contact Joinus 4 Globel Education for direct mbbs admission in Nepal and cheapest mbbs in Nepal for indian students to obtain the best quality of education from study MBBS in Nepal for Indian Students. Important to know direct admission in mbbs in nepal for indian students, Great opportunity for every one to study mbbs in Nepalfor indian students .