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saleguides.us
January 12, 2020 January 12, 2020 editor
The Good The Samsung Galaxy Note 9 is a big, beautiful phone with top-tier specs including a massive battery and internal storage that starts at 128GB. The new S Pen doubles as a wireless remote for taking long-distance selfies.
The Bad It's expensive and offers few real innovations over last year's Note 8. The fingerprint reader is uncomfortably close to the camera.
The Bottom Line The ultrapricey Note 9 is one of the year's best phones, but wait until Samsung announces its Galaxy Note 10 successor to make a decision.
It's Wednesday, and Samsung is about to announce its Galaxy Note 10 phone, the successor to the Galaxy Note 9. The Note, a line aimed at power users, typically represents the best Samsung has to offer, but depending what Samsung trots out — and how much it costs — the Note 9 could also benefit from a boost of attention.
It's a strange year for the Note line. While Samsung is expected to show off a phone with a new and improved design, S Pen stylus and internal specs, the Note 10 is also surrounded by more competition than ever before, much of it coming from among Samsung's ranks. There are four Galaxy S10 phones, including the 5G variety, and the Galaxy Fold, whose screen problems Samsung said are now fixed, and which is slated to sell in September.
If the Galaxy Note 10 stirs up enough interest in Note phones, and Samsung, carriers and retailers slash the Note 9 price enough, 2018's device could pick up additional sales, especially if potential customers who were on the fence decide they'd rather save a hundred bucks or two than invest in Samsung's next big thing.
We'll report on the specs Samsung plans to tantalize us with in its Note 10, but the Note 9 was an excellent device in its own right. Sure, we felt that Samsung was holding back some of the finer points that turn a phone from good to great or great to fantastic, but all in all, we'd happily use a Note 9 to do everything from navigate around (hello, less smudgy screen) to jot down memos in the conference room or on the go.
Follow along with Samsung's Galaxy Note 10 announcement. Until then, remind yourself that the Galaxy Note is the reason phone screens today are so big.
Update, Aug. 7, 2019: Below is CNET's Galaxy Note 9 review, originally published March 8, 2018, and last updated Aug 24. 2018.
I picked up the Galaxy Note 9, popped out the S Pen stylus and started to write on the black screen in tart lemon-colored digital ink. It hit me: This phone is more fun to use than your phone. When I'm using the Note 9, I feel more inspired to write, draw, take precise screenshots using the tool, snap selfies with the S Pen's remote shutter, and playfully annotate photos to send to friends.
But the Note 9 is no mere toy. It's also powerful as hell, with a 6.4-inch screen, 4,000-mAh battery, Qualcomm Snapdragon 845 processor and either 128GB or a whopping 512GB (!!!) of onboard storage, plus a microSD card if you want more, more, more.
Yet the fizzing question at the center of it all, the one that's pounding away at your grey matter, is this: Are the power and fun of Samsung's new Galaxy Note 9 worth $1,000 of your hard-earned cash? (That's £899 and AU$1,499 — or $1,250, £1,099 and AU$1,799 for the 512GB version.) It's the same question Apple fans have been asking themselves since the iPhone X first hit the $1,000 mark last year.
For Android fans who want the best, the answer is "yes."
The Galaxy Note 9 looks great from every angle.
Angela Lang/CNET
The Galaxy Note 9's specs are top-notch. This is an everything phone, one of the absolute best you can buy with All The Things. It will carry you through the next two or three years with excellent photos, Android updates and all that jazz.
And while the price is eye-wateringly high compared to last year's Note 8 (unless you live in Australia, in which case it costs the same), promotions, preorder savings and future holiday deals can knock hundreds off the Note 9's cost, making it suddenly much more "affordable."
Read: Sorry, your iPhone or Android phone is getting more expensive
So, is there any reason to not get the Note? Well, yes. For starters, it lacks a certain "wow" factor. Apart from the tad higher battery capacity and double the storage, there's not all much different from the Galaxy S9 Plus, or really from the Galaxy Note 8 before it.
Bonkers storage is good, but you could also buy a cheaper phone and scoop up external memory for much less than the cost of a new Note 9 — there are fewer phones with this option, but the Galaxy S9 and S9 Plus have it. And while the Note 9's battery life will take you from morning to late night on a single charge, is it really worth the price of 300 cappuccinos? Can you find two hours each day to charge from 0 to 100 percent? And 15 minutes extra if you need an emergency top-up?
As for the Note 9's new, cool S Pen stylus — Bluetooth turns it into a remote control, but it feels forced to use it in day-to-day life.
We didn't get a rumored in-screen fingerprint reader like what some other Android phones have, or a 3D front-facing camera like the iPhone X and Oppo Find X ($678 at Amazon). And the Note 9 can't latch on to insanely fast 5G data speeds whose networks will start bubbling up in 2019. This phone feels like Samsung's holding back for next year's Galaxy S10.
The Note 9's S Pen gets a Bluetooth antenna for a brain.
Read: 'Missing' Galaxy Note 9 features that Samsung may be saving for the Galaxy S10
So yes, buy the Galaxy Note 9 if you're upgrading from an older phone today and want the most feature-rich, super-powerful, large-screen Android phone out there. But if your current phone is in good shape and you don't care two clicks about the S Pen stylus — then wait. 2018's iPhones and Google's Pixel 3 ($290 at Amazon) are on their way in weeks, not months, and next spring's Galaxy S10 should help kick off a larger revolution with 5G.
Keep reading for the Note 9's new features, and how it compares to rival phones like the iPhone X. Here are the Galaxy Note 9 specs compared with the Galaxy S9, S9 Plus and Note 8.
Note 9 highlights
Battery life is strong in real-world testing. I'll continue to keep an eye on long-term drain.
The S Pen stylus' new Bluetooth features work as advertised.
As a natural note taker, I love being able to jot things down. I've made so many lists to pin to the lock screen.
I tested in both blue and purple, and the Note 9's bold colors stand out. It also sells globally in black and metallic copper.
Every Note phone (except the recalled Note 7) from the very beginning.
Note 9 low points
You won't be able to write or draw to the screen edge without the S Pen falling off the curved sides.
The fingerprint reader is too close to the camera array. Why hasn't Samsung figured out the optimal placement yet?
The Note 9's new AI camera tool works more slowly than I'd like for identifying scenes and optimizing settings for the best shot.
Bixby 2.0 is expanded, but the button on the Note 9's left side still only maps to Bixby, as it does with the Galaxy S8 and newer.
If you write on the phone screen with the S Pen's signature color (yellow, purple or copper), any notes you save will save in that color "ink" on a white background, which can be hard to read.
Where and how to buy the Galaxy Note 9
You can preorder the Galaxy Note 9 now. The phone goes on sale generally on Aug. 24. Keep an eye out for trade-in deals, promotions and bundled gifts. These values can change region by region.
Here's everything you need to know about buying the Galaxy Note 9 in the US.
Galaxy Note 9 looks smooth and oh-so-familiar
I've been using the Galaxy Note 9 in ocean blue and lavender purple, both of which I like. This phone is classic modern Samsung, with deep color and shine, a glossy finish that effortlessly picks up prints, and gracefully curved edges. It's a gorgeous device, even if very little of the Note's design feels fresh.
The Galaxy Note 9 is the only Galaxy phone to pack a digital pen.
That's because the phone's 83 percent screen-to-body ratio (18.5:9) hasn't changed all that much from last year's Note 8. There are a few minor exceptions, including the fingerprint reader that now sits just under the camera module, and a slightly larger S Pen holder for a slightly larger S Pen.
I found the tall, narrow Note 9 easy enough to hold and use in my smaller hands, but I really had to lunge to hit that fingerprint reader. Those with larger hands will have more luck, but may find the target a little small. It's a good idea to unlock the phone interchangeably with the iris scanner as well (both are secure enough for mobile payments).
More than one coworker or friend pointed out that the towering Note 9 was threatening to drop out of my back pocket. I typically carry it there when it's not hitching a ride inside my purse pocket.
I also kept inadvertently hitting the Bixby Voice button on the phone's left edge, which calls up Samsung's version of Google Assistant and Apple's Siri. It's a minor design problem, but still.
Who wouldn't want a headphone jack?
The Galaxy Note 9's new S Pen picks up Bluetooth tricks
The new S Pen is the Galaxy Note 9's standout feature by far. You can still write, draw, navigate around and create live messages. But now, the S Pen's button can trigger different actions: You can open the camera app, flip the camera around and take a photo, all in a couple clicks. Of all the things you can do with the S Pen, this impressed me the most.
The Pen also lets you control a presentation — you can advance slides, play or fast-forward songs in a music player and shuffle through photos in a kind of gallery slideshow. The new skills rely on Bluetooth Low Energy (there's an antenna inside the pen body), and you can customize the actions in a dedicated settings menu. Your options are limited for now, but Samsung wants to get other app makers on board.
The S Pen remote is said to work up to 30 feet away from the phone, but I was still taking photos from 100 feet away. Standby time is either 200 clicks or 30 minutes, and the S Pen recharges inductively in under a minute, starting when you reinsert the stylus into the holster.
Your phone will warn you if it's been without the S Pen for too long. If you lose the tool, you'll have to buy a replacement — there's no Find my S Pen feature.
Here's another fun perk: The Note 9 will now write in the color of its S Pen when you're jotting notes on the black lock screen (the feature is called Screen-off memo). So, that's yellow, purple, copper and white (for the black version). You can also switch to white "ink" if you'd prefer. At this point, though, why can't you pick any color you like?
Note 9's new AI camera features
Sadly, Samsung's new AI camera software for it dual 12-megapixel cameras isn't anything to get excited about.
The AI software analyzes a scene and quickly detects if you're shooting a flower, food, a dog, a person or something else entirely. There are 20 options, including snowflakes, cityscapes, fire — you get it. Then the camera optimizes white balance, saturation and contrast to make photos pop.
Food mode pops on automatically.
It works fine, and you'll see some big differences when photographing your lunch for Instagram photos. But the scene optimizer often takes a beat to kick in, and you can't dismiss the suggestions with a swipe the way you can on the Huawei P20 Pro. It's either on, or off.
Samsung is following the way the industry is going with AI camera software. But others got there first, so it feels like a minimal addition to what is essentially the S9 Plus' camera app. It's hardly worth calling out.
Yes, you can draw with it, too.
Another tiny featurette: flaw detection. If someone blinks an eye or moves, or the lens is smudged or there's too much backlight getting in the way of a clean shot, a dialog box pops up right after you take the picture, prompting you to retake the picture.
Apart from these software additions, the Note 9 keeps the S9 phones' main camera lens, which automatically changes aperture when it detects the need for a low-light shot. (Samsung calls this dual aperture.) There's also portrait mode and an 8-megapixel front-facing camera for your selfies. Photos from both are very good, though any movement in low light shots can cause unwanted blur.
Read: Our epic phone camera shootout with the Galaxy S9 Plus, iPhone X, Huawei P20 Pro and Pixel 2
Battery life, internal speeds, speakers
Battery life was pretty terrific on the Galaxy Note 9. Sometimes, for whatever reason, a phone doesn't always live up to its battery capacity. This one does. The US version of the Note 9 lasted for an average of 19 hours and 20 minutes in CNET's looping video drain test, involving five tests so far with two different devices.
For chipset nerds, our test of the Note 9 running on Samsung's own processor, the Exynos 9810, is still underway. It's topping 20 hours in our battery drain test so far.
In real-world observation, the Note 9 is holding its own, and I haven't yet gotten nervous that the phone will die on me if I stay out late. I'll continue to keep an eye on battery drain day-to-day in the coming weeks.
Keep in mind that all batteries are at their most efficient when you start using your device, and then drain faster (they won't last as long between charges) as time goes on. It isn't ideal, but it's the way things will go until or unless battery tech changes dramatically. Regardless, with the Note 9, you're off to a good start.
Internal speeds are just as fast on benchmark tests and real life usage as other phones using Qualcomm's Snapdragon 845 chipset. Some tasks will still take time because all your photo processing, and PUBG– and Fortnite-playing take their toll on system resources. We did notice that the Note 9 didn't get nearly as hot as the Galaxy S9 while playing Fortnite, and you won't notice undue lag while you're in the business of using your phone.
One final shout-out goes to the Note 9's surround speakers, which are (the same) and just as loud as those on the Galaxy S9. Just remember not to cover the bottom speaker if you hold the phone to your ear.
Galaxy Note 9 versus iPhone X, Galaxy S9 Plus, Google OnePlus 6
Samsung Galaxy Note 9 Samsung Galaxy S9 Plus iPhone X OnePlus 6
Display size, resolution
6.4-inch Super AMOLED; 2,960×1,440 pixels 6.2-inch; 2,960×1,440 pixels 5.8-inch; 2,436×1,125 pixels 6.28-inch OLED; 2,280×1,080 pixels
516ppi 529ppi 458 ppi 402ppi
6.37×3.01×0.35 in 6.22×2.91×0.33 in 5.7×2.79×0.30 in 6.13×2.97×0.31 in
Dimensions (Millimeters)
161.9×76.4×8.8 mm 158.1×73.8×8.5 mm 143.6×70.9×7.7 mm 155.7×75.4×7.75 mm
Weight (Ounces, Grams)
7.09 oz.; 201g 6.66 oz; 189g 6.14 oz; 174 g 6.2 oz; 177 g
Android 8.1 Oreo Android 8.0 Oreo iOS 11 Android 8.1 Oreo
Dual 12-megapixel (wide), 12-megapixel (telephoto) Dual 12-megapixel Dual 12-megapixel 16-megapixel standard, 20-megapixel telephoto
Front-facing camera
8-megapixel 8-megapixel 7-megapixel 16-megapixel
4K 4K 4K 4K
Qualcomm Snapdragon 845 octa-core processor Octa-core Qualcomm Snapdragon 845 processor (2.8GHz + 1.7GHz), or Octa-core Samsung Exynos 9810 (2.7 GHz + 1.7 GHz) Apple A11 Bionic 2.8GHz Qualcomm Snapdragon 845
128GB, 512GB 64GB, 128GB, 256GB 64GB, 256GB 64GB, 128GB, 256GB
6GB, 8GB 6GB 3GB 6GB, 8GB
512GB 400GB None None
4,000mAh 3,500mAh ? 3,300mAh
Back Back None (Face ID via TrueDepth camera) Back of phone
USB-C USB-C Lightning USB-C
Yes Yes No Yes
Water resistant (IP68); wireless charging; S-Pen with Bluetooth connectivity; Iris and facial scanning Dual-aperture camera, water-resistant (IP68); super slo-mo video; wireless charging; iris scanning Water resistant (IP67); wireless charging; Face ID 3D unlock, Animoji Portrait mode, notifications toggle, dual-SIM, Dash Charging
Price off-contract (USD)
$1,000 (128GB), $1,250 (512GB) Varies: $840-$930 (64GB) $999 (64GB), $1,149 (256GB) $529 (64GB), $579 (128GB), $629 (256GB)
Price (GBP)
£899 (128GB), £1,099 (512GB) £869 £999 (64GB), £1,149 (256GB) £469 (64GB), £519 (128GB), £569 (256GB)
Price (AUD)
AU$1499 (128GB), AU$1,799 (512GB) AU$1,349 (64GB), AU$1,499 (256GB) AU$1,579 (64GB), AU$1,829 (256GB) AU$702 (64GB), AU$769 (128GB), AU$835 (256GB)
Samsung Galaxy Note 10 Plus review
January 9, 2020 January 9, 2020 editor


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African spells / Binding spells / Breakup spells / Effective love spells
MAKE HER MADLY IN LOVE WITH YOU SPELL.
- November 4, 2017 - powerfulspells
Make her madly in love with you spell that works instantly
Everyone at some point in their lives falls in love, and also wants the other person to feel the same. But on many occasions, love is unrequited, which causes the person who loves to become depressed and feel bad for this love that leads nowhere. But with a spell to infuriate a woman's love madly, all these anxieties and sufferings may remain in the past. If you are a man who has been suffering because that woman was failing to love, turn the odds today using make her madly in love with you. She has been telling you lies. There are many questionable phone numbers stored in her phone. she claims to be related to are many. Every time you propose to her, she meanders and never gives you a straight answer. You can make that girl super-loving and committed
Make her madly in love with you-Long distance relationship
Looking for a spell for long distance relationship? You are in the right place then. Managing a long distance relation is very difficult. Your long distance lover can start cheating on you. However, anything can be possible with magic. In magic, there is a premise "The works of magic do not know time or distance," this is true because the spirit knows no time and this can be seen for example in a person who is in our heart a cousin or uncle or better a brother whom you have spent 10 or 15 years without seeing. When you meet such a person after the said period, the feeling will be the same the very way it was the last time. The work of magic is the same. The work of magic is to help remind your partner of the relationship that exists between the two of you. The person starts to think and dream about you every 24 hours after casting this make her madly in love with you spell.
BEGINNERS SPELLS THAT REALLY WORK INSTANTLY
SIMPLE EASY BREAK-UP LOVE SPELLS THAT WORK

redpajama_set_name: RedPajamaCommonCrawl


Composing a thesis and diploma on construction and architecture - Técnicos en Actividades Comerciales.
The particulars of composing an ongoing work is based on the popular features of the medical discipline. Give consideration to all of the subtleties of composing a graduation project for construction and architecture specialties.
The style associated with the title web page is dependent upon the standards and methodological recommendations adopted at the university. In the title page, indicate the name associated with organization, specialty, membership into the division, the subject of work, the title and clinical title associated with curator for the graduation Project, the true title and band of the pupil.
Feedback – a description associated with work, which can be compiled by the supervisor associated with the thesis or diploma project. Feedback is attached to your workplace.
A review of work is an obligatory document, that will be connected to the thesis work. A description is contained by the review and an evaluation regarding the work – it really is written by a graduate having a qualification in a specific specialty. It is desirable that a reviewer features a degree that is scientific. Many Often, the authors of the review are the relative heads of businesses and enterprises, where students practiced before composing a diploma.
Annotation to the diploma – the description of this thesis, is published by the writer. Provides the primary provisions associated with the diploma task, this issue, the actual quantity of https://customwriting.org/ work, issues that the author has touched while writing the work, the stages associated with the study, the outcome and conclusions.
The structure associated with ongoing work is in line with the logic of this research process and mirror all the features associated with the procedure of learning issues. The job is composed of a plan, an introduction, the part that is main which sets out of the theoretical and practical bases of research, conclusions, variety of abbreviations, literature and applications.
Introduction (3-5 pages) is first a demonstration of this amount of illumination of dilemmas in literary sources.
The urgency of this subject formulates the known degree of need and clinical novelty of this research. You will find 2 directions for the growth of relevance: this issue is not examined after all or perhaps the outcomes of the thesis project can be used for further medical research.
The amount of clinical elaboration is formulated. Summary about this point may be made on a scientific foundation, makes it possible for to research the issue.
Introduction requires the formula for the object while the topic of research. The thing could be the sphere associated with researcher's interests – is really what is planned to present in the ongoing work, the sphere of knowledge which allows to understand the given problematics. To properly figure out the thing of research, it's important to analyze the literary works in the chosen subject. The topic is really a narrower concept than an object.
The purpose of the ongoing work is the desired outcome, what type wants to attain as a consequence of processing literature of a particular orientation.
The tasks associated with diploma task are split stages in the act of realization for the goal, intermediate actions that serve an effective research. Each task should lead to the understanding regarding the goal that is main of thesis.
The methodological foundation of research is an obligatory area of the research, which contains a collection of all practices used to understand the goals and goals.


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Senate Approves Plan to Reduce State Personnel Costs
Home Op-Eds & Press Releases Senate Approves Plan to Reduce State Personnel Costs
STATE HOUSE, BOSTON 02133-1053
(Boston) –The Senate on Wednesday voted to provide the executive branch with a set of tools to achieve payroll savings, including the continuation of a current hiring freeze, buyouts and retirement incentives, announced Senator Joan Lovely. The bill offers options for the Executive Office of Administration and Finance to reduce state personnel costs by $325.1 million in gross savings, in order to make $171.9 million available in the Fiscal Year 2016 budget.
"First and foremost, the early retirement proposal aims to generate cost-savings for the Commonwealth and its taxpayers for the upcoming fiscal year and reduce the likelihood of large-scale layoffs," said Senator Joan Lovely. "The Senate added measures to enhance reporting, monitor the program's effectiveness, and give the Administration the flexibility to achieve greater efficiencies within its branch of government."
"Through a collaborative process, the Senate has crafted a strong bill that offers tools to achieve savings without undermining the operation of state government," said Senator Karen Spilka (D-Ashland), Chair of the Senate Committee on Ways and Means. "The state's fiscal situation is tight, and these savings will allow us to pass a budget for Fiscal Year 2016 that reflects our priorities and advances families and communities across the Commonwealth."
"Thank you Senator Spilka and the Committee on Ways and Means for crafting a bill that achieves the same amount of savings as the House's early retirement bill while also protecting the ability of agencies to effectively deliver services," said Senate President Stan Rosenberg (D-Amherst). "The Committee strengthened the bill by incorporating ideas from other Senators and I commend them for their hard work on this important legislation."
The bill sets a cap of 4,500 executive department employees who are eligible to retire or resign and requires detailed reporting to the Legislature and State Retirement Board to ensure accountability and transparency during the implementation process.
Reporting requirements would allow the Legislature and State Retirement Board to monitor the number of employees retiring and the impact on agency operations. The bill also directs the Administration to designate "critical" positions key to the mission of state agencies that would be exempt from staff reductions and "priority fill" positions that would be the first to be rehired during the backfill process.
The bill offers several tools for the Administration to use to reduce personnel costs, such as authorizing buyouts of targeted employees. Additionally, the Administration's current hiring freeze could be continued to achieve savings without the possible financial liability associated with a retirement incentive program.
The retirement incentive program would be open to executive department employees who are Group 1 members of the state retirement system and vested with a minimum of 20 years of service or are at least 55 years old on the date of retirement. Employees not funded from the state payroll, elected officials and certain other employees are not eligible for the program. The eligibility limits are intended to guarantee expected savings by giving the executive branch direct control over the backfill of positions.
The application period would be from April 27 through May 29, 2015, with a retirement date of June 30, 2015.
To allow for sufficient knowledge transfer from departing employees, the bill allows retiring employees to be rehired as consultants for up to 90 days.
In March, the House of Representatives passed a similar bill establishing an employee retirement incentive program. The Senate and House will now produce a compromise bill for final passage and the Governor's signature.


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p0691 BC-OLY-Beijing-BritishA 02-10 0322
British athletes must agree to avoid protests, gestures at Beijing Olympics
Eds: VERSION also moved in BC-Names in the Game. Moving on general news and sports services.
By THOMAS WAGNER
LONDON (AP) -- British Olympic athletes must sign a new clause in their contracts prohibiting politically sensitive remarks or gestures during the Beijing Games.
"The reality is, given the level of political scrutiny of the world's media on these games and the way China will handle them, the BOA felt it was sensible and proper to flag that rule to our athletes," British Olympic Association communications director Graham Mewson said Sunday.
The International Olympic Committee already has a rule that states that "no kind of demonstration or political, religious or racial propaganda is permitted in any Olympic sites, venues or other areas."
In January, Belgian athletes were told they would be prohibited from raising human rights or other political issues at Olympic venues. Outside the sports venues and Olympic village, however, they will be free to speak their mind.
The BOA is making the change because China's government is widely regarded as sensitive to criticism over issues such as its human rights record and Tibet, Mewson said.
British athletes have been required to sign the 32-page BOA contract for 20 years before competing in the Olympics, but this is the first time such a clause is being added.
Mewson said the clause will not bar British athletes from "honestly answering" questions they are asked during interviews at the Aug. 8-24 games about "politically sensitive issues."
"An athlete who decides to lift up his team shirt to show a 'Free Tibet' one below it, that's very different," Mewson said.


Shridhara Temple, a 19th century shrine in West Bengal, lies to the north-west of Madan-Mohan temple at Bishnupur. The temple features nine spires - known as navratna - and has beautiful carvings on the walls. The dancing image of Lord Shiva at the entrance of the temple is a fine example of Bengal"s rich sculptural heritage.


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A Chorus Line (1985)
Nicole Fosse as Kristine
Sharon Brown as Kim
Michael Douglas as Zach
Gregg Burge as Richie
Michael Blevins as Mark
Tony Fields as Al
Vicki Frederick as Sheila
Michelle Johnston as Bebe
Jan Gan Boyd as Connie
Yamil Borges as Morales
Cameron English as Paul
Janet Jones as Judy
Drama, Music, Musical
Show business is the only business that reminds us there is no business like it. And it never tires of that message. If there were as many books about books as there are musicals about musicals, there wouldn't be room on the shelf for books about anything else.
"A Chorus Line" is the quintessential backstage musical, a celebration of the lives and hard times of the gypsy dancers who turn up by the hundreds to audition for a handful of jobs on Broadway. It takes years of brutal hard work to become a good-enough dancer to dare to go to an audition, and then the reward usually is a brusque "thank you" and a sweaty ride home on the subway. In order to succeed as a Broadway dancer, applicants need a limitless capacity to absorb rejection, and "A Chorus Line" celebrates that masochism in song and dance.
"A Chorus Line" is now in its 11th year on stages all over the world; its story is by now well-known. A choreographer is casting eight dancers for a new musical he hopes to stage, and during one long and truthful day he auditions dozens of dancers before he makes his final selection. Richard Attenborough's film treatment of this story sticks to the outlines of the stage version, by and large, although he leaves the stage to fill in the details of the choreographer's old romance, and he leaves out some of the original songs to make room for some new ones.
The result may not please purists who want a film record of what they saw on stage, but this is one of the most intelligent and compelling movie musicals in a long time - and the most grown up, since it isn't limited, as so many contemporary musicals are, to the celebration of the survival qualities of geriatric actresses.
Most of the scenes take place inside a theater. Zach (Michael Douglas), the choreographer, sits behind a writing platform somewhere out there in the darkness. Occasionally he lights a cigarette, and the ash glows as he takes the measure of the dancers on the stage. He can see them. They can't see him. He communicates by microphone. They step hesitantly to the edge of the stage, blinded by the spotlight, and talk into the void. Well, if that isn't the life they wanted, why did they volunteer for it?
Platoons of dancers are brought on stage, winnowed, dismissed. Finally there are 16 left, and Zach asks each one of them to talk on a personal level - talk about when they were born, and where, and what their lives have been like, and what their dreams are. Many of the dancers have the most extraordinary difficulties in doing this, and one of them is frank: "Give me the lines, and I can play anybody. Just don't ask me to talk about myself."
Meanwhile, a backstage drama is taking shape. An unexpected dancer has appeared for the auditions - Sheila (Vicki Frederick), Zach's former girlfriend. They met in the theater, courted in the theater, broke up because Zach's job left no time for a personal life. Sheila was a star, but now she simply needs a job.
Unlike the play, the movie opens up by going offstage for flashbacks to their affair, but the flashbacks are notable mostly for the way they focus on the theatrical lives of this couple - the way their private lives seem valid only to the degree that they reflect acceptance from the audience.
The underlying tension in the movie circles around Zach's eventual decision: Will his heart or his profession make the eventual decision about Sheila? Douglas plays Zach on a staccato, harsh note; this is a workaholic who walks around with a lot of anger. That makes it all the more effective when he occasionally relents and gives one of the dancers a break; softening momentarily before putting on his mask again.
I thought Zach's most revealing moment came when he made the cut from 16 dancers to eight, reading out eight names and then, when the eight were assembled downstage with smiles on their faces, thanking them and dismissing them; he had chosen the eight he did not name. Was this a misguided attempt to tell the rejected eight that they were also winners? Or was it simply cruelty? We are left to answer for ourselves. Such questions are intercut with song and dance, with virtuoso solo numbers (my favorite was Charles McGowan's "I Can Do That!") and ensemble production numbers, leading up to a big and splashy finale, in which all of the dancers who originally auditioned are back on stage, together once again. That leads to my one major difference with Attenborough's approach.
Since "A Chorus Line" is a musical about itself, and since the whole hard, bitter, romantic truth of the story is that many are called but few are chosen, the roll call at the end strikes a false note of triumph. Better, perhaps, to have eight dancers on stage, and then cut to the others putting on their street clothes, waiting at bus stops, explaining to friends how they didn't get the job, or going to their dance classes yet once again. I think the message of the play is that you don't get called back for a grand finale; you simply go to another audition.
by Nell Minow
by Brian Tallerico
Richard Attenborough: 1923-2014
by Peter Sobczynski


Have you read the latest from our current Housing Minister? According to BBC's Radio 5 Live, John Healey says, "for some people, having their home repossessed, can be the best option". What poppycock! And I can assure you, that's me doing my best to be polite!
I'm assuming he doesn't know first hand what it's like to be so severely in debt that you face losing your home? Or the sleepless nights so many people around Britain have, wondering how it all went wrong and what on earth they'll do if they can't provide a roof over their family's heads? Mmmm, I didn't think so.
So how is it, that he feels it appropriate, to make statements implying losing your home and being out on the streets is in anyone's best option?
In my opinion, just having the knowledge and fear that your debt has got so out of control, that it jepordises the security of your home, can substantially lower the number of people experiencing reposessions. How do we we make this effective? One thing that might help would be to bring in earlier repossession proceedings, enabling the problem to be tackled before things get out of hand and families are left homeless on the streets.
What the government has done is put in place measures that seem to protect consumers struggling with mortgage payments – but in real life, these don't really help. All they do is delay the evil day when the family finds itself on the street, because they don't look at the family's debt situation in the round.
Still following? Let's break it down to more simple terms – pay attention Mr Healey 😉 If you were struggling to make ends meet, what's the last thing you'd fall behind with – month to month? Your mortgage, of course. The most important security in your life, is the last thing you would ever place in jepordy. So, meanwhile, you rob Peter to pay Paul, get behind with the gas bill, or your overdraft – or forget to pay that 0% credit card you had – which means the interest you pay, following the default, will whizz upwards to an eye-watering level and somewhere in between all of this, you hit rock bottom and miss a mortgage payment, or two, or three.
Taking into account the above synopsis, in my experience, I've found that when someone misses a mortgage payment, this is a symptom of much worse financial distress. They probably have many other debts and are struggling across the board. If you leave them till they've missed a number of mortgage payments – which is encouraged by the current system, then – in my view – it's much more likely there wil be no alternative but actual repossession, and maybe homelessness.
Surely it's preferable for repossession proceedings to be issued automatically the first time people miss a mortgage payment – and then immediately suspended whilst the homeowner puts in place a plan to repay what they can afford across all their debts?
It may seem harsh, but it will catch the average debtor much earlier in their crisis than the current system – forcing them to take action on all their debts whilst they can still afford to, and giving them both the carrot (of being safe whilst they repay what they can REALLY afford) and the stick (the possibility of losing their home remains) – keeping them motivated to climb back to financial security.


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text:
\section{Introduction}

The application of tools and concepts of statistical physics in the modeling of social behavior has received considerable attention in recent times (for a recent review see, e.g., Castellano et al. 2009). There are instances, in social and biological contexts, of regularities emerging on the collective scale even when the interactions of any given individual are confined to a very small subset of the population, and these examples share similarities with cooperative phenomena in other physical systems. Such social interactions may well take the form of \emph{learning}, describable in very general terms as mimicking, or adopting, the behavior of other individuals.
From the perspective of game theory, this would be seen as an adoption of a particular \emph{strategy}, whose result may or may not be associated with a favorable \emph{outcome}.  It is quite reasonable to expect that the effectiveness of a strategy in yielding favorable outcomes would influence how likely it is to persist, and spread through the population.

Against the backdrop of the above ideas, a model of strategic learning was introduced in Mehta and Luck (1999) which considers a population of interacting agents on a regular lattice, with one of two possible strategies, or types, being associated with each. At every time step of the dynamics, each agent independently makes a choice to either remain unchanged in type or switch to the other type, based on two elementary rules: a \emph{majority-based} updating, reflecting every agent's tendency to align with the local neighborhood, followed by a \emph{performance-based} updating depending on the [random] outcomes of \emph{its neighbors} of both types in some `game'. An outcome is categorized as being either a success (with a fixed probability $p_{\pm}$ for each type) or a failure; every individual revises its choice, adopting the more successful of the two types among its neighbors. This model, in its two versions, interfacial and cooperative, was investigated in depth in Mehta and Luck (1999) at \emph{coexistence}, i.e., when $p_+$ = $p_-$ = $p$. Using an analytical pair approximation as well as numerical simulation, the corresponding phase diagrams were constructed. In the case of the interfacial model, the phase diagram was found to possess a regime of disorder for intermediate values of the performance parameter $p$, flanked by frozen phases on either side; a correspondence was exhibited between regimes close to $p=0$ and $p=1$.

In what follows, a variant of the interfacial learning model is considered with the addition of one extra ingredient: the introduction of self-coupling in the performance-based updation rule. This is a natural extension of the previous work on this model, and would perhaps bring it closer to capturing the dynamics of a group of individuals in contexts where some form of learning by observation may be involved. In certain situations, choices made by individuals between available alternatives might be based not only on which of the alternatives is more widespread in the population, but also on which of them is \emph{perceived} to be more `successful' at doing whatever they are supposed to do. If agents are going to decide between sticking to their own type and changing to the other type depending on the relative outcomes, then it is reasonable to suppose that every agent would observe and factor in the result of its \emph{own} chosen course \emph{as well as} the realizations of its neighbors, in making these choices. As plausible real-world examples which such a model might approximate, one can think of consumers being influenced by the experiences of friends in deciding which brand of product to buy, or farmers opting for a particular agricultural practice/crop variety on the basis of their neighbors' successes, or even perhaps, the competition that may have existed between prevalent hunting strategies in pre-historic social groups. We mention here that the kernel of the above idea has appeared earlier in some work on the learning-based diffusion of technology (Chatterjee and Xu 2004). However, that study was limited to a one-dimensional setting, and in its details differs considerably from the model to be analyzed here.

This paper proceeds as follows: In the next section, the modified version of the learning model is introduced, and three possible ways of incorporating the self-coupling are described. Subsequently, the results of the numerical analysis of these models on a two-dimensional square lattice are presented in Sec. III. This is followed in Sec. IV by a recapitulation and some discussion of the relationship of the present model with the cooperative model of Mehta and Luck (1999). We conclude by putting our work in the context of previous models of learning in Sec. V.

\section{The Model}

In the standard interfacial model as defined in Mehta and Luck (1999), individuals are located at the sites of a regular lattice with coordination number $z$ (equal to 4 for a square lattice in two dimensions) and every individual is associated with an \emph{efficiency} $\eta_i$, which can take one of two possible values: +1 or -1. These alternatives will be referred to as `$+$' types and `$-$' types respectively. The dynamical evolution proceeds via a two-step process: a majority-based updation (step 1) followed by a performance-based rule (steps 2 and 3). In the latter step, every \emph{interfacial} site compares the `successes' of its neighbors of both types (more specifically, the ratios $r_+$ and $r_-$ representing the \emph{fractions} of successful neighbors of each type) averaged over some observation period, and based on their relative values, decides to either convert to the other type or remain unchanged. \emph{Bulk} sites, having all neighbors of a single type, are left untouched by the outcome-based rule in this model, as its name suggests.

Mehta and Luck (1999) also introduce a \emph{cooperative} variant of the above model. Very briefly, the cooperative model involves the same update rules as the interfacial model for interfacial sites, but introduces the additional possibility that, in the event of a majority of the neighbors of a bulk site failing, they are \emph{persuaded} to convert to the other type \emph{along with} the central site. (If the central site is \emph{already} of the other type, then of course it stays the same.)

Here, it will be assumed that step 1 of the interfacial model remains unaffected, and as before, the efficiencies are updated according to the following majority rule:
\begin{eqnarray}
\eta_i (t+1) &=& +1 ~~\textrm{if}~~ h_i(t)>0, \nn \\
\eta_i (t+1) &=& +1 ~\textrm{or}~ -1 ~~\textrm{w.p.}~~ 1/2 ~~\textrm{if}~~ h_i(t)=0, \nn \\
\eta_i (t+1) &=& -1 ~~\textrm{if}~~ h_i(t)<0. \nn \\
\end{eqnarray}
Following Mehta and Luck (1999), $h_i(t)$ denotes the \emph{local} field of site $i$, defined as the sum of efficiencies of its neighbors.

It is only the second step which will be altered by now taking every site's \emph{own} outcome into account too: In addition to its neighbors on the lattice, the realizations of the central site are also included in the determination of the corresponding ratios $r_+$ and $r_-$. A departure from the properties of the original model may thus be expected.

For a start, the inclusion of outcomes of one extra site in the performance-based updation rule changes the form of the \emph{transition probabilities} $w_{\pm}(h_i)$. These encapsulate the net effect of steps 2 and 3 in the limit when the time scale on which individuals change \emph{type} is much slower than the rate at which the \emph{outcomes} of individuals are updated. In the symmetric case ($p_+ = p_-$), these probabilities are now given by
\begin{eqnarray}
\eta_i (t) = +1 ~&\rightarrow&~ \eta_i (t+1) = +1 ~~\textrm{w.p.}~~ w_+ (h_i), \nn \\
\eta_i (t) = -1 ~&\rightarrow&~ \eta_i (t+1) = +1 ~~\textrm{w.p.}~~ w_- (h_i)  \nn
\end{eqnarray}
where
\begin{eqnarray}
w_+ (+4) &=& 1 \nn \\
w_- (+4) &=& 1 - p - (1-p)^5 \nn \\
w_+ (+2) &=& p^5 + 1 - p \nn \\
w_- (+2) &=& 1 - (1-p)^5 - p^2 - 2 p (1-p)^3 (1+2p) \nn \\
w_+ (0) &=& 1 - p^2 (1-p^3) - 2 p (1-p)^3 (1+2p) \nn \\
w_- (0) &=& 1 - p^5 - (1-p)^5 - 3p(1-p)^4 \nn \\
&&- (1-p^2)(3p^2 - 2p^3) \nn \\
w_+ (-2) &=& 1 - 3p(1-p)^4 - (1-p^2)(3p^2 - 2p^3) \nn \\
w_- (-2) &=& p - p^5 \nn \\
w_+ (-4) &=& p +(1-p)^5 \nn \\
w_- (-4) &=& 0.   \label{scm1}
\end{eqnarray}
Here, we have followed the convention of Mehta and Luck (1999) and continue to treat the probabilities as functions of the local field $h_i$, even though their determination would now involve input coming from the central site as well. These probabilities satisfy the condition $w_+(+h_i)+w_-(-h_i)=1$, which follows from symmetry under a global reversal of signs. (The derivation of the above expressions is summarized in the Appendix.) The extra feature being introduced here, compared with the interfacial model, is the possibility of conversion even when a site might be surrounded by all neighbors of a single type: For example, if $\eta_i (t) = -1$ and is surrounded by all neighbors of the `$+$' type, there is a non-zero probability, given by $w_- (+4)$, that it will switch to the `$+$' state at step 3 (this cannot happen in the interfacial model of Mehta and Luck (1999)). Loosely speaking, the self-coupling increases the effective noise in the system by adding to the randomness in the dynamics of the outcome-based update rule -- `noise' here referring to changes which do not obey a simple majority rule. A widening of the disordered paramagnetic phase may thus be expected.

The above modification (referred to hereafter as SCM1 $\equiv$ Self-Coupled Model 1), however, does not introduce the potential for changes in a \emph{sea} of individuals belonging to the same type. (To avoid ambiguity, it is clarified here that our notion of a `sea', in which the central site \emph{as well as} its neighbors are identical, is distinct from `bulk', by which we only mean that all the neighbors of a given site have the same type, but are not necessarily identical to it.) In fact, the probabilities $w_- (+4)$ and $w_+ (-4)$ only act to \emph{increase} the order locally. One could therefore think of an alternative modification of the following kind: If a site is surrounded by neighbors all of which are of its own type, then it will decide to flip to the other type if, and only if, \emph{all} of them (including itself) fail. In this particular situation, the central site is unable to make a comparison between the two types since all its neighbors as well as itself have the same type. This rule can thus be thought of as the central site deciding to take a risk and changing to the other type \emph{only} in the worst-case scenario of total failure of its own type (i.e. itself \emph{and} all its neighbors). It may be noted that this ``all-or-nothing'' rule is different from step 3 of the \emph{cooperative} model of Mehta and Luck (1999) in two respects: One, only the central site flips here, so there is no \emph{persuasion} of the neighbors to convert; and secondly, this is not a `majority-based' rule, in the sense that the central site flips \emph{only if} \emph{all} $z+1$ sites fail, not just if more than half of them fail (which is the case for the cooperative model).

This second version of self-coupling modifies only the probabilities $w_{+} (4)$ and $w_- (-4)$, which are now given by
\begin{eqnarray}
w_+ (+4) &=& 1 - (1-p)^5 \nn \\
w_- (-4) &=& (1-p)^5,  \label{scm2}
\end{eqnarray}
and this variant will be referred to as SCM2. All other transition probabilities remain the same as in the SCM1 model. One can immediately see from Eq. (\ref{scm2}) that SCM2 introduces the possibility of changes in a sea of the same type; this effect is expected to increase as site failures become more probable, which would happen at sufficiently small values of $p$. This conversion of the central site to the other type could `propagate' in the course of the updating according to step 3, before the next round of step 1 (the majority rule) has a chance to `reset' it back again.

Taking the above reasoning a step further, one can possibly consider a third variant (SCM3), which implements a majority-based rule at step 3: If a site as well as all its neighbors are of the same type, the central site decides to flip to the other type if a majority of them (which on a square lattice would be equal to or greater than 3 out of 5) fail. The transition probabilities for sites in a sea now assume the form
\begin{eqnarray}
w_+ (+4) &=& 1 - (1-p)^5 - 5p(1-p)^4 - 10 p^2 (1-p)^3, \nn \\
w_- (-4) &=& (1-p)^5 + 5p(1-p)^4 + 10 p^2 (1-p)^3  \label{scm3}
\end{eqnarray}
with the rest of the probabilities remaining the same as in SCM1 and SCM2. At $p=0$, the expressions in Eq. (\ref{scm3}) obviously coincide with the corresponding probabilities for SCM2 in Eq. (\ref{scm2}), since here \emph{all} the sites fail in any case. However, for any non-zero value of $p$, SCM3 introduces a greater likelihood of site conversions inside ordered domains.

There are thus two kinds of effective noise being introduced by the dynamics of the outcome-based update rule: the \emph{interfacial} noise which affects sites having neighbors of \emph{both} types, and \emph{bulk} noise which is associated with the updating of sites all whose neighbors are of one type. Referring to the models defined here, SCM1 contains only interfacial noise, but in the SCM2 and SCM3 models, bulk noise is also present.

An attempt to understand the resulting dynamical phases of the above models will be made in the following sections. Two-dimensional square lattices of size $N \times N$ are considered and periodic boundary conditions (with the identification of opposite edges) are imposed. In order to quantify the nature of the collective dynamics and identify various phases, the averages of two quantities will be looked at. These are the \emph{magnetization} $M(t)$, defined as the mean efficiency over the whole lattice, and the \emph{energy} $E(t)$, given by the fraction of \emph{disparate}, or active, bonds (i.e. those which connect sites with unequal efficiencies):
\begin{eqnarray}
M &=& \frac{1}{N^2} \sum_i \eta_i \\
E &=& \frac{1}{4N^2}\sum_{(ij)} (1-\eta_i \eta_j).
\end{eqnarray}
The second summation above is taken over all possible pairs of nearest-neighbor sites. With this definition, the energy is zero when full consensus is reached in the population (corresponding to $M = \pm 1$), and maximized (equal to +1) in an \emph{antiferromagnetic} state.

At a qualitative level, an idea of the behavior of these models may be had by making a correspondence with a generalized two-parameter stochastic model studied in Drouffe and Godr\`{e}che (1999) and Oliveira et al. (1993), as was done in Mehta and Luck (1999). The transition probabilities in this class of nonequilibrium spin models are parametrized by two variables $p_1$ and $p_2$, interpreted respectively as measures of interfacial and bulk noise. Specific points in the $p_1-p_2$ plane correspond to particular cases, like the zero-temperature Ising-Glauber model ($p_1 = p_2 = 1$), and the voter model ($p_1 = 3/4, p_2 = 1$) (Holley and Liggett 1975; Frachebourg and Krapivsky 1996; Dornic et al. 2001). The analysis in Drouffe and Godr\`{e}che (1999) and Oliveira et al. (1993) showed that the phase diagram is divided in a ferromagnetic region near the $p_1 = 1, p_2 = 1$ point and a disordered paramagnetic phase, which are separated by a line of \emph{continuous} phase transitions terminating at one end at the voter point. In Mehta and Luck (1999), the interfacial model with no bulk noise was associated with the $p_2 = 1$ line, and the critical point was shown to have the character of a \emph{first-order} transition belonging to the same universality class as the voter model, which shows criticality in two dimensions.

Considered in the above context, each self-coupled model, in a rough sense, may be represented by a curve parametrized by $p$ in the $p_1-p_2$ plane. These curves would all converge at the zero-temperature Ising point when $p=1$, and cross the critical line into the disordered phase at different points (equivalently, at different values of $p < 1$). In SCM1, bulk noise (taken to mean a non-zero probability for a site surrounded by all neighbors of its type to flip to the opposite type) is absent. Thus, this version would follow the $p_2 = 1$ line just as the interfacial model, going from fully ordered ferromagnetic to paramagnetic behavior; the corresponding critical point may be expected to be voter-like with a discontinuity in the value of order parameter. As for SCM2 and SCM3, there is also the presence of bulk noise in addition to interfacial noise, implying that the order-disorder transition in these models would be of the continuous type. Moreover, while the interfacial noise is identical in all the three models -- allowing a direct correspondence to be made with $p$ -- there is an overall enhancement of bulk noise in SCM3 compared to SCM2. Referring back to the phase diagram and the critical line of the generalized model suggests that the critical point of SCM2 would occur at higher interfacial noise, i.e. at a smaller value of $p$, relative to SCM3 (this is also anticipated on general grounds). These expectations will be borne out by the analysis in the next section.

\section{Numerical results}

The time-evolution of the lattice dynamics is implemented numerically. In every run, the population is assumed to start out in a random initial configuration of efficiencies, in which each site is independently chosen to be of the `$+$' type or `$-$' type with equal probability. As for the updating procedure to be employed, different options exist. A commonly used scheme in the literature for problems of this type (which, however, normally involve only a single-step update rule) is \emph{random} sequential updating (see, e.g., Christensen and Moloney 2005; Landau and Binder 2005) in which one site is picked at random at every moment and its state updated; this means that any given site in the lattice gets updated once every $N^2$ time steps only \emph{on an average}. But here, following Mehta and Luck (1999), the updating will be carried out in an \emph{ordered} sequential manner, which consists of ``sweeping'' through the lattice in a regular fashion (left to right and top to bottom, say) so that adjacent sites along any one direction get updated in succession. One complete round of updating is implemented by first sweeping through the lattice updating sites according to step 1, followed by a sweep of updating based on the probabilities $w_{\pm}$. These two sweeps together will be taken to constitute one ``time step'' of the dynamics.

We first consider the SCM1 model. Figs.~\ref{fig:scm1a} and \ref{fig:scm1b} show the evolution of the average energy with time for different values of $p$, for a square lattice of size 500 $\times$ 500 over $T=1,000$ time steps. Each curve is an average over 100 runs (with different starting seeds). The narrow range of $p$ values displayed in Fig.~\ref{fig:scm1a} has been chosen to bring out the crossover occurring in the behavior of the system between $p = 0.81$ and $0.82$, which is reflected in the change in the asymptotic nature of the curves. The occurrence of a similar transition near $p \approx 0.5$ can be inferred from Fig. \ref{fig:scm1b}.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{scm1_E-vs-t_curves1.eps}
\caption{\label{fig:scm1a} Plots of the inverse of the mean energy $E(t)$ as function of time in the vicinity of $p=0.8$ for the self-coupled model 1 (SCM1). System size is $500 \times 500$.}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.55]{scm1_E-vs-t_curves2.eps}
\caption{\label{fig:scm1b} Evolution of the mean energy with time for $p$ values around 0.5 for the SCM1 model. System size is $500 \times 500$.}
\end{center}
\end{figure}

This behavior shows up in the plots in Figs. \ref{fig:scm1E} and \ref{fig:scm1M} of the average energy and average magnetization as functions of the probability $p$; the corresponding results for the interfacial model are also shown for comparison. A square lattice of size $100 \times 100$ has been used here (and in the subsequent results); this system size is found to be large enough to give reliable averages, and the deviations of its predictions from larger lattices are insignificant. For obtaining the data points, runs lasting $T=50,000$ time steps each have been carried out with averaging being done over the last 10,000 steps. These graphs clearly indicate a paramagnetic phase with zero net magnetization occurring between $p \approx 0.5-0.8$, with macroscopic order in the remainder of $p$-space. The transition from disorder to fully ferromagnetic is quite sharp, consistent with the expectation of a first-order phase transition for this model. These results bear a resemblance to the interfacial model in Mehta and Luck (1999); the similarity is expected, given that in both models it is only the interfacial noise which is present. However, the interval of disorder in SCM1 is almost doubled in comparison with the interfacial model, which may be attributed to an overall enhancement of the interfacial noise due to the self-coupling.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{scm1im_E-vs-p.eps}
\caption{\label{fig:scm1E} Comparison of mean energy vs. $p$ for the interfacial model (IM) and self-coupled model 1 (SCM1). Lattice size is $100 \times 100$.}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{scm1im_M-vs-p.eps}
\caption{\label{fig:scm1M} Comparison of mean magnetization vs. $p$ for the interfacial model (IM) and self-coupled model 1 (SCM1). (Lattice size is $100 \times 100$.)}
\end{center}
\end{figure}

The nature of the ordered phases was probed a little further by looking at the time series of individual runs (starting from random initial configurations) with different choices for $p$. In all the runs carried out, the system was observed to always get completely ordered when evolution was allowed to proceed for sufficiently long times. Also, since bulk noise is absent in this particular model of self-coupling -- that is, individuals cannot break out of a sea -- the fully ordered state is absorbing. At the two end points $p=0$ and $p=1$, however, it is found that the system can also end up in frozen \emph{stripe} states which are only partly ordered and remain stable over time. At these end points, where all sites succeed or all of them fail with certainty, the outcome-based rule in step 3 becomes irrelevant and evolution is driven solely by the zero-temperature Ising update rule of step 1. The frozen states seen here are consistent with earlier results for quenching and phase ordering in zero-temperature Ising models (Spirin et al. 2001a, b; Barros et al. 2009). Based on this, it is conceivable that sufficiently close to $p=0$ and $p=1$ where the interfacial noise introduced by step 3 would be very weak, the system may not get fully ordered for some initial seeds. It could, for example, remain trapped in partially ordered configurations indefinitely; in such cases, while the interfacial noise might induce some activity at the boundaries it could be too weak to induce transitions to the fully ferromagnetic state. This possibility has not, however, been looked into in detail in the present work.

The second version with self-coupling, SCM2, is considered next. Figs. \ref{fig:scm2E} and \ref{fig:scm2M} are plots of the mean energy and mean magnetization as functions of $p$ displayed along with the corresponding curves for SCM1 (again for a 100$\times$100 lattice with every run lasting $T=50,000$ time steps and averaging being done over the last $10,000$ steps). Visually, from the plot of the magnetization, only one clear-cut transition can be made out, and it is located very near to the critical point of SCM1. But recalling the analogy with the generalized spin model of Drouffe and Godr\`{e}che (1999) and Oliveira et al. (1993), the presence of bulk noise here is expected to change the nature of the transition to a continuous type. To locate the critical point, the susceptibility, defined in terms of the variance of the net magnetization as $\chi = N^2 (\langle M^2 \rangle - \langle M \rangle^2)$ (Christensen and Moloney 2005), is considered in the vicinity of the transition and is plotted in Fig. \ref{fig:scm2_sus} (the points correspond to averages over the last 20,000 time steps in runs up to $T=100,000$). From the peak in the data, which is expected at the disorder-order transition, the critical point is located at $p_c \approx 0.815$, confirming the second-order nature of this transition.

Shifting attention to the other side, i.e. for smaller values of $p$, the magnetization is observed to remain nearly zero. The energy is far from minimized, and in fact, it steadily increases in the $p \to 0$ limit, suggestive of a preference for an antiferromagnetic state. However, it is found that even in the $p \to 0$ limit the proportion of active bonds does not reach unity, so the system does not appear to get antiferromagnetically aligned in its entirety. Figs. \ref{fig:scm2runE} and \ref{fig:scm2runM} show the time evolution of energy and magnetization over one run each for the SCM1 and SCM2 models (with random initial seeds) when $p=0$, at which point the difference between the two models is maximum. It can be seen that the SCM1 model gets stuck in a \emph{frozen} stripe state (at $t \approx 1,700$). In the case of SCM2, the energy asymptotes to some steady-state value slightly less than unity (with small fluctuations) and the net magnetization is zero, indicating an antiferromagnetic state. Based on several individual runs, it is confirmed that this asymptotic value of energy is independent of the initial conditions, and is a function of $p$ alone. This is illustrated in Fig. \ref{fig:scm2_E-t_comparison} with three different starting states, including two random seeds and the fully ordered state (which after a single time step converts to antiferromagnetic).

A glimpse at the snapshot of the lattice at $p = 0$ (taken after a sufficiently long time is allowed to elapse starting from a random initial seed) shows that an \emph{almost} completely antiferromagnetic configuration is reached, consistent with the increase in the energy of the system towards the $p \to 0$ end. However, some patches of disorder are found to persist along the edges. We have checked that these are due to finite-size effects. Furthermore, in the antiferromagnetic domain, collective flipping of sites at every time step is observed.

The preference for antiferromagnetic ordering seen in this model can be understood by noting that close to $p=0$ failures become increasingly likely, so that an individual surrounded by all neighbors of the opposite type would tend to \emph{retain} its type. In contrast, an individual in a sea of identical types would tend to do the exact opposite, choosing to convert because of all-round failure of its own type. Also, there would be no interfacial noise at $p=0$. The majority rule in step 1 acts to increase the local order. The bulk noise due to step 3 following it would then come into play and neighbors would tend to anti-align, thus favoring a configuration made up of active bonds all over.

The above behavior is expected to be seen also for the third variant, SCM3, which implements a majority rule at step 3 and thus introduces even more bulk noise. Fig. \ref{fig:scmsE} displays the result for behavior of the energy, plotted against the earlier self-coupled models. SCM2 and SCM3 show qualitatively similar behavior in the small $p$ regime, but in the latter case, the antiferromagnetic alignment is comparatively stronger at any given value of $p$. (In the strict limit of $p \to 0$, however, this difference disappears as the models become essentially identical.) From the plot of Fig. \ref{fig:scmsE}, as well as from lattice snapshots taken at large times, it appears that even here, the system does not reach a fully antiferromagnetic state at $p=0$. A finite-size scaling analysis (Christensen and Moloney 2005) of this model, however, indicates that in the limit of infinite lattice size, the system energy does tend to unity. A steady reduction in the relative magnitude of the disordered area to the lattice size is implicit in this result, leading to a completely ordered antiferromagnetic state. Fig. \ref{fig:scm3_E_vs_N} shows the results for different lattice sizes ranging from $N=50$ to $N=400$ on a log-log plot, and a linear fit to the data points yields the scaling relation $1-E(N) \propto N^\delta$ with $\delta = -0.959 \pm 0.008$.

Towards the other end of the phase diagram, a continuous ferromagnetic transition is found to occur beyond a critical value of $p$ which is discernible in the plot in Fig. \ref{fig:scmsM}. The corresponding data for the susceptibility as a function of $p$ in the transition region (see Fig. \ref{fig:scm3_sus}) shows a peak at $p_c = 0.86$, which is associated with the critical point. This value is higher than the corresponding value of $p_c$ for the SCM2 model. Also, the transition from disorder to ferromagnetic (i.e. complete) ordering is more gradual relative to SCM2. These differences may be put down to the comparatively higher level of bulk noise introduced by the majority rule of step 3 here (in place of the all-or-nothing rule of SCM2), which persists for larger values of $p$, and introduces greater disorder.

\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{scm12_E-vs-p.eps}
\caption{\label{fig:scm2E} Comparison of mean energy vs $p$ for the two self-coupled models SCM1 and SCM2, for system size of $100 \times 100$.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.55]{scm12_M-vs-p.eps}
\caption{\label{fig:scm2M} Comparison of mean magnetization vs $p$ for the self-coupled models SCM1 and SCM2 (again for a 100 $\times$ 100 lattice).}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{scm2_sus.eps}
\caption{\label{fig:scm2_sus} The susceptibility, defined in terms of the variance of $M$, as a function of $p$ for the SCM2 model (system size is 100 $\times$ 100). The capped spike occurs at the disorder-order transition.}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{scm12_E-vs-t.eps}
\caption{\label{fig:scm2runE} Comparison of time evolution of energy over one run at $p=0$ for the two self-coupled models SCM1 and SCM2 (for a 100 $\times$ 100 lattice).}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.55]{scm12_M-vs-t.eps}
\caption{\label{fig:scm2runM} Comparison of time evolution of the mean magnetization over one run at $p=0$ for the self-coupled models SCM1 and SCM2 (for a 100 $\times$ 100 lattice).}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.55]{scm2_E-vs-t.eps}
\caption{\label{fig:scm2_E-t_comparison} Result for three runs with different seeds (including a fully ordered starting state) for the SCM2 model at $p=0$. (Lattice size = 100 $\times$ 100.)}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{scm123_E-vs-p.eps}
\caption{\label{fig:scmsE} Comparison of plots for mean energy vs $p$ for all three models with self-coupling, for a 100$\times$100 lattice. The SCM3 model implements a majority rule at step 3.}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.55]{scm3_fss.eps}
\caption{\label{fig:scm3_E_vs_N} Log-log plot showing scaling of the mean energy at $p=0$ with lattice dimension $N$ for the SCM3 model. The best-fit straight line gives $1 - E(N) \propto N^{\delta}$ with $\delta = -0.959 \pm 0.008$.}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.55]{scm123_M-vs-p.eps}
\caption{\label{fig:scmsM} Plots of the net magnetization vs $p$ for all three models with self-coupling, for a 100$\times$100 lattice.}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.55]{scm3_sus.eps}
\caption{\label{fig:scm3_sus} Dependence of susceptibility on $p$ for the SCM3 model (system size is 100 $\times$ 100). The capped spike indicates location of an order-disorder transition.}
\end{center}
\end{figure}

\section{Discussion: Quantitative analysis of results}

The model of competitive learning introduced in Mehta and Luck (1999) involves agents which make choices based only on observing their local neighborhood, and do not take into consideration the realizations of their own actions. In the preceding sections, the consequence of introducing this additional feature for the collective behavior of the population has been explored. The motivation for doing this needs hardly any justification: since agents in the original model were assumed to be sophisticated enough to learn by temporal integration of information from their local environment, it seems quite natural to \emph{also} assume some capacity for self-observation (or self-reflection), which should then have a bearing on the learning dynamics. Having said that, this particular extension is to be regarded as only one among the many possible directions in which the original model of Mehta and Luck (1999) can be taken further, and needless to say, several other possibilities remain to be explored.

Nonetheless, the analysis carried out here suggests that the incorporation of a self-term can indeed introduce new features into the dynamics of the model. In general, the self-coupling as considered here adds to the interfacial noise, and also brings in some bulk noise. The question of which update rule to adopt when an individual happens to be at a bulk site was explored systematically with three possible choices; the corresponding phase diagrams do indeed show some significant differences. On allowing for the possibility of conversions inside a sea of individuals of the same type, as has been done in the SCM2 and SCM3 models, a preference for antiferromagnetic ordering is seen in the lattice as the $p=0$ end point is approached, in which limit failures become increasingly more frequent. An antiferromagnetic state would be interpreted as a tendency of individuals to continually maintain a position which is at odds with their local neighborhood. Additionally, introduction of bulk noise into the system changes the character of the phase transition occurring at the high-$p$ end, replacing the discontinuity in SCM1 with a continuous transition from paramagnetic to ferromagnetic behavior. This is a feature that the models SCM2 and SCM3 share in common with the \emph{cooperative model}, which was introduced in Mehta and Luck (1999). In going from SCM1 to SCM3, as the effective bulk noise being introduced into the system increases, the location of the critical point as well as the onset of full ferromagnetic ordering are pushed further towards the $p=1$ end.

It may also be pointed out that the self-coupled versions are very different from the cooperative model at the low-$p$ end of the phase diagram. The cooperative model incorporating a hard rule (Mehta and Luck 1999) predicts a phase of \emph{oscillatory coarsening} in a narrow interval near $p=0$, which is not reproduced here merely by addition of self-coupling to the original interfacial model. The existence of this time-dependent oscillatory phase is presumably related to the fact that in the cooperative model, not only can a bulk site change, but it can also persuade its neighbors to convert to the other type. This guess is supported by looking at the implementation of the cooperative model with the \emph{soft} rule, according to which \emph{only} the failed sites among its neighbors accompany the central site in converting to the other type. This version also shows oscillations near $p=0$. If one takes the soft rule a step further, and implements an even weaker version of it in which \emph{only} the central bulk site converts [provided that a majority of its neighbors fail, as before], it turns out that the oscillatory phase disappears and, in fact, gets replaced by antiferromagnetic behavior at the low-$p$ end. It may be noted that this weaker-than-soft scheme is actually quite similar to the SCM3 model (leaving aside the self-coupling of the latter). The above results thus suggest the interpretation that persuasion of neighbors to convert \emph{collectively}, a feature which is present in both the hard and soft versions of the cooperative model, but not in the other models, may be necessary to sustain the peculiar \emph{oscillatory} behavior.

Going a bit further, the variation of mean energy with $p$ has been plotted for the three versions of the cooperative model in Fig. \ref{fig:cms}. For comparison, the result for the weaker-than-soft version of the cooperative model is also shown along with the corresponding curve for SCM3 in Fig. \ref{fig:cmsr2scm3}. The onset of ordering in the case of the weaker-than-soft version occurs at a smaller value of $p$ relative to the other cases. At a qualitative level, these differences are traceable to the differences in the effective `noise' being introduced by the outcome-related rule in the various models, and this may be seen as follows: At $p=1$, all models are ferromagnetically ordered. As $p$ decreases, the strengths of both the bulk and the interfacial noise grow (starting from zero at $p=1$) in all the models. However, at any particular value of $p$, both sources of noise are \emph{stronger} in SCM3 compared to the weaker-than-soft version of the cooperative model. That is, with decreasing $p$, the effective noise grows faster in the SCM3 model. If a transition to the disordered regime is taken to happen once the noise in the system crosses a certain `threshold', then it is natural to expect that the onset of the paramagnetic phase would occur in SCM3 \emph{before} it occurs in the weaker-than-soft cooperative model, or, in other words, closer to the $p=1$ end point.

A similar comparison may also be made between the different versions of the cooperative model: the hard/soft rule on one hand and the weaker-than-soft rule on the other. All the three variants introduce identical \emph{interfacial} noise, because the outcome-based rules for conversion of the interfacial sites are the same for all of them. In addition, they all introduce the same \emph{transition probability} for the \emph{bulk} sites. However, in the weaker-than-soft case, the neighbors of the bulk site which is being updated are not persuaded to convert; as a result, updating of a bulk site would involve a conversion of \emph{only that} site (if at all). In contrast, when the hard version of the cooperative model is considered, the corresponding update rule for bulk sites involves not just the site which is currently being updated, but also a conversion of its \emph{neighbors} due to persuasion. In a sense, the conversion of a \emph{cluster} of sites in a sea (i.e., the neighbors \emph{plus} the updated bulk site) which happens in the hard version, is a bigger `fluctuation' than the conversion of \emph{just} a single site in a sea, which happens in the weaker-than-soft version; the probabilities in the two cases are, however, the same. If one then makes the assumption that the level of effective noise is `measured' not just in terms of how \emph{probable} changes are, but also by their `size', i.e., the \emph{number} of sites affected by update of a site, it would follow that the effective noise is uniformly stronger over the entire range of $p$ when persuasion is present. Thus, the hard version of the cooperative model may be anticipated to turn paramagnetic nearer to $p=1$ relative to its weaker-than-soft variant. The interpretation in the last line is lent support by recalling the analogy with the two-parameter generalized model (Drouffe and Godr\`{e}che 1999; Oliveira et al. 1993), which predicts a critical point at smaller value of interfacial noise (here, at larger $p$) when more bulk noise is present in the system.

\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{cms_E-vs-p.eps}
\caption{\label{fig:cms} Comparison of mean energy vs $p$ curves for all three versions of the cooperative model. (Lattice size = 100$\times$100.)}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.55]{scm3cmsr2_E-vs-p.eps}
\caption{\label{fig:cmsr2scm3} Comparison of weaker-than-soft version of the cooperative model with SCM3 (Lattice size = 100$\times$100.)}
\end{center}
\end{figure}

To conclude this section: the extension of the competitive learning model of Mehta and Luck (1999) to additionally include self-performance in the revising of the agents' individual choices is found to bring about some changes in the macroscopic dynamics of the population. The three choices for the update rule of bulk sites which were considered in this context introduce some systematic modifications, and these are reflected in the corresponding phase diagrams.

\section{Discussion: Implications of our results for models of learning }

In this section, we will put our results in the context of existing models of learning, and analyze their implications.  Chatterjee and Xu (2004) have in a scholarly paper, mentioned several papers of relevance in the context of human behavior: models of cultural evolution (Bisin and Verdier 2001) which represent social conformity, and models of learning by
boundedly rational agents (Ellison and Fudenberg 1995,
 Eshel et al. 1998), which incorporate the tendency to learn from the successes of others. A realistic example of this can be found in agricultural innovation, where farmers employ fertilisers in response to their neighbors' successes (Conley and Udry 2000).

Chatterjee and Xu (2004) build on these ideas to come up with
 two learning rules: one where the agent never switches type
 unless he fails, and one where he always looks around. In both cases the agent learns from the successes of his neighbours, as in the model of Mehta and Luck (1999). However in the model of Chatterjee and Xu (2004), there is no mechanism to change to a more successful type in a sea of failures, which exists both in
 the present model (SCM2 and SCM3), and in the cooperative model
of Mehta and Luck (1999). Another difference is that the analysis of Chatterjee and Xu (2004) was done not at coexistence, but where one of the probabilities of success was greater than the other; the dynamics of Ising spins makes it clear that in the general case, the magnetization will correspond to the sign of the dominant spin, i.e. the better technology will spread with probability 1, as was found in that paper. Analyzing the behavior at coexistence, as we have done, makes the generic behavior clearly visible in the phase diagrams presented here as well as in Mehta and Luck (1999). A third difference is that
the agents in the paper of Chatterjee and Xu (2004) make decisions based on immediate outcomes; in both this paper and that of Mehta and Luck (1999), we have set timescales such that an agent can make a decision based on the average outcome over a time period. It is easy to show (De Munshi et al. 2009) that the phase diagram depends critically on this choice; the conclusions of Bala and Goyal (1998) agree with this. Also, the stochastic selection of outcomes at any given timestep in our models is similar to that of Ellison and Fudenberg (1995).

To the best of our knowledge, none of the models mentioned above include the weighting of choices which include the agent's own performance, which is the main contribution of the present paper. In the case of the SCM1 model, the self-term acts to increase local order (the social conformity referred to above) by allowing the agent to switch types; while in the case of SCM2 and SCM3, the agent can, to varying extents, flip to a more successful type if it is in a sea of failures ('learning from one's failures'). This is midway between the inertia of earlier models in the case of failures (Chatterjee and Xu 2004) and the persuaded flipping of unsuccessful clusters via the hard and soft rules of the cooperative model of Mehta and Luck (1999). We view this as a rather realistic example of human behavior, which, when confronted by the failure of its own choices -- imposed by social conformity -- decides to change course individually.

So far, all our examples of learning were taken from cognitive/behavioral examples, which are in general those to which the present model is most applicable. There is however an interesting parallel to be found in the phenomenon of motor learning, where both learning and forgetting, as adaptive processes with different timescales have been shown to be important mechanisms in the phenomenon of reaching (Smith et al. 2006). In the original model of Mehta and Luck (1999), both learning and forgetting are part of the cognitive arsenal of a given agent, and it would be interesting to see, in future work, if the simple ideas of this paper could be used for modelling motor learning in humans and animals.

\section*{Appendix: Derivation of the probabilities $w_{\pm}(h)$}

The transition probabilities are calculated based on reasoning similar to that followed in arriving at the probabilities in Mehta and Luck (1999), but it is to be remembered that now the type and outcomes of the central site also play a role, and have to be included in calculating the ratios $r_+$ and $r_-$.

Let us begin with the derivation of the expression for $\bf w_+ (0)$: this is just the probability that $r_+ \geq r_-$, given that $\eta_i (t) = +1$ and $h_i=0$ (i.e. with a total of three `+' and two `$-$' sites \emph{including} the central site), and may be symbolically written as $P(r_+ \geq r_- \mid \eta_i = +1, h=0)$. Now, $P(r_+ = r_- \mid \eta_i = +1, h=0)$ is the probability that all sites succeed or all of them fail, which is just equal to $p^5 + (1-p)^5$. Next, to get $P(r_+ > r_- \mid \eta_i = +1, h=0)$, all pairs $(r_+,r_-)$ for which $r_+ > r_-$ have to be enumerated and their probabilities added up. The set of all such possible pairs is $\lbrace (1/3,0/2), (2/3,0/2), (2/3,1/2), (3/3,0/2), (3/3,1/2)\rbrace$, and their weighted sum is thus given by
\begin{eqnarray}
&&P(r_+ > r_- \mid \eta_i = +1, h=0) = \nn \\
&& 3p(1-p)^2 \cdot(1-p)^2 \nn \\
&& +~ 3(1-p)p^2 \cdot [(1-p)^2 + 2p(1-p)] \nn \\
&& +~  p^3 \cdot [(1-p)^2 + 2p(1-p)] \nn \\
&& = 3p(1-p)^4 + (3p^2 - 2p^3)(1-p^2), \nn
\end{eqnarray}
so $w_+ (0)$ finally reads
\begin{eqnarray}
w_+ (0) &=& P(r_+ \geq r_- \mid \eta_i = +1, h=0) \nn \\
&=& p^5 + (1-p)^5 + 3p(1-p)^4 \nn \\
&& +~ (3p^2 - 2p^3)(1-p^2).
\end{eqnarray}
This expression can be rearranged to end up with the form given in Eq. (\ref{scm1}). All other $w$'s can be derived in a similar manner: \newline
$\bf w_+(+2)$: This is equal to $P(r_+ \geq r_- \mid \eta_i = +1, h = 2)$ with 4 `+' and 1 `$-$' sites, and is given by
\begin{eqnarray}
w_+(+2) &=& P(r_+ \geq r_- \mid \eta_i = +1, h = 2) \nn \\
&=& 1 - P(r_+ < r_- \mid \eta_i = +1, h = 2) \nn \\
&=& 1 - p \cdot (1-p^4) = 1 - p + p^5
\end{eqnarray}
and the third line above makes use of the fact that $r_+ < r_-$ is possible only when the `$-$' type site succeeds and at least one `$+$' type site fails. \newline
$\bf w_-(+2)$: This is the probability that a `$-$' site will \emph{flip} its state, and is given by $P(r_- < r_+ \mid \eta_i = -1, h = 2)$ with 3 `+' and 2 `$-$' sites. The possible pairs $(r_+,r_-)$ for which this holds are $\lbrace(1/3,0/2),(2/3,0/2),(3/3,0/2),(2/3,1/2),(3/3,1/2)\rbrace$, and the total probability is thus
\begin{eqnarray}
w_-(+2) &=& P(r_- < r_+ \mid \eta_i = -1, h = 2) \nn \\
 &=& (1-p)^2 \cdot [1-(1-p)^3] \nn \\
& & + 2 \cdot p (1-p)\cdot[3p^2(1-p) + p^3] \nn \\
&=& (1-p)^2 - (1-p)^5 \nn \\
&& +~ 2p(1-p)(3p^2 - 2p^3)
\end{eqnarray}
which can be brought into the form given in Eq. (\ref{scm1}). \newline
$\bf w_+(-2)$ and $\bf w_-(-2)$ can be directly obtained from the above probabilities: $w_+(-2) = 1- w_-(+2)$ and $w_-(-2) = 1-w_+(+2)$. \newline
$\bf w_+(+4)$($\bf w_-(-4)$) would be equal to 1(0) since the central site has no way to make a comparison between the outcomes of the two types. On the other hand, if $\eta_i (t) = -1$ and $h= +4$, the central site can make such a comparison by considering its own outcome too; it will flip to the other type if it fails \emph{and} at least one of its neighbors (all of which are `+') succeeds, so
\begin{eqnarray}
w_-(+4) &=& P(r_- < r_+ \mid \eta_i = -1, h = 4) \nn \\
&=& (1-p)\cdot[1-(1-p)^4] \nn \\
&=& 1 - p - (1-p)^5
\end{eqnarray}
and this consequently gives ${\bf w_+(-4)}=1-w_-(+4)=p+(1-p)^5$.

\begin{acknowledgements}
G.M. would like to thank Nirmal Thyagu for helpful discussions in the course of this work. G.M. was supported by a grant from DST (Govt. of India) through the project ``Generativity in Cognitive Networks''.
\end{acknowledgements}

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