text:
Book, Lyrics and Music by Jonathan Larson
Script Consultant David Auburn
Vocal Arrangements and Orchestrations by Stephen Oremus
Jonathan Larson's semi-autobiographical tale about a composer living the bohemian lifestyle and taking a hard look at his life on the eve of his 30th birthday. His girlfriend's desire to marry and leave New York City for the suburbs, possible career changes and all sorts of other pressures cause him to question his ultimate dream -- writing the great American musical. Containing the kind of powerhouse musical numbers that helped make Rent a smash hit, tick, tick... BOOM! is an endearing, engaging look at persevering to make your dreams a reality. Photos by Michael Brunk | NWlens.com.
Adam Minton (Jon) Adam is thrilled to be getting back into the theater scene and to be joining SecondStory Rep for this amazing show. After growing up in New Jersey, Adam majored in Theatre For Youth at Shenandoah Conservatory, and has numerous credits in acting, directing, and design. Favorite on-stage roles include: Pippin (Pippin), Charlie Brown (You're A Good Man...), Applegate (Damn Yankees), Tobias (Sweeney Todd), and most recently, The Baker in Into The Woods this past summer in Shoreline. He would like to thank his loving husband, his Mother and Father for encouraging his theater passion all his life, SecondStory for this phenomenal opportunity, and this rockin' cast and production team!
Faith Howes (Susan) is delighted to be making her debut with SecondStory Rep. Recent credits include: Commedia Cinderella (Columbia) at Sea Side Rep,The Tortoise and the Hare (Buddy/ Osgood) with Missoula Children's Theatre, Spring Awakening (Adult Woman,) and Robin Hood (Allen A'dale) at Emerson College. Faith also is a proud member of The Dicken's Carolers and a current Teaching Artist Apprentice with Seattle Children's Theatre. She holds a B.A. in Theater Education and Performance from Emerson College. Thank you to Kate and Amy for all your support! More at About.me/faithhowes.
Ryan Lile (Michael) is extremely excited to be performing in his first full length show since his sequin-clad days performing on cruise ships in 2008. These days he spends most of his time recording audiobooks and teaching yoga around the Capitol Hill neighborhood in Seattle. Thanks so much to SSR for the opportunity to perform in a show he's been geeking out over since he was 17.
Benjamin Bentler (Music Director) joins SecondStory Rep for the first time with tick, tick…BOOM! He previously music directed, conducted, or provided additional orchestrations for 5th Avenue, Marriott Theatre, Spokane Civic, and various theaters throughout the midwest. Favorite projects include Wonderful Town, A Year with Frog and Toad, The Drowsy Chaperone, In the Heights, Tarzan, and Mary Poppins. His work has been heard in Chicago at the Marriott Theatre and Porchlight Music Theatre and in Seattle at the 5th Avenue and Village Theatre KIDSTAGE. Other professional work includes Resident Music Director at Spokane Civic Theatre, Director/Music Director at ICCT, and has been a private vocal teacher with hundreds of students across the country. Mr. Bentler is a graduate of University of Iowa and is a member of NATS. Thanks to my fiancé Tyler for creating art together in Seattle and my loving parents for helping me reach for my dreams.
Aaron Alonso (Stage Manager) is thrilled to work with SecondStory Repertory! Recent stage management credits include Little Shop of Horrors at Mount Baker Theatre, Shrek: The Musical at Village Theatre KIDSTAGE Everett, and Equus and RENT at Western Washington University. Upon graduating from Western Washington University, Aaron also had a brief stint this summer at the renowned Eugene O'Neill Theater Center in Connecticut. He thanks his friends and family for all of their support in his crazy theatrical endeavors!
Carmen Olmedo (Costume Designer) is thrilled to be costuming for SecondStory Rep once more. She'd like to thank her cast, fellow production team, friends, family, and absolutely stupendous boyfriend for all their support throughout this process. Previous credits include Alice's Anthem, 14/48: The World's Quickest Theater Festival, Fiddler on the Roof, Kiss of the Spider Woman, and Jerry Springer the Opera.


What does the writing say on the tremolo base plate? That would provide a big clue.
That's called the tone or sustain block. The base plate is what the saddles sit on. It looks like there is writing on the opposite side of where the arm socket is.
If this is what you meant, it doesn't seem anything is written on it.
Yes there is it says Floyd Rose. You didn't have to take the saddles off though.
Ok. I was expecting some number there. It's a Floyd Rose II by Schaller.
Ok. One more thing. Does it have a metal plate on the back where the neck meets the body?
No metal plate, and there's nothing written on the neck either.
What you have is an EX model with serious upgrades. The EX was the budget model preceding the RX & GIO. Tuners, trem, pickups have all been upgraded. In addition the FR has had a tone block replacement. Play the hell outta it. If you're looking to sell it. The only thing that would catch someone's eye is the parts of the upgrade. There's is no demand for the EX itself.
Thank you so much for this info. Any idea of the year of production, more or less? How much would you sell it for? Sorry for all these questions but I'm considering selling it and have no idea about its true value.
Maybe you could list by it for $200 & haggle from there. The tremolo is quite pitted. & The pickup poles are rusty. You'll need to see if the pups at least still work. Anytime I see rust on the poles I worry about corrosion inside the pickup which will short out the windings.


After having my iPhone 3G for almost a year, I am now able to record videos on it!!! Thanks to the recent apps that were released on the App Store! The picture above shows the most recent apps I've downloaded there (the ones on the red box). They are mainly for the enhancement of the camera feature of the iPhone 3G.
First, the iVideoCamera and the Camcorder. Both of them are just $0.99 each and with that price you can already record videos on your iPhone 3G or 2G. It's quality is obviously not as good as the iPhone 3GS. It's just enough to make you see moving frames although the developers stated, "We will be enhancing the frame rate and quality soon." So with that stated, I am expecting it to be much better in the next few months perhaps. :) Although the Camcorder does seem to have a better quality than the iVideoCamera, i would still prefer the iVideoCamera just because I can actually import the file to the computer and even upload it directly on YouTube or Facebook. Pretty nice feature. Here is a sample of a vid I took earlier.
The 5.0 MPX app you see above, as its name implies, turns your iPhone into a 5 megapixel camera. My iPhone 3G only has a 2MP camera so it does improve the picture quality really well.
It would really come in handy if I would want to take better looking photos. It only cost $0.99 too.
Lastly, the PS Mobile or the Photoshop Mobile app. It is free but the features are limited to crop, exposure, some effects, tint, rotate, flip and sketch. I don't know if I missed some but those features are the basic ones that you would probably frequently use for photo enhancement.
So there, iPhone 3G or 2G users, these apps will greatly improve the camera feature of your phone. Grab it now while it is still $0.99 on the app store!


We're really excited that our guest on the next #JobAdviceSA will be Tshepo 1Million (@Tshepo1Million) which is a Gauteng Provincial Government youth empowerment initiative designed to break down barriers for youth to participate in the economy with any education level, from Grade 10 to Graduates. "Phanda" is the name of the show that Tshepo 1Million gives to work seekers at their activation events, so we are looking forward to doing an online Phanda with them!
Tshepo 1Million provides hope to young people in Gauteng who are unemployed and looking for a way to access the skills and learning that will prepare them for the world of work. If you are between the ages of 18-34 with at least Grade 10 and are a first-time work seeker with no more than 12 months work experience then you can register by going to www.tshepo.mobi using any mobile phone or device that can connect to the internet and following a simple five step registration process.
Q1. What is the first thing young people should do when they start looking for work?
Q2. How should they go about drafting their cover letter and/or CV?
Q3. How should they prepare for an interview?
Q4. Can you give young people some general tips on interview etiquette?
Feel free to answer any of the questions yourselves using A1, A2, A3, etc. or ask Tshepo 1Million any other questions you might have.
We look forward to chatting with you on Monday!
A few job search tips for you!
Copyright © 2019 Job Advice SA. All rights reserved.


monster high house and school. My husband and I created this monster high house and school for our granddaughter for क्रिस्मस 2012. HD Wallpaper and background images in the मॉन्स्टर हाइ club tagged: monster high school draculara lagoona cleo clawdeen frankie house.


text:
[Blu-Ray Review] Days Of Thunder (1990) (Paramount Presents); Now Available From Paramount
June 18, 2020 · by Kevin Lovell	· in Movies, Movie Reviews, Reviews.	·
Days Of Thunder (1990) (Paramount Presents) [Blu-Ray]
Director: Tony Scott
Cast: Tom Cruise, Nicole Kidman, Robert Duvall
Newly remastered from a 4K film transfer supervised by producer Jerry Bruckheimer, Days Of Thunder explodes with the most spectacular racing action ever captured. Tom Cruise plays race driver Cole Trickle, whose talent and ambition are surpassed only by his burning need to win. Discovered by businessman Tim Daland (Randy Quaid), Cole is teamed with legendary crew chief and car-builder Harry Hogge (Academy Award winner Robert Duvall) to race for the Winston Cup at the Daytona 500. After a fiery crash nearly ends Cole's career, he turns to a beautiful doctor (Nicole Kidman) to regain his nerve and the true courage needed to race, to win, and to live.
Be sure to also check out our Blu-ray reviews for the other releases in the 'Paramount Presents' Blu-ray line below!
Paramount Presents #1: Fatal Attraction (1987) (Blu-ray Review)
Paramount Presents #2: King Creole (1958) (Blu-ray Review)
Paramount Presents #3: To Catch A Thief (1955) (Blu-ray Review)
Paramount Presents #4: Flashdance (1983) (Blu-ray Review)
It's time to hit the racetrack once again with the racing classic 'Days of Thunder' which tells the tale of pro-driving newcomer Cole Trickle (Cruise) who joins a businessman's new team only to impress and improve almost every step of the way, quickly finding himself a key player on the road to the biggest event around. But after suffering a brutal crash that he was lucky to survive, he finds a new passion and drive through a budding love with the young doctor treating him, soon allowing him to believe he can return to driving better than before. Directed by Tony Scott, produced by Don Simpson and Jerry Bruckheimer and starring Tom Cruise, 'Days of Thunder' is a wildly enjoyable and fast paced ride that revisits much of what made 'Top Gun' work so well, yet in a new arena and environment, and it still holds up as an enjoyable and generally well made movie with a healthy fan base thirty years after its debut.
The tense, high-speed classic 'Days of Thunder' returns home newly remastered on Blu-ray as the fifth entry in the new 'Paramount Presents' numbered Blu-ray line of Paramount's various film classics from the studio's expansive library with collectible packaging and new 4K film transfers, plus more. Bringing back together the winning 'Top Gun' teaming of director Tony Scott, producers Don Simpson and Jerry Bruckheimer, and of course star Tom Cruise in the film's lead, 'Days of Thunder' borrows its share of concepts and general ideas from 'Top Gun' while taking the sky-soaring action and excitement to the ground this time around. Some obvious similarities aside, it definitely still conjures up a thoroughly exciting, often intense and always fun ride with plenty to keep you glued to the screen.
Complimented by a wonderful supporting cast that includes Robert Duvall, Nicole Kidman and Randy Quaid, 'Days of Thunder' remains nearly as fun and thrilling as it did upon its initial release and fans hoping for a new and fully remastered release of the car racing hit are in for quite a treat. Hardcore fans of Cruise, Scott or the film 'Days of Thunder' in particular will all likely want to at least consider picking up a copy of this newly remastered Blu-ray release (or the 4K Ultra HD release which was released the same day and marks the first time ever available on the new format) and shouldn't be disappointed with the result, albeit somewhat lacking when it comes to extras.
Overall, the new 'Paramount Presents' Blu-ray release of the high-speed racing drama 'Days of Thunder' should altogether satisfy fans who still don't own the movie on Blu-ray or don't mind double dipping for an upgrade. Serving as the fifth entry in the collectible 'Paramount Presents' Blu-ray line of Paramount film classics, the Tom Cruise led hit returns home with a quite dazzling new high definition video presentation from a new 4K film transfer that was supervised by producer Jerry Bruckheimer (those of you who are fully 4K capable may also want to look into the new 4K Ultra HD release which hit stores the same day as this Blu-ray edition and marks the film's first time ever available on the format). Further boosted by an immersive multichannel soundtrack and a couple of extras, this new 'Paramount Presents' Blu-ray release of 'Days of Thunder' is recommended for anyone already considering a purchase.
The 'Paramount Presents' Blu-ray release of 'Days of Thunder' features a new full 1080p High Definition presentation of the film with a 2.40:1 Cinemascope Aspect Ratio from a new 4K film transfer supervised by producer Jerry Bruckheimer. This newly remastered video presentation looks fantastic and serves up a smooth, sharp and nicely detailed presentation from start to finish that still retains its share of grain and other welcome artifacts reflecting the film source. It holds up admirably even during the various fast moving and darkly lit sequences, never allowing anything onscreen to become negatively affected or rendered indiscernible. Overall, this is a wonderful new high definition video presentation that should easily thrill the film's fans.
The 'Paramount Presents' Blu-ray release features a 5.1 channel Dolby TrueHD soundtrack. While I'm not entirely certain whether this is a new or carryover audio presentation, this multichannel soundtrack still delivers the goods quite impressively regardless. It regularly takes full advantage of all five available channels in order to send racing effects, along with crowd chatter, music and plenty more zipping throughout the various speakers rather frequently, while always making certain that no dialogue or other audio elements that might be occurring simultaneously ever become distorted or rendered inaudible in the process. Overall, this is a great 5.1 channel Dolby TrueHD soundtrack that never disappoints and repeatedly contributes some additional fun and excitement to the viewing experience.
The new 'Paramount Presents' Blu-ray release of 'Days of Thunder' comes home sporting the usual collectible artwork that's been a standard for the 'Paramount Presents' Blu-ray line-up, featuring a slipcase that opens up to showcase the film's original theatrical artwork and more photos on the interior of the Blu-ray case insert (as always with these releases we've included a beauty shot of the artwork from the studio below to provide a better look). The release also offers a couple of extras on the disc itself including an 'Isolated Score' track for the film, along with the brand new Featurette 'Filmmaker Focus: Days of Thunder with Producer Jerry Bruckheimer' (running approximately 6 minutes in length) created specifically for this release.
'Days Of Thunder' is Now Available to Own, Newly Remastered on 4K Ultra HD for the First Time Ever, or as part of the 'Paramount Presents' Blu-ray line from Paramount Home Entertainment
Tags: Blu-Ray, Blu-ray reviews, Days of Thunder, Days of Thunder BD, Days of Thunder Blu ray, Days of Thunder Blu ray review, Days of Thunder Paramount Presents, Days of Thunder Paramount Presents Blu ray, Days of Thunder Paramount Presents Blu ray review, Days of Thunder Paramount Presents Review, Movies, Paramount, Paramount Home Entertainment, Paramount Presents, Paramount Presents Blu ray, Paramount Presents Blu ray reviews, Racing Movies, Reviews, Tom Cruise, Tony Scott
← Deathstroke: Knights & Dragons; The Animated Movie Arrives On Digital August 4 & On Blu-ray & DVD August 18, 2020 From DC & Warner Bros
DEG Watched At Home Top 20 List For 06/18/20: The Hunt, Yellowstone: Season 1 & More →

redpajama_set_name: RedPajamaCommonCrawl


text:
Remanufacturing Benefits
Vision – Mission – Values
Commercial Vehicles – Europe
Current Positions – US
Current Positions – UK
North America Plant Manager Retires After 45 Years
On April 8th, 2021
Rodney Traylor grew up on a farm with ten brothers and sisters. He wanted to be a pilot. He served in the Army for 2 years, and then in 1976 at age 19 he applied to Fred Jones and was hired tearing down water pumps with hot oil dripping on his head, for $1.65 per hour.
Forty five years later, Rodney is retiring from ATC Drivetrain (formerly Fred Jones). After he started in water pumps, he tore down heads, then was the head painter, lathe operator and crank grinder, and then he was promoted to the lead person in cranks. He was then promoted to the supervisor on the 2nd floor downtown location for brake shoe and window lifts. Later he went to transmissions where he was promoted to a supervisor. He transferred to engines as a supervisor and was then promoted to a plant manager, the role which he has held for many years.
Rodney talks fondly of when he started, and a lot of memories from the company picnics with the employees and their families. Also in his earliest memories of his career was the onsite cafeteria where he would buy breakfast and hang out with his coworkers.
Reflecting back on 45 years, Rodney says the cleaning process and chemicals have been the thing that has changed the most, while the people are what has changed the least. "We have always had great people," he stated. He even lists some of his previous managers as personal inspiration to him – Bill Berry, Sean Tate, Debra Shumar and Cathy Walker.
While Rodney feels that he's missed out on events with his family over the years due to his dedication to work, he now plans on spending time with his four grandchildren and working on some farm equipment.
When asked what he would miss the most, Rodney replied quickly, "the people. I will miss all the people that I've worked with over the years. I will miss folks giving me a hard time about being 100 years old."
Always smiling and often adding a joke or a quick remark, Rodney Traylor's presence at ATC will be missed greatly.
For the company newsletter announcing Rodney's welcome to Fred Jones in 1976, click here.
For Rodney's full interview, click here.
9901 W Reno Ave
customerservice@atcdt.com
ATC Drivetrain © 2022


Judas Iscariot the Apostle with a Special place in Hell part 2 video. Judas sold out Jesus for thirty pieces of silver (KJV). What would today's value be of thirty silver coins, or thirty pieces of silver (KJV)? It would be about $15,000 on today's market. Too many today have their own "thirty pieces of silver" reason they sell out Jesus. Do you have one – a relationship, addiction, greed, laziness? Is there a special place in hell waiting?
Judas Iscariot is the most colossal failure in human history as he did the most horrific deed ever done in human history by betraying the Son of God for money. Do you know anyone who's ever turned their back on Jesus because of money?
The thing about Judas Iscariot is he is the most pungent example of how despicable the human heart can be. Judas Iscariot was called to the greatest calling a person could be called, that of an apostle of Jesus Christ. While the other eleven grew transforming every day becoming sons of God, Judas grew transforming every day becoming a son of hell.


Our Elite Performance Program, where the skies (and opportunities) await!
The South Australian Circus Centre has a structured Performance Program Divided into 5 tiers. Keen Beanz, Live Wirez, Tribe, Troupe and an independent artist program. Each tier focuses on developing different attributes of the young artist's journey. Periodically children from the general circus classes are invited into the program based on technical skills, creativity, leadership and enthusiasm as well as natural aptitude.
These classes are arranged into groups of like-minded peers and each focuses on certain characteristics of developing young artists. Students are able to fast track their skill development, develop performance skills and begin to focus on their preferred specialities. Members of these classes are given optional, public performance opportunities at events including school fetes and community events. Artists are also supported to realise their own vision for their creative future. This may mean creating small companies or shows, or using the skills of visioning, planning and reflection in whatever career they want. Several or our graduates are travelling the world with companies like Gravity and Other Myths, Circa, Circus Oz as well as winning awards at fringe festivals around the country.
The next workshop audition for the elite performance program will be held in early April 2018 after the Fringe. If your child is interested in beginning their circus journey pathway, please call us on 8346 5735 to discuss.
Members of the Live Wirez class are the fourth tier of the performance program run by our new 'Artist Development Coordinator' Jonathan (Jono) Dragt. Jono is an old Cirkidz member who has recently returned from working with children in a Belgium Circus School. His position will be to follow the journey of all the performance students through the school. Your trainer will be Joel Osbourne - who is one of our most experienced trainers at Cirkidz and has been a key part in developing young artists and nurturing their journey into the Tribe and Troupe Program.
Members of the Keen Beanz class are the fifth tier of the performance program run by our new 'Artist Development Coordinator' Jonathan (Jono) Dragt. Jono is an old Cirkidz member who has recently returned from working with children in Circus Schools in Belgium and England. His position will be to follow the journey of all performance students through their journey. Jono will also be the trainer for the class.
Tuesdays from 5:00pm – 7:30pm, AND Saturdays 1:30pm – 4:30pm.
Members of the Tribe are the second Tier of the performance program run by our new 'Artist Development Coordinator' Jonathan (Jono) Dragt. Jono is an old Cirkidz member who has recently returned from working with children in a Belgian Circus School. His position will be to follow the journey of all the performance students through the school. Tribe is where intensive training begins, there will be 4 coaches available for the tribe training sessions, and students will be required to choose one of two Major training focuses (trained every Tuesday and Saturday), and one of two Minor training focuses (trained every Saturday). These intensive blocks last six weeks before new training focuses are chosen.
Tuesdays from 5:00pm – 7:30pm, Thursdays from 5.00pm - 7.30pm AND Saturdays 1:30pm – 4:30pm.
The Troupe is made up of our most dedicated performers who audition each year for entry and train three times per week. Here the focus is all about highly skilled, artistic excellence and the love of the artform. Troupe members make up the cast of Cirkidz annual major production, travel on regional tours and high profile performances.
The Independent Artist program allows experienced circus artists to train at the centre at a significantly reduced cost in return for their active participation in the circus community. The program offers the artist access to the SA Circus Centre for self directed learning at approved times which is unavailable to the general public. All applicants must be approved by the Artistic Director and are required to have current Duck 4 Cover Public Liability, workshop and equipment cover. The program is a wonderful opportunity for artists to connect and collaborate in a friendly and supportive circus community hub.
Registering for a Circus School class couldn't be easier.
Take a peek at our newsletter. You can sign up right here.
Come and check out our space!


Financing needed care is often a challenge. Our Financial Coordinator will work with you to ensure that our care will fit into your budget. Our office accepts checks, major credit cards or we can arrange financing using several other options. We will file predetermination statements with your insurance carrier and arrange with you for the copayment. Payment ultimately is the responsibility of the patient, but with creativity and cooperation we can help you afford needed treatment.
At Aspen Periodontics our goal is to help you in every way with your dental health and smile. We know that some of the procedures which we offer are not covered by standard dental benefits and we would like you to know that there are a number of ways to pay for your care. You can select whichever one is most convenient for you.
Surgeries may be paid for in two (2) installments. The first payment of 50% would be made the day of surgery, with the balance to be paid in 30 days.


text:
[VIDEO] Silk Reveals Possible Reason For Diamond's Death… Shocks Everyone
As you all know, "Diamond" of "Dimond and Silk" passed away recently. There's been a lot of confusion about her death. The left claims she died from COVID, and was unvaccinated, and therefore deserved to die. You know, the typical "loving" stuff we often hear from those tolerant communists. NBC News reported that Diamond of the pro-Trump entertainers Diamond and Silk has died, the former president as well as the duo's verified Twitter account announced Monday night. She was 51.
Both women, sisters whose names are Lynette Hardaway and Rochelle Richardson, were born in 1971, with Diamond making her debut on Thanksgiving Day, according to their 2020 book, "Uprising."Calling her death "really bad news for Republicans," former President Donald Trump said on his Truth Social platform, "Our beautiful Diamond of Diamond and Silk has just passed away at her home in the state she loved so much, North Carolina."
The tribute to Diamond's life was amazing and beautiful. She was an extraordinary woman, full of life and sass, and she left this world too soon…
But what happened to her?
Her sister, Silk, shed some light on the situation, and from what it sounds like, many people believe that Silk is saying that Diamond was indeed vaccinated, and was a victim of this new horrific "died suddenly" phenomenon.
You can watch the video below:
BREAKING: Silk reveals the possible reason for Diamond's death. WATCH! #RipDiamond pic.twitter.com/EvWh1PlJsj
— RSBN 🇺🇸 (@RSBNetwork) January 21, 2023
Here's what people online are saying:
"Omg, was she vaccinated?"
"They were vaccinated. What else could Silk have been referring to? I didn't think they took it, but it sounds like they did…"
"What an AMAZING celebration"
"I pray for all of them and their families. I watched diamond & silk all the time and my heart hurts for silk right and the rest of the family. "
"I hope they find out what really happened to her. She deserves that. May she rest in peace."
"Things about to get real , real soon ."
"A lot of people were scared. They didn't believe in mandates, but they believed the fear mongering from media. Sad."
I am sure this isn't the last we've heard about this. Let's hope that the family finds out why Dimond died, so they can begin to have closure.
The opinions expressed by contributors and/or content partners are their own and do not necessarily reflect the views of WayneDupree.com
I'm glad you're here, WayneDupree.com comments! Please maintain polite and on-topic conversations. You could see comments from our Community Managers, who will be identified by a "WD Staff" or "Staff" label, in order to promote fruitful and civil discussions. We stop accepting comments on articles three days after they are posted in order to provide the optimal user experience. The conversations forums on WayneDupree.com welcome comments for an unlimited period of time. For further information, please refer to our community policies.
SIGN UP HERE and join us!
Follow Wayne on Rumble!
Classified Documents Mishandling is One Thing; Dem Mayors Angry At Biden Over Border Crisis
WTH? Nancy Pelosi Had Priests Perform "Exorcism" After "Hammer Attack," Daughter Says...


text:
Mohamed Amroune (Algiers, 10 maart 1989) is een Algerijns voetballer die momenteel uitkomt voor MC Alger. Hij speelde sinds 2008 38 wedstrijden voor die club en scoorde daarin 11 goals. Amroune speelt als aanvaller.

Hij maakte vroeger ook deel uit van de meeste nationale jeugdelftallen, maar speelde nog nooit één wedstrijd voor het Algerijns voetbalelftal.

Algerijns voetballer

redpajama_set_name: RedPajamaWikipedia


TRENTON, NJ— Lawmakers on a joint legislative panel expressed concerns and questioned Gov. Phil Murphy's rush to phase out a statewide standardized exam, calling for meetings to be held and more data to be provided.
State education commissioner Lamont Repollet appeared before a joint hearing of the Senate and Assembly Education committees on Monday to testify and answer questions about the future of the Partnership for Assessment of Readiness for College and Careers (PARCC) exam.
The skepticism among lawmakers was bipartisan, and it appears unlikely that the Repollet will be able to scale back the use of the PARCC exam as quickly as he had been expecting.
His appearance before the joint committee came less than a week after the state board of education tabled a vote that would cut four out of six English and math standardized tests in high school level.
"I think this is a reset, a restart button, and one that … affords us all an opportunity to really set the bar, as New Jersey has often done, to create the next generation of tests," Ruiz said, adding that she did not expect to see the matter on the Board of Education's October meeting agenda.
The hearing at times was contentious as the Commissioner was challenged to explain the rush to end tests that he said were going to be replaced by "next generation" tests that have yet to be identified. Repollett acknowledged that transition out of PARCC to a new statewide exam would be two- to three-year process.
However, Ruiz said the approach to just end PARCC without empirical evidence was problematic.
Murphy, who ran on a platform of eliminating the statewide exam on day one of his tenure, announced in July plans to eliminate all but the federal minimum allowed for testing high school students. Under Repollet's proposal, the PARCC exam would largely remain the same for students in third through eighth grade with adjustments only being made to testing time.
Repollet said the scaling back of the exam allows for high school students to prepare for life after graduation.
"We feel they've been tested enough," Repollet said, adding that currently, high school students are taking two to three standardized tests a year and that the PARCC test did not measure college and career readiness.
But some legislators weren't having it.
"We get too caught up sometimes in just the conversation around tests and forgetting that there are generations of kids who are not prepared to succeed in this world, and if we keep fighting over what the name of a test is called, there are still generations of kids who are not prepared to succeed in this world," Singleton said, calling into question what the elimination of the tests might mean for the state's most vulnerable students.
"Being able to close the achievement gaps based on social economics and racial demographics should be what keeps many of us up at night," Singleton said.
Lampitt also questioned the Education Department's plan to reduce teacher evaluation based on Student Growth Percentile (SGP) down from 30 percent to 5 percent, rather than helping the teachers reach the standard.
Repollet said only about 20 percent of teachers statewide are evaluated using the SGP from the PARCC exam. "It's not fair to those other educators," Repollet said.
Following the hearing, a coalition of organizations that included the New Jersey Education Association, Save our Schools New Jersey and the Education Law Center held a press conference offering their reaction to the hearing, and called for the state Board of Education to vote on the Murphy administration's proposal.
"We're very concerned about how these assessments have narrowed the curriculum in our schools, have changed what our children and learn," Julie Borst, executive director of Save our Schools said.


text:
\subsection{Kernel of the type of Green's function}\label{subsection:2.1}

Let $r \geq 0$   be an integer and
assume that the kernel $\kappa$ of the integral operator $\mathcal {K}$ defined in (\ref{eq:1.1}) has the following properties.

\begin{enumerate}
\item Let $ \Psi = [0, 1] \times [0, 1] \times \R.$ The partial derivative
$ \displaystyle {\ell (s, t, u) = \frac {\partial \kappa ( s, t, u) } { \partial u}}$
is continuous for all $ (s, t, u) \in \Psi. $

\item Let
$ \Psi_1 = \{ (s, t, u): 0 \leq t \leq s \leq 1, \; u \in \R \},\;\;\;
 \Psi_2 = \{ (s, t, u): 0 \leq s \leq t \leq 1, \; u \in \R \}.$
There are functions $\ell_i \in C^{r+1}  ( \Psi_i ), i = 1, 2, $ with
$$
\ell (s,t, u) = \left\{ {\begin{array}{ll}
\ell_1 (s, t, u), \;\;\; (s, t, u) \in \Psi_1,   \\
\ell_2 (s, t, u), \;\;\; (s, t, u) \in \Psi_2.
\end{array}}\right.
$$

\item
There are two functions $\kappa_i \in C^{r+1} ( \Psi_i ), i = 1, 2, $ such that
$$
\kappa (s,t, u) = \left\{ {\begin{array}{ll}
\kappa_1 (s, t, u), \;\;\; (s, t, u) \in \Psi_1,   \\
\kappa_2 (s, t, u), \;\;\; (s, t, u) \in \Psi_2.
\end{array}}\right.
$$

\item
$ \displaystyle {\frac { \partial^2 \kappa} {\partial u^2} \in C ( \Psi).}$
\end {enumerate}
Following Atkinson-Potra \cite{AtkP1}, if the kernel  $ \kappa$ satisfies the above conditions,
then we say that $ \kappa$ is of class $\mathcal{G}_2 ({r+1}, 0).$

Let $ f \in C^{r+1} [0, 1]$, then by the Corollary 3.2 of Atkinson-Potra \cite{AtkP1}, it follows that $\varphi \in C^{r+1} [0, 1].$ If $ r = 0,$ then it is assumed that $ f \in C^{2} [0, 1]$ so that $\varphi \in C^{2} [0, 1].$ We assume that $\mathcal{K}$ is twice Fr\'echet  differentiable and that $1$ is not an eigenvalue of $\mathcal {K}' (\varphi).$

\subsection{Nystr\"{o}m  approximation}
Let $m \in \mathbb{N}$ and
 consider the following uniform partition of $[0, 1]:$
\begin{equation}\label{eq:2.1}
0  <  \frac{1} {m}  < \cdots <   \frac{m-1} {m}   <  1.
\end{equation}
Let
$\displaystyle { \tilde{h} = \frac {1} {m} \;\; \mbox {and} \;\; s_i = \frac {i} {m},  \;\; i = 0, \ldots, m.}$
   Consider a basic quadrature rule of the form
  \begin{equation}\nonumber
  \int_0^1 f (t) d t \approx \sum_{q=1}^{\rho} w_q f (\mu_q),
  \end{equation}
  where the weights $w_q > 0$ and the nodes $\mu_q \in [0, 1].$ It is assumed that the
  quadrature rule is exact
   at least for polynomials of degree $\leq 2 r. $
   Then $\displaystyle{\sum_{q=1}^{\rho} w_q  =  1}.$

A composite integration rule with respect to the partition  (\ref{eq:2.1})
is then defined as
\begin{eqnarray}\label{eq:2.2}
 \int_0^1 f (t) d t &=& \sum_{i =1}^m \int_{s_{i -1}}^{s_i} f (t) d t
  \approx  \tilde h \sum_{i=1}^m  \sum_{q =1}^{\rho} w_q \; f (\zeta_q^i ), \;\;\; \zeta_q^i = s_{i -1} + \mu_q \tilde{h}.
\end{eqnarray}
We replace the integral in (\ref{eq:1.1}) by the numerical quadrature formula (\ref{eq:2.2}) and define
the Nystr\"{o}m operator as
\begin{equation}\nonumber
\mathcal{K}_m (x) (s)  =  \tilde {h}  \sum_{i=1}^m  \sum_{q=1}^\rho w_q \;  \kappa \left (s,  \zeta_q^i, x \left (\zeta_q^i \right ) \right), \;\;\; s \in [0, 1].
\end{equation}
Note that $\mathcal{K}_m$ is twice Fr\'echet  differentiable and
\begin{equation}\nonumber
\mathcal {K}_m'  (x) v (s)  =  \tilde {h}  \sum_{i=1}^m  \sum_{q=1}^\rho  w_q \;  \frac {\partial \kappa } {\partial u} (s,  \zeta_q^i, x (\zeta_q^i)) v (\zeta_q^i), \;\;\; s \in [0, 1],
\end{equation}
and
\begin{equation}\nonumber
\mathcal {K}_m''  (x) v (s)  =  \tilde {h}  \sum_{i=1}^m  \sum_{q=1}^\rho  w_q \;  \frac {\partial^2 \kappa } {\partial u^2} (s,  \zeta_q^i, x (\zeta_q^i)) v (\zeta_q^i), \;\;\; s \in [0, 1].
\end{equation}
For $ \delta_0 > 0, $ let
$ \displaystyle {\mathcal{B}(\varphi ,\delta_0) = \{ \psi \in \mathcal{X}: \| \varphi - \psi \|_\infty <
\delta_0 \}.}$
Define
$$  C_1 =   \max_{\stackrel {s, t \in [0, 1]}{|u| \leq \|\varphi \|_\infty + \delta_0 }}
\left | \frac {\partial \kappa } {\partial u} (s, t, u) \right | \; \mbox {and} \;\;\; C_2 =   \max_{\stackrel {s, t \in [0, 1]}{|u| \leq \|\varphi \|_\infty + \delta_0 }}
\left | \frac {\partial^2 \kappa } {\partial u^2} (s, t, u) \right |. $$
Then  for $x, y \in \mathcal{B}(\varphi ,\delta_0),$
\begin{eqnarray}\label{eq:2.3}
\|\mathcal {K}_m' (x) v \|_\infty \leq  C_1 \|v\|_\infty
\end{eqnarray}
and
\begin{eqnarray}\label{eq:2.4}
\|\mathcal {K}_m'  (x) - \mathcal {K}_m'  (y)\| \leq  C_2  \|x - y \|_\infty.
\end{eqnarray}
Even though the numerical quadrature is assumed to be exact for polynomials of degree $\leq 2 r,$ since the kernel
lacks smoothness along $s = t,$ we only have the following order of convergence from Atkinson-Potra \cite {AtkP2}:
If $x \in C^2 [0, 1],$ then
\begin{equation}\label{eq:2.5}
\|\mathcal{K} (x) - \mathcal{K}_m (x) \|_\infty = O \left (\tilde{h}^2 \right ).
\end{equation}
In the Nystr\"{o}m method,  (\ref{eq:1.2}) is approximated by
\begin{equation}\nonumber
x_m - \mathcal{K}_m (x_m) = f.
\end{equation}
For all $m$ big enough, the above equation   has a unique solution $\varphi_m$ in $\mathcal{B} (\varphi, \delta_0 )$
and
\begin{eqnarray}\label{eq:2.6}
\|\varphi - \varphi_m \|_\infty  \leq C_3  \|\mathcal{K} (\varphi) - \mathcal{K}_m (\varphi ) \|_\infty = O \left (\tilde{h}^{2}\right).
\end{eqnarray}
See Atkinson \cite{Atk1}.
We quote the following result from Krasnoselskii et al  \cite{KraV}  for future reference:

If $v_1, v_2 \in \mathcal{B} (\varphi, \delta_0),$ then by the generalized Taylor's theorem,
\begin{eqnarray}\label{eq:2.7}
\mathcal{K}_m (  v_2 ) (s) -  \mathcal{K}_m ( v_1 ) (s) - \mathcal{K}_m' ( v_1) (v_2 - v_1) (s) &= &
 R  (v_2 - v_1)  (s), \; s \in [0, 1],
 \end{eqnarray}
 where
 \begin{eqnarray}\nonumber
&&R  (v_2 - v_1)  (s)
 = \int_0^1  {(1 - \theta) }   \mathcal {K}_m^{''} \left (v_1 + \theta
 (v_2 - v_1) \right ) (v_2 - v_1)^2  (s)    d \theta.
\end{eqnarray}
It then follows that
\begin{equation}\label{eq:2.8}
\| R  (v_2 - v_1)  \|_\infty \leq  C_2  \|v_2 - v_1\|_\infty^2.
\end{equation}
\subsection {Discrete orthogonal projection}

Let $n \in \mathbb{N}$ and consider the following uniform partition of $[0, 1]:$
\begin{equation}\label{eq:2.9}
\Delta: 0  <  \frac{1} {n}  < \cdots <   \frac{n-1} {n}   <  1.
\end{equation}
Define
\begin{equation}\nonumber
t_j = \frac {j} {n}, \;\;\;\Delta_j = [t_{j-1}, t_j]   \;\;\; \mbox {and} \;\;\; h = t_{j} - t_{j-1} = \frac {1} {n}, \;\;\; j = 1, \ldots, n.
\end{equation}
For $r \geq 0,$ let $\mathcal{X}_n$ denote the space of piecewise polynomials of degree $\leq r $ with respect to
the partition of (\ref{eq:2.9}) of $[0, 1].$ Assume that the values at
$t_j-, \; j = 1, \ldots, n,$ are defined by continuity. Then the dimension of $\mathcal{X}_n$ is $ n (r+1).$

\noindent
For $\eta = 0, 1, \ldots, r,$ let $L_\eta $ denote the Legendre polynomial of degree $\eta$ on $[-1, 1].$ Define
\begin{eqnarray}\nonumber
\varphi_{1, \eta} (t) &=& \left\{ {\begin{array}{ll}
\sqrt {\frac {2} { h}} L_\eta \left ( \frac {2 t - t_1 - t_0} {h} \right ), \;\;\;  t \in [t_{0}, t_1],   \\
0, \;\;\;  \mbox{otherwise}.
\end{array}}\right.
\end{eqnarray}
For $j = 2, \ldots, n,$ and for  $\eta = 0, 1, \ldots, r,$ define
\begin{eqnarray}\nonumber
\varphi_{j, \eta} (t) &=& \left\{ {\begin{array}{ll}
\sqrt {\frac {2} { h}} L_\eta \left ( \frac {2 t - t_j - t_{j-1}} {h} \right ), \;\;\;  t \in (t_{j-1}, t_j],   \\
0, \;\;\;  \mbox{otherwise}.
\end{array}}\right.
\end{eqnarray}
Note that $\displaystyle { \|\varphi_{j, \eta} \|_\infty = \max_{t \in [t_{j-1}, t_j]} |\varphi_{j, \eta} (t)|  = \sqrt {\frac {2} { h}} \|L_\eta\|_\infty.}$

From now onwards we assume that $m = p n$ for some $p \in \N.$ Thus each interval $\Delta_j$ is
divided into $p$ equal parts and the integral over each interval $\Delta_j$ is approximated by using the quadrature formula (\ref {eq:2.2}) restricted to the interval $\Delta_j.$
For $f, g \in C (\Delta_j),$
define
\begin{equation}\label{eq:2.10}
\inp {f} {g}_{\Delta_j} = \tilde {h} \sum_{\nu = 1}^p \sum_{q = 1}^{\rho}   w_q ~ f {\left (\zeta_q^{(j-1) p + \nu} \right )} ~ g{\left (\zeta_q^{(j-1) p + \nu} \right )},
\end{equation}
where $\zeta_q^{(j-1) p + \nu}$ are defined in (\ref{eq:2.2}).
Note that $\displaystyle {\inp {f} {g}_{\Delta_j} = \inp {g} {f}_{\Delta_j}.}$
Define
\begin{equation}\nonumber
\| f \|_{\Delta_j, \infty} = \max_{t \in [t_{j-1}, t_j]} |f (t)|.
\end{equation}
Then
\begin{equation}\label{eq:2.11}
\left |\inp {f} {g}_{\Delta_j} \right | \leq  \| f \|_{\Delta_j, \infty} \| g \|_{\Delta_j, \infty} h.
\end{equation}
Since the quadrature rule is exact for polynomials of degree $\leq 2 r ,$  it follows that
$$ \inp {\varphi_{j, \eta}}{\varphi_{j, \eta'}} = \int_{0}^{1}  \varphi_{j, \eta} (t) \varphi_{j, \eta'} (t) d t =
  \inp {\varphi_{j, \eta}}{\varphi_{j, \eta'}}_{\Delta_j}.$$
Thus,
$\displaystyle {\{\varphi_{j, \eta},  j = 1, \ldots, n, \; \eta = 0,  \ldots, r \}}$
forms an orthonormal basis for  $\mathcal{X}_n \subset L^\infty [0, 1].$
Let $\mathcal{P}_{r, \Delta_j}$ denote the space of polynomials of degree $\leq r $  on $\Delta_j.$
Define the discrete orthogonal projection $Q_{n,j}: C (\Delta_j) \rightarrow \mathcal{P}_{r, \Delta_j}$ as follows:
\begin{equation}\nonumber
Q_{n, j} x = \sum_{\eta = 0}^r \inp {x} {\varphi_{j, \eta}}_{\Delta_j} \varphi_{j, \eta}.
\end{equation}
It follows that
\begin{eqnarray}\nonumber
\inp {Q_{n, j} x} {y}_{\Delta_j}  = \inp { x} {Q_{n, j} y}_{\Delta_j}, \;\;\; Q_{n, j}^2 = Q_{n, j} \;\;\; \mbox {and} \;\;\; Q_{n, j} Q_{n, i} = 0 \;\;\; \mbox {for} \;\;\; i \neq j.
\end{eqnarray}
Also,
\begin{eqnarray}\nonumber
\|Q_{n, j} x\|_{\Delta_j, \infty} &\leq& \sum_{\eta = 0}^r \left | \inp {x} {\varphi_{j, \eta}}_{\Delta_j} \right |
\|\varphi_{j, \eta}\|_{\Delta_j, \infty}
 \leq   \left (2 \sum_{\eta = 0}^r  \|L_\eta\|_\infty^2 \right ) \|x\|_\infty.
\end{eqnarray}
A discrete orthogonal projection $Q_n:  C[0, 1] \rightarrow \mathcal{X}_n$ is defined as follows:
 \begin{eqnarray}\label{eq:2.12}
  Q_n x = \sum_{j=1}^n Q_{n, j} x.
  \end{eqnarray}
  Using the Hahn-Banach extension theorem, as in Atkinson et al \cite{AtkG}, $Q_n$ can be extended to $L^\infty [0, 1].$
 Then
\begin{eqnarray}\label{eq:2.13}
Q_n^2 = Q_n \;\;\;\mbox{and} \;\;\; \|Q_n\| \leq 2 \sum_{\eta = 0}^r  \|L_\eta\|_\infty^2 = C_4.
\end{eqnarray}
The following estimate is standard:
If $f \in C^{r+1} (\Delta_j),$ then we have,
\begin{eqnarray}\label{eq:2.14}
  \|(I - Q_{n,j})f \|_{\Delta_j, \infty} &\leq& C_5 \|f^{(r+1)} \|_{\Delta_j, \infty}  h^{r+1}.
\end{eqnarray}
Thus, if $f \in C^{r+1} [0, 1],$ then
\begin{eqnarray}\label{eq:2.15}
  \|(I - Q_{n})f \|_{ \infty} &=& O \left (  h^{r+1} \right ).
\end{eqnarray}
   \subsection{Discrete Projection Methods}

   We define below the discrete versions of various projection methods given in Section 1 by replacing the
   integral operator $ \mathcal{K}$ by the Nystr\"{o}m operator $\mathcal{K}_m$ and the orthogonal projection
   $\pi_n$ by the discrete orthogonal projection $Q_n.$

 \noindent
Discrete Galerkin Method:
\begin{equation}\nonumber
z_n^G - Q_n \mathcal{K}_m  (z_n^G) = Q_n f.
\end{equation}
Discrete Iterated Galerkin Method:
\begin{equation}\nonumber
{z}_n^S -  \mathcal{K}_m  (Q_n {z}_n^S) =  f.
\end{equation}
The discrete modified projection operator is defined as
\begin{equation}\label{eq:2.16}
\tilde {\mathcal{K}}_n^M (x) = Q_n \mathcal{K}_m (x) + \mathcal{K}_m (Q_n x) - Q_n \mathcal{K}_m (Q_n x).
\end{equation}
Discrete Modified Projection method:
\begin{equation}\label{eq:2.17}
{z_n^M - \tilde {\mathcal{K}}_n^M (z_n^M) = f.}
 \end{equation}

\noindent
Discrete Iterated Modified Projection method:
\begin{equation}\label{eq:2.18}
{\tilde{z}_n^M =  {\mathcal{K}}_m (z_n^M) + f.}
\end{equation}

\setcounter{equation}{0}

\section{Piecewise polynomial approximation : $\mathbf{ r \geq 1 }$ }

In this section we consider the case $r \geq 1$ and obtain orders of convergence in the discrete modified projection method and its iterated version.

\subsection{Preliminary results}

In Proposition 3.1 we first obtain an error estimate for the term
$\| \mathcal {K}_m'  (\varphi) (I - Q_n) v \|_\infty, $  where $v \in C^{r+1} [0, 1].$ Note that the term
$\mathcal {K}_m'  (\varphi) (I - Q_n) v  (s)$ needs to be treated differently depending upon whether $s$ is a partition point of the partition (\ref{eq:2.9}) or $s \in (t_{i-1}, t_i)$ for some $i.$  Using this result, we obtain an error estimate for the term
$\| \mathcal {K}_m'  (\varphi) (I - Q_n) \mathcal {K}_m'  (\varphi) (I - Q_n)v \|_\infty $ in
Proposition 3.2. These estimates are used in obtaining orders of convergence of $z_n^M$ and $\tilde{z}_n^M.$

Let
\begin{equation}\nonumber
\ell_* (s, t) =   \ell (s, t, \varphi (t)), \;\;\; 0 \leq s, t \leq 1,
\end{equation}
where $ \displaystyle {\ell (s, t, u) = \frac {\partial \kappa ( s, t, u) } { \partial u}}$. Then
$$
\ell_* (s,t) = \left\{ {\begin{array}{ll}
\ell_{1, *} (s, t) = \ell_1 (s, t, \varphi (t)), \;\;\; 0 \leq t \leq s \leq 1, \\
\ell_{2, *} (s, t) = \ell_2 (s, t, \varphi (t)), \;\;\; 0 \leq s \leq t \leq 1.
\end{array}}\right.
$$
Since  $\varphi \in C^{r+1} [0, 1],$ it follows that
 $$ \ell_{1, *} \in C^{r+1} ( \{ 0 \leq t \leq s \leq 1\}) \; \mbox {and} \; \ell_{2, *} \in C^{r+1} ( \{ 0 \leq s \leq t \leq 1\}).$$

\noindent
We introduce the following notation.
For a fixed $s \in [0, 1],$ define
\begin{equation}\nonumber
\ell_{*, s} (t) = \ell_* (s, t), \; t \in [0, 1].
\end{equation}
Note that
\begin{eqnarray}\nonumber
\mathcal {K}_m'  (\varphi) v (s)  & = &  \tilde {h}  \sum_{j = 1}^n \sum_{\nu =1}^p  \sum_{q=1}^\rho  w_q \;
 \ell_* (s,  \zeta_q^{(j-1) p + \nu} ) v (\zeta_q^{(j-1) p + \nu}) = \sum_{j=1}^n \inp {\ell_{*, s}} {v}_{\Delta_j}.
\end{eqnarray}
 Let
$$ C_6 = \max_{1 \leq j \leq r+1} \left \{ \max_{\stackrel { 0 \leq t \leq  s \leq 1 }{|u| \leq \|\varphi \|_\infty  }}
\left | D^{(0,j, 0)} \ell_{1} (s, t, u)  \right |, \max_{\stackrel { 0 \leq s \leq  t \leq 1 }{|u| \leq \|\varphi \|_\infty }}
\left | D^{(0, j, 0)} \ell_{2}  (s, t, u)  \right |   \right \}.$$

The following proposition is crucial. It will be used several times in what follows.

\begin{proposition}\label{prop:3.1}
 If  $v \in C^{ r + 1} [0, 1],$
 then
\begin{eqnarray}\label{eq:3.1}
\| \mathcal {K}_m'  (\varphi) (I - Q_n) v \|_\infty
& \leq &  (C_5)^2 C_6  \|v^{(r+1)} \|_\infty  h^{r +3}.
\end{eqnarray}
\end{proposition}

\begin{proof}
For $s \in [0, 1],$
\begin{eqnarray*}
\mathcal {K}_m'  (\varphi)  (I - Q_n ) v (s)
& = & \sum_{j=1}^n \inp {  \ell_{*, s}} {(I - Q_{n,j}  )v}_{\Delta_j} = \sum_{j=1}^n \inp {(I - Q_{n,j}  ) \ell_{*, s}} {(I - Q_{n,j}  )v}_{\Delta_j}.
\end{eqnarray*}
Case 1:  $ s = t_i$ for some $i \in \left\{0, 1, \ldots, n \right\}.$ Then
$\ell_{*, s} \in C^{r+1} (\Delta_j)$
for $j = 1, \ldots, n.$ Since $v \in C^{r+1} [0, 1],$
it follows  from (\ref{eq:2.14}),
\begin{eqnarray}\nonumber
\max_{0 \leq i \leq n} |\mathcal{K}_m (I - Q_n ) x (t_i) | &\leq&  \sum_{j=1}^n \|(I - Q_{n,j}  ) \ell_{*, s}\|_{\Delta_j, \infty}
\|(I - Q_{n, j}) v\|_{\Delta_j, \infty} h\\\label{eq:3.2}
& \leq &  (C_5)^2 C_6 \|v^{(r+1)}\|_\infty   h^{2 r + 2}.
\end{eqnarray}
Case 2: $s \in (t_{i-1}, t_i)$ for some $i \in \left\{1, 2, \ldots, n \right\}.$ We write
\begin{eqnarray}\nonumber
\mathcal {K}_m'  (\varphi)  (I - Q_n ) v (s) & = &
\sum_{\stackrel {j=1} {j \neq i}}^n \inp { (I - Q_{n,j}  ) \ell_{*, s}} {(I - Q_{n,j}  )v}_{\Delta_j}\\\label{eq:3.3}
&&~ +  \inp { (I - Q_{n,i}  ) \ell_{*, s}} {(I - Q_{n,i}  )v}_{\Delta_i}.
\end{eqnarray}
For $j \neq i,$ $\ell_{*, s} \in C^{r+1} (\Delta_j)$ and $v  \in C^{r+1} (\Delta_j).$
Hence
\begin{eqnarray}\label{eq:3.4}
\left |\sum_{\stackrel {j=1} {j \neq i}}^n \inp { (I - Q_{n,j}  ) \ell_{*, s}} {(I - Q_{n,j}  )v}_{\Delta_j} \right |
\leq (C_5)^2 C_6 \|v^{(r+1)}\|_\infty  (n-1) h^{2 r + 3}.
\end{eqnarray}
We now consider the case $j = i.$ Note that $\ell_{*, s}$ is only continuous on $[t_{i-1}, t_i].$
Define a constant function:
$$ g_i (t) = \ell_{*, s} (s) , \;\; t \in [t_{i-1}, t_i].$$
Note that
$\hspace*{0.5 cm}
\displaystyle {\inp { (I - Q_{n,i}  ) \ell_{*, s}} {(I - Q_{n,i}  )v}_{\Delta_i}   =  \inp { \ell_{*, s} - g_i} {(I - Q_{n,i}  )v}_{\Delta_i}.}$
For $t \in [t_{i-1}, t_i],$
\begin{eqnarray*}
 \ell_{*, s} (t) - g_i (t)
& = & \left\{ {\begin{array}{ll}
D^{(0,1)} \ell_{1, *} (s,  \theta_t) (t - s),  \;\;\;  \theta_t \in (t,   s),   \\
D^{(0,1)} \ell_{2, *} (s, \eta_t ) (t - s),  \;\;\;    \eta_t \in (s, t).
\end{array}}\right.
\end{eqnarray*}
Thus,
\begin{eqnarray}\label{eq:3.5}
\left |\inp { (I - Q_{n,i}  ) \ell_{*, s}} {(I - Q_{n,i}  )v}_{\Delta_i} \right |
& \leq &  C_5  C_6 \|v^{(r+1)} \|_\infty h^{r+3}.
\end{eqnarray}
Without loss of generality, let $C_5 \geq 1.$
 From (\ref{eq:3.3}), (\ref{eq:3.4}) and (\ref{eq:3.5}) we obtain,
\begin{eqnarray}\nonumber
|\mathcal {K}_m'  (\varphi)  (I - Q_n ) v (s)|
& \leq &  (C_5)^2 C_6   \|v^{(r+1)} \|_\infty  h^{r + 3}.
\end{eqnarray}
Combining  the above estimate with (\ref{eq:3.2}) we obtain the required result.
\end{proof}

\begin{proposition}\label{prop:3.2}
If  $v \in C^{ r + 1} [0, 1],$
 then
\begin{equation}\label{eq:3.6}
\|  \mathcal {K}_m'  (\varphi) (I - Q_n )  \mathcal {K}_m'  (\varphi) (I - Q_n) v \|_\infty =
O \left ( h^{r + 5} \right ).
\end{equation}
Also,
\begin{eqnarray}\label{eq:3.7}
\left \| \mathcal {K}_m'  (\varphi) (I - Q_n ) \mathcal {K}_m'  (\varphi)   \right \| = O \left (   h^{ 2 } \right ).
\end{eqnarray}
\end{proposition}

\begin{proof}
The proof of (\ref{eq:3.6}) is similar to that of (\ref{eq:3.1}). For $s \in [0, 1],$ we write
\begin{eqnarray*}
\mathcal {K}_m'  (\varphi)  (I - Q_n ) \mathcal {K}_m'  (\varphi)  (I - Q_n ) v (s)
& = & \sum_{j=1}^n \inp { (I - Q_{n,j}  ) \ell_{*, s} } {    \mathcal {K}_m'  (\varphi) (I - Q_n ) v}_{\Delta_j}.
\end{eqnarray*}
If $s = t_i,$ for some $i,$ then using (\ref{eq:2.14}) and (\ref{eq:3.1})
we obtain
\begin{eqnarray}\nonumber
 \left | \mathcal {K}_m'  (\varphi) (I - Q_n ) \mathcal {K}_m'  (\varphi)  (I - Q_n ) v (t_i) \right |
 \leq  (C_5)^3 (C_6)^2   \|v^{(r+1)} \|_\infty  h^{2 r + 4}.
\end{eqnarray}
If $s \in (t_{i-1}, t_i),$ then we write
\begin{eqnarray*}
\mathcal {K}_m'  (\varphi)  (I - Q_n )  \mathcal {K}_m'  (\varphi)  (I - Q_n ) v (s)
& = &  \sum_{\stackrel {j=1} {j \neq i}}^n \inp { (I - Q_{n,j}  ) \ell_{*, s}} {   \mathcal {K}_m'  (\varphi)  (I - Q_n )   v}_{\Delta_j}\\
 & + &  \inp {   \ell_{*, s} - g_i }  {  \mathcal {K}_m'  (\varphi)  (I - Q_n ) v}_{\Delta_i}.
\end{eqnarray*}
Proceeding as in the proof of Proposition 3.1, we obtain
\begin{eqnarray}\nonumber
|\mathcal {K}_m'  (\varphi) (I - Q_n ) \mathcal {K}_m'  (\varphi) (I - Q_n ) v (s) |
 &\leq & (C_5)^3 (C_6)^2  \|v^{(r+1)} \|_\infty h^{ r + 5 }.
\end{eqnarray}
The estimate (\ref{eq:3.6}) follows from the above two estimates.

In order to prove  (\ref{eq:3.7}), consider $v \in C [0, 1].$ Let
 $ s = t_i$ for some $i.$ Then
\begin{eqnarray}\label{eq:3.8}
 \left | \mathcal {K}_m'  (\varphi) (I - Q_n )  v (s) \right |
& \leq & C_5 C_6  \|v \|_\infty  h^{ r + 1 }.
\end{eqnarray}
Now
let $s \in (t_{i-1}, t_i). $ We write
\begin{eqnarray}\nonumber
\mathcal {K}_m'  (\varphi)  (I - Q_n )    v (s)
=  \sum_{\stackrel {j=1} {j \neq i}}^n \inp { (I - Q_{n,j}  ) \ell_{*, s}} {    v}_{\Delta_j}
 +  \inp {  (I - Q_{n,i}  ) (\ell_{*, s} - g_i) }  {v}_{\Delta_i}.
\end{eqnarray}
and obtain
\begin{eqnarray}\nonumber
|\mathcal {K}_m'  (\varphi) (I - Q_n ) v (s) |
 &\leq &   (1 + C_4 + C_5) C_6    \|v \|_\infty h^{ 2 }.
\end{eqnarray}
Combining (\ref{eq:3.8}) and the above estimate, we obtain
\begin{eqnarray}\label{eq:3.9}
\|\mathcal {K}_m'  (\varphi) (I - Q_n ) v \|_\infty
 &\leq &   (1 + C_4 + C_5) C_6  \|v \|_\infty h^{ 2 }.
\end{eqnarray}
Since from (\ref{eq:2.3}),
$  \| \mathcal {K}_m'  (\varphi)  v\|_\infty \leq  C_1  \|v\|_\infty,$
we obtain
\begin{eqnarray}\nonumber
\|\mathcal {K}_m'  (\varphi) (I - Q_n ) \mathcal {K}_m'  (\varphi)  v \|_\infty
 &\leq & \; C_1 (1 + C_4 + C_5) C_6  \|v \|_\infty h^{ 2}
\end{eqnarray} and the required result follows taking the supremum over unit ball in $C[0, 1].$
\end{proof}

\subsection{Error in the  discrete modified projection method}

As in Grammont \cite{Gram1}, it can be shown that there is a $\delta_0 > 0$ such that (\ref{eq:2.17}) has a unique
solution $z_n^M$ in $\mathcal{B} (\varphi, \delta_0)$ and that
\begin{eqnarray}\nonumber
\| z_n^M - \varphi\|_\infty
& \leq & 6  \left \| \left (I - \mathcal{K}' (\varphi) \right )^{-1} \right \| \left ( \|\mathcal{K} (\varphi) -
\mathcal{K}_m (\varphi)\|_\infty + \|\mathcal{K}_m (\varphi) -\tilde{\mathcal{K}}_n^M (\varphi)\|_\infty \right ).\\\label{eq:3.10}
\end{eqnarray}
In the following theorem, we obtain the order of convergence of the discrete modified projection solution.
\begin{theorem}\label{thm:3.3}
Let  $r \geq 1,$  $ \kappa$ be of class $\mathcal{G}_2 ({r+1}, 0)$
and $ f \in C^{r+1} [0, 1].$
Let $\varphi$ be the unique solution of (\ref{eq:1.2}) and assume that $1$ is not an eigenvalue of $\mathcal{K}' (\varphi).$
Let $\mathcal{X}_n$ be the space of piecewise polynomials of degree $\leq r $ with respect to the partition (\ref{eq:2.9})
and $Q_n$ be the discrete orthogonal projection defined by
(\ref{eq:2.12}). Let $z_n^M $ be the discrete modified projection solution in $\mathcal{B} (\varphi, \delta_0 ).$ Then
\begin{equation}\label{eq:3.11}
\|z_n^M - \varphi \|_\infty = O (\max \{\tilde{h}^{2}, h^{r + 3}\}).
\end{equation}
\end{theorem}
\begin{proof}
From (\ref{eq:2.5}),
\begin{eqnarray}\label{eq:3.12}
\|\mathcal{K} (\varphi) -  \mathcal{K}_m (\varphi)\|_\infty = O \left ( \tilde{h}^{2} \right ).
\end{eqnarray}
Since $\varphi \in C^{r+1} [0, 1],$ it follows from (\ref{eq:2.15}) that $\displaystyle {\| Q_n \varphi - \varphi \|_\infty = O (h^{r+1}).}$
Note that
\begin{eqnarray}\nonumber
\|\mathcal{K}_m (\varphi) - \tilde{\mathcal{K}}_n^M (\varphi)  \|_\infty
& \leq & \| (I - Q_n) (\mathcal{K}_m (Q_n\varphi) - \mathcal{K}_m (\varphi) - \mathcal {K}_m'  (\varphi) (Q_n \varphi - \varphi ) )\|_\infty\\\nonumber
&& + \| (I - Q_n) \mathcal {K}_m'  (\varphi) (Q_n \varphi - \varphi )\|_\infty.
\end{eqnarray}
From (\ref{eq:2.7}), (\ref{eq:2.8}) and (\ref{eq:2.15}),
\begin{eqnarray}\nonumber
\| \mathcal{K}_m (Q_n\varphi) - \mathcal{K}_m (\varphi) - \mathcal {K}_m'  (\varphi) (Q_n \varphi - \varphi ) \|_\infty \leq    {C_2 }  \| Q_n \varphi - \varphi \|_\infty^2 = O (h^{2 r + 2}).
\end{eqnarray}
By (\ref{eq:2.13}) and Proposition \ref{prop:3.1},
\begin{eqnarray}\nonumber
\| (I - Q_n) \mathcal {K}_m'  (\varphi) (Q_n \varphi - \varphi )\|_\infty
\leq (1 + C_4) \| \mathcal {K}_m'  (\varphi) (Q_n \varphi - \varphi )\|_\infty  = O (h^{r + 3}).
\end{eqnarray}
Since $r \geq 1,$ it then follows that
\begin{eqnarray}\nonumber
\|\mathcal{K}_m (\varphi) - \tilde{\mathcal{K}}_n^M (\varphi)  \|_\infty = O (h^{r + 3}).
\end{eqnarray}
The required result follows from (\ref{eq:3.10}), (\ref{eq:3.12}) and the above estimate.
\end{proof}

\begin{remark}
It can be  shown that
\begin{equation}\label{*}
 \|z_n^G - \varphi \|_\infty = O \left (\max \left \{\tilde{h}^{2}, h^{r + 1 } \right \} \right ),
 \;\;\; \|z_n^S - \varphi \|_\infty = O \left (\max \left \{\tilde{h}^{2}, h^{r +3} \right \} \right ).
\end{equation}
Thus the order of convergence of $z_n^S$ and $z_n^M$ is the same. We prove  the estimate (\ref{eq:3.11}) as
it is needed for obtaining the order of convergence in the iterated discrete modified projection method.
\end{remark}

\subsection{Error in the  discrete iterated modified projection method}

Note that
\begin{equation}\nonumber
\tilde{z}_n^M - \varphi_m =  \mathcal {K}_m ({z}_n^M)  - \mathcal {K}_m (\varphi_m).
\end{equation}
From (\ref{eq:2.7}) and (\ref{eq:2.8}),
\begin{eqnarray}\nonumber
\mathcal {K}_m ({z}_n^M) - \mathcal {K}_m (\varphi_m)  = \mathcal {K}_m' (\varphi_m) (z_n^M - \varphi_m)  +
O \left ( \|z_n^M - \varphi_m\|_\infty^2 \right ).
\end{eqnarray}
From (\ref{eq:2.6}) and Theorem \ref{thm:3.3}, we obtain
\begin{equation}\nonumber
\|z_n^M - \varphi_m \|_\infty \leq  \|z_n^M - \varphi \|_\infty + \|\varphi - \varphi_m \|_\infty = O (\max \{\tilde{h}^{2},
h^{r + 3}\}).
\end{equation}
Thus,
\begin{eqnarray}\label{eq:3.13}
\tilde{z}_n^M - \varphi_m =  \mathcal {K}_m' (\varphi_m) (z_n^M - \varphi_m) + O (\max \{\tilde{h}^{2},
h^{r + 3}\}^2).
\end{eqnarray}
We quote the following result from Kulkarni-Rakshit \cite{Kul1}:
\begin{eqnarray}\nonumber
&&\mathcal{K}_m'   (\varphi_m) ( z_n^M - \varphi_m ) \\\nonumber
& = &
 - \left [ I - {\mathcal{K}}_m'   (\varphi_m) \right]^{-1}  \mathcal{K}_m'   (\varphi_m) \left \{  \mathcal {K}_m (\varphi_m) -  \tilde{\mathcal{K}}_n^M (\varphi_m) \right \}\\\nonumber
&&+ \left [ I - {\mathcal{K}}_m'   (\varphi_m) \right]^{-1} \mathcal{K}_m'   (\varphi_m) \left \{
\tilde{\mathcal{K}}_n^M (z_n^M) - \tilde{\mathcal{K}}_n^M (\varphi_m)  - \left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m)
(z_n^M - \varphi_m) \right \}\\\label{eq:3.14}
&& + \left [ I - {\mathcal{K}}_m'   (\varphi_m) \right]^{-1} \mathcal{K}_m'   (\varphi_m) \left \{
\left (\left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m)  -  \mathcal{K}_m'   (\varphi_m) \right  ) (z_n^M - \varphi_m) \right \}.
\end{eqnarray}
We obtain below orders of convergence for the three terms in (\ref{eq:3.14}).

\begin{proposition}\label{prop:3.4}
Let $\varphi_m$ be the Nystr\"{o}m solution. Then
\begin{eqnarray}\nonumber
 \left \| \mathcal {K}_m'  (\varphi_m) \left (\mathcal {K}_m (\varphi_m) -  \tilde{\mathcal{K}}_n^M (\varphi_m) \right ) \right \|_\infty = O \left ( h^{4} \max \{\tilde{h}^{2}, h^{r +1}\} \right ).
 \end{eqnarray}
\end{proposition}
\begin{proof}
Let $v \in C[0, 1].$ Then from (\ref{eq:2.4}),  (\ref{eq:2.6}), (\ref{eq:2.13}) and (\ref{eq:3.9}),
\begin{eqnarray}\nonumber
\| \mathcal{K}_m' (\varphi_m)(I - Q_n) v \|_\infty &\leq& \| \left [ \mathcal{K}_m' (\varphi_m) -  \mathcal{K}_m' (\varphi) \right ]
(I - Q_n) v \|_\infty
+  \| \mathcal{K}_m' (\varphi)(I - Q_n) v \|_\infty \\\nonumber
&\leq & C_2  (1 + C_4) \|v\|_\infty \|\varphi_m - \varphi \|_\infty
+  (1 + C_4 + C_5) C_6  \|v\|_\infty h^{2}\\\label{eq:3.15}
&\leq & C_7 \|v\|_\infty  h^{2}.
\end{eqnarray}
Note that
\begin{eqnarray}\nonumber
\mathcal{K}_m (\varphi_m) - \tilde{\mathcal{K}}_n^M (\varphi_m)
& = & - (I - Q_n) (\mathcal{K}_m (Q_n\varphi_m) - \mathcal{K}_m (\varphi_m) - \mathcal {K}_m'  (\varphi_m) (Q_n \varphi_m - \varphi_m ) )\\\label{eq:3.16}
&& -  (I - Q_n) \mathcal {K}_m'  (\varphi_m) (Q_n \varphi_m - \varphi_m ).
\end{eqnarray}
Let
\begin{eqnarray}\nonumber
y_n = \mathcal{K}_m (Q_n\varphi_m) - \mathcal{K}_m (\varphi_m) - \mathcal {K}_m'  (\varphi_m) (Q_n \varphi_m - \varphi_m )
= R (Q_n \varphi_m - \varphi_m ) .
\end{eqnarray}
Then by (\ref{eq:3.15})
\begin{eqnarray}\nonumber
\| \mathcal{K}_m' (\varphi_m)(I - Q_n) y_n \|_\infty &\leq& C_7 \|y_n\|_\infty  h^{2}.
\end{eqnarray}
By (\ref{eq:2.8})
\begin{eqnarray}\nonumber
\|y_n \|_{\infty} = \|R (Q_n \varphi_m - \varphi_m )\|_{ \infty } \leq
 C_2  \|Q_n \varphi_m - \varphi_m \|_\infty^2.
\end{eqnarray}
Using (\ref{eq:2.6}),  (\ref{eq:2.13}) and (\ref{eq:2.15}),   we obtain
\begin{eqnarray}\label{eq:3.17}
\|Q_n \varphi_m - \varphi_m \|_\infty
= O (\max \{\tilde{h}^{2}, h^{r+1 }\}).
\end{eqnarray}
Thus,
\begin{eqnarray}\label{eq:3.18}
\| \mathcal{K}_m' (\varphi_m)(I - Q_n) y_n \|_\infty = O \left (  h^{2}  \max \{\tilde{h}^{2}, h^{r +1}\}^2 \right ).
\end{eqnarray}
Using (\ref{eq:3.6}) it can be checked
  that
 \begin{eqnarray*}\nonumber
&&\left \|\mathcal {K}_m'  (\varphi_m)   (I - Q_n) \mathcal {K}_m'  (\varphi_m)   (I - Q_n) \varphi_m \right \|_{\infty}
 = O \left ( h^{4} \max \{\tilde{h}^{2}, h^{r +1}\} \right ).
\end{eqnarray*}
The required result then follows from (\ref{eq:3.16}),  (\ref{eq:3.18}) and the above estimate.
\end{proof}

\begin{proposition}\label{prop:3.5}
Let $\varphi_m$ be the Nystr\"{o}m solution and $z_n^M$ be the discrete modified projection solution. Then
\begin{eqnarray}\nonumber
&&\left \|\tilde {\mathcal{K}}_n^M (z_n^M) - \tilde{\mathcal{K}}_n^M (\varphi_m)  - \left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m)
(z_n^M - \varphi_m) \right \|_\infty = O \left (\max \left \{ \tilde{h}^{2},  h^{ r + 3} \right \}^2 \right).
\end{eqnarray}
\end{proposition}
\begin{proof}
Note that  for $m$ and $n$ big enough,
 $\displaystyle {\varphi_m, z_n^M \in \mathcal {B} \left (\varphi, \delta_0 \right ). }$
By the generalized Taylor's theorem,
\begin{eqnarray*}
&&\tilde {\mathcal{K}}_n^M (z_n^M) (s) - \tilde{\mathcal{K}}_n^M (\varphi_m)  (s) - \left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m)
(z_n^M - \varphi_m) (s) \\
&& \hspace*{0.5 in} =
\int_0^1 ( 1 - \theta) \left (  \tilde{\mathcal{K}}_n^M \right )'' \left (\varphi_m + \theta (z_n^M - \varphi_m) \right ) (z_n^M - \varphi_m)^2
(s) \; d \theta.
\end{eqnarray*}
Hence
\begin{eqnarray}\nonumber
&&\left \|\tilde {\mathcal{K}}_n^M (z_n^M) - \tilde{\mathcal{K}}_n^M (\varphi_m)  - \left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m)
(z_n^M - \varphi_m) \right \|_\infty  \\\nonumber
&& \hspace*{0.5 in} \leq \frac {1} {2} \max_{0 \leq \theta \leq 1}
\left \|\left (\tilde {\mathcal{K}}_n^M \right )'' \left  (\varphi_m + \theta (z_n^M - \varphi_m) \right ) \right \| \| z_n^M - \varphi_m\|_\infty^2.
\end{eqnarray}
It can be shown that
\begin{eqnarray}\nonumber
\max_{0 \leq \theta \leq 1}
\left \|\left (\tilde {\mathcal{K}}_n^M \right )'' \left  (\varphi_m + \theta (z_n^M - \varphi_m) \right ) \right \|
\leq  C_8.
\end{eqnarray}
We skip the details.
The required result then follows from Theorem 3.3.
\end{proof}

\begin{proposition}\label{prop:3.6}
Let $\varphi_m$ be the Nystr\"{o}m solution and $z_n^M$ be the discrete modified projection solution. Then
\begin{eqnarray}\nonumber
&&\left \|\mathcal{K}_m'   (\varphi_m)
\left (\left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m)  -  \mathcal{K}_m'   (\varphi_m) \right  ) (z_n^M - \varphi_m)
\right \|_\infty
= O \left ( h^2 \max \left \{ \tilde{h}^{2},  h^{r + 3} \right \} \right).
\end{eqnarray}
\end{proposition}

\begin{proof}
Note that
\begin{eqnarray*}
\mathcal{K}_m'   (\varphi_m) \left (\left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m)  -  \mathcal{K}_m'   (\varphi_m) \right  )
& = &\mathcal{K}_m'   (\varphi_m) (I - Q_n) (\mathcal{K}_m'  (Q_n \varphi_m )  -  \mathcal{K}_m'   (\varphi_m)) Q_n \\
& - & \mathcal{K}_m'   (\varphi_m)  (I - Q_n) \mathcal{K}_m'   (\varphi_m) (I - Q_n).
\end{eqnarray*}
Using (\ref{eq:2.3}) and (\ref{eq:3.15}) it can be shown that
$$\| \mathcal{K}_m'   (\varphi_m)  (I - Q_n) \mathcal{K}_m'   (\varphi_m) \| = O (h^{ 2}).$$
By (\ref{eq:2.6}) and (\ref{eq:3.17}),
\begin{eqnarray*}
\|\mathcal{K}_m'  (Q_n \varphi_m )  -  \mathcal{K}_m'   (\varphi_m)\|\leq  C_2 \|Q_n \varphi_m - \varphi_m \|_\infty =
O (\max \{\tilde{h}^2, h^{r+1} \}).
\end{eqnarray*}
Since
by (\ref{eq:2.3}),
$\displaystyle { \|\mathcal{K}_m'   (\varphi_m) \| \leq C_1},$
it follows that
\begin{eqnarray}\nonumber
\left \|\mathcal{K}_m'   (\varphi_m) \left (\left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m)  -  \mathcal{K}_m'   (\varphi_m) \right  ) \right \|= O (h^2).
\end{eqnarray}
The required result follows using the estimate for $\|z_n^M - \varphi\|_\infty $ from  Theorem 3.3.
\end{proof}

We now prove our main result about
 the order of convergence in the discrete iterated modified projection method.

\begin{theorem}\label{thm:3.7}
Let   $r \geq 1,$ $ \kappa$ be of class $\mathcal{G}_2 ({r+1}, 0)$
and $ f \in C^{r+1} [0, 1].$
Let $\varphi$ be the unique solution of (\ref{eq:1.2}) and assume that $1$ is not an eigenvalue of $\mathcal{K}' (\varphi).$
Let $\mathcal{X}_n$ be the space of piecewise polynomials of degree $\leq r $ with respect to the partition
(\ref{eq:2.9})
and $Q_n$ be the discrete orthogonal projection defined by
(\ref{eq:2.12}). Let $\tilde{z}_n^M$ be the discrete iterated modified projection solution defined by (\ref{eq:2.18}).
 Then
\begin{eqnarray}\label{eq:3.19}
\|\tilde{z}_n^M - \varphi \|_\infty = O \left ( \max \left \{\tilde{h}^2,
h^{r + 5} \right  \}\right).
\end{eqnarray}
\end{theorem}
\begin{proof}
We have from (\ref{eq:3.13})
\begin{eqnarray}\nonumber
\tilde{z}_n^M - \varphi_m =  \mathcal {K}_m' (\varphi_m) (z_n^M - \varphi_m) + O (\max \{\tilde{h}^{2},
h^{r + 3}\}^2).
\end{eqnarray}
From (\ref{eq:3.14}) recall that
\begin{eqnarray}\nonumber
&&\mathcal{K}_m'   (\varphi_m) ( z_n^M - \varphi_m ) \\\nonumber
& = &
 - \left [ I - {\mathcal{K}}_m'   (\varphi_m) \right]^{-1}  \mathcal{K}_m'   (\varphi_m) \left \{  \mathcal {K}_m (\varphi_m) -  \tilde{\mathcal{K}}_n^M (\varphi_m) \right \}\\\nonumber
&&+ \left [ I - {\mathcal{K}}_m'   (\varphi_m) \right]^{-1} \mathcal{K}_m'   (\varphi_m) \left \{
\tilde{\mathcal{K}}_n^M (z_n^M) - \tilde{\mathcal{K}}_n^M (\varphi_m)  - \left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m)
(z_n^M - \varphi_m) \right \}\\\nonumber
&& + \left [ I - {\mathcal{K}}_m'   (\varphi_m) \right]^{-1} \mathcal{K}_m'   (\varphi_m) \left \{
\left (\left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m)  -  \mathcal{K}_m'   (\varphi_m) \right  ) (z_n^M - \varphi_m) \right \}.
\end{eqnarray}
By Proposition 4.2 from Kulkarni-Rakshit \cite{Kul1}, we have
$$ \left \|\left [ I - {\mathcal{K}}_m'   (\varphi_m) \right]^{-1} \right \| \leq
4 \left \| \left ( I - \mathcal{K}' (\varphi ) \right )^{-1} \right \|.$$
Hence by Proposition \ref{prop:3.4},
  Proposition \ref{prop:3.5},
 and Proposition \ref{prop:3.6},
 \begin{eqnarray}\nonumber
 &&\left \| \mathcal{K}_m'   (\varphi_m) ( z_n^M - \varphi_m )  \right \|_\infty
 = O \left (  \max \left \{  h^4 \max \{\tilde{h}^{2}, h^{r +1}\},    h^{2} \max \{\tilde{h}^{2}, h^{r +3}\}\right \} \right ),
 \end{eqnarray}
 It follows that
  \begin{eqnarray}\nonumber
 &&\left \| \tilde{z}_n^M - \varphi_m  \right \|_\infty =
 O \left (     h^{2} \max \{\tilde{h}^{2}, h^{r +3}\} \right ).
 \end{eqnarray}
 Since, $\tilde{z}_n^M - \varphi = \tilde{z}_n^M - \varphi_m + \varphi_m - \varphi $ and $ \left \| \varphi - \varphi_m \right \|_\infty = O \left ( \tilde{h}^{2} \right ),$
 the required result  follows.
\end{proof}

\setcounter{equation}{0}

\section{Piecewise constant polynomial approximation : $\mathbf{ r = 0 }$}

In this section we assume that $ \kappa$ is of class $\mathcal{G}_2 ({2}, 0).$
If we follow the development in Section 3, then we obtain the following orders of convergence:
 \begin{equation}\nonumber
  \| z_n^M - \varphi \|_\infty = O (h^2 ), \;\;\;
\| \tilde{z}_n^M  - \varphi \|_\infty = O ( \max \{\tilde{h}^2, h^3 \} ).
\end{equation}
But by looking at the proofs more carefully, we are able to improve the above estimates.
More specifically, while for $r \geq 1,$  if $v \in C^{r+1} [0, 1],$ then both $\| \mathcal {K}_m'  (\varphi) (I - Q_n) v \|_\infty $
and $\|(I- Q_n) \mathcal {K}_m'  (\varphi) (I - Q_n) v \|_\infty $ are of the same order, we could show that if $r=0,$ then
$$ \| \mathcal {K}_m'  (\varphi) (I - Q_n) v \|_\infty = O (h^2) \;\;\; \mbox{and} \;\;\;
\| (I - Q_n) \mathcal {K}_m'  (\varphi) (I - Q_n) v \|_\infty = O (h^3).$$
This is the essential point in proving the estimates (\ref{eq:1.5}).

Consider $\mathcal{X}_n$ to be the space of piecewise constant functions with respect to the partition (\ref{eq:2.9}).  Thus, $r = 0.$
We
choose Gauss 2 point rule as a basic quadrature rule:
  \begin{equation}\nonumber
  \int_0^1 f (t) d t \approx  w_1 f (\mu_1) + w_2 f (\mu_2),
  \end{equation}
  where
  \begin{eqnarray*}
  w_1 = w_2 = \frac {1} {2}, \; \mu_1 = \frac {1} {2} - \frac {1} {2 \sqrt{3}}, \;
  \mu_2 = \frac {1} {2} + \frac {1} {2 \sqrt{3}}.
  \end{eqnarray*}
A composite integration rule with respect to the fine partition  (\ref{eq:2.1})
is then defined as
\begin{eqnarray}\nonumber
 \int_0^1 f (t) d t &\approx&  {\tilde h}  \sum_{i =1}^m   \sum_{q =1}^{2} w_q  \; f (\zeta_q^i ),   \;\;\;
  \;\;\;  \zeta_q^i = s_{i -1} + \mu_q \tilde{h}.
\end{eqnarray}
Since $m = n p,$ the above
rule can be written as
\begin{equation}\nonumber
  \int_{0}^1 f (t) d t \approx
  \tilde {h} \ \sum_{j=1}^n \sum_{\nu = 1}^p \sum_{q = 1}^{2} w_q  f \left (\zeta_q^{(j-1) p + \nu} \right ).
  \end{equation}
The Nystr\"{o}m operator is defined as
\begin{eqnarray}\label{eq:4.1}
\mathcal{K}_m (x) (s)  & = & \tilde {h} \sum_{j=1}^n \sum_{\nu = 1}^p \sum_{q = 1}^{2}   w_q ~\kappa \left (s, \zeta_q^{(j-1) p + \nu},  x{\left ( \zeta_q^{(j-1) p + \nu} \right )} \right ).
\end{eqnarray}
Recall from (\ref{eq:2.10})  that
for $f, g \in C (\Delta_j),$
\begin{equation}\nonumber
\inp {f} {g}_{\Delta_j} = \tilde {h} \sum_{\nu = 1}^p \sum_{q = 1}^{2}   w_q ~ f{\left (\zeta_q^{(j-1) p + \nu}\right )} ~ g{\left (\zeta_q^{(j-1) p + \nu} \right)}.
\end{equation}
The discrete orthogonal projection $Q_{n,j}: C (\Delta_j) \rightarrow \mathcal{P}_{0, \Delta_j}$ is defined
as follows:
\begin{eqnarray}\nonumber
(Q_{n, j} v) (t)  =
 \frac {1} {p }  \left [ \sum_{\nu = 1}^p \sum_{q = 1}^2
   w_q v \left (\zeta_q^{(j-1) p + \nu} \right )  \right ], \;\;\; t \in (t_{j-1}, t_j], \;\;\;
\end{eqnarray}
and
\begin{eqnarray}\nonumber
(Q_{n, 1} v) (0)
&=& \frac {1} {p }  \left [ \sum_{\nu = 1}^p \sum_{q = 1}^2
   w_q v \left (\zeta_q^{ \nu} \right )  \right ].
\end{eqnarray}
A discrete orthogonal projection $Q_n:  C[0, 1] \rightarrow \mathcal{X}_n$ is defined as
\begin{eqnarray}\label{eq:4.2}
{Q_n v = \sum_{j=1}^n Q_{n, j} v.}
\end{eqnarray}

The following result is crucial in obtaining improved orders of convergence in the discrete modified
projection method and its iterated version.

\begin{proposition}

If $v \in C^1 [0, 1],$ then
\begin{eqnarray}\label{eq:4.3}
\|(I - Q_{n} ) \mathcal{K}_m'(\varphi) (I - Q_n ) v \|_\infty = O (h^3),
\end{eqnarray}
\begin{eqnarray}\label{eq:4.4}
\left \| \mathcal{K}_m'(\varphi) (I - Q_n ) \mathcal{K}_m'(\varphi) (I - Q_n ) v \right \|_\infty = O (h^4).
\end{eqnarray}
\end{proposition}
\begin{proof}
Note that
\begin{eqnarray*}
\| (I - Q_n ) \mathcal{K}_m'(\varphi) (I - Q_n ) v \|_\infty
& = & \max _{1 \leq i \leq n} \max_{s \in [t_{i-1}, t_i]}
\left |(I - Q_{n, i} ) \mathcal{K}_m'(\varphi) (I - Q_n ) v (s) \right|.
\end{eqnarray*}
   For $s \in [t_{i-1}, t_i],$
  \begin{eqnarray} \nonumber
&&(I - Q_{n, i} ) \mathcal{K}_m'(\varphi) (I - Q_n ) v (s) \\\nonumber
   && \hspace*{1 cm} = \frac {1} {p } \sum_{\nu = 1}^p \sum_{q = 1}^2
   w_q \left \{ \mathcal{K}_m'(\varphi) (I - Q_n ) v (s) - \mathcal{K}_m'(\varphi) (I - Q_n ) v \left (\zeta_q^{(i-1) p + \nu} \right ) \right \}
   \\\nonumber
   && \hspace*{1 cm} = \frac {1} {p } \sum_{\nu = 1}^p \sum_{q = 1}^2   \sum_{ \stackrel {j =1} {j \neq i}}^n
   w_q  \inp {\ell_{*,s} - \ell_{*,\zeta_q^{(i-1) p + \nu}} } {(I - Q_{n, j} ) v}_{\Delta_j}\\\label{eq:4.5}
   && \hspace*{1 cm} + \frac {1} {p } \sum_{\nu = 1}^p \sum_{q = 1}^2
   w_q  \inp {\ell_{*,s} - \ell_{*,\zeta_q^{(i -1) p + \nu}} } {(I - Q_{n, i} ) v}_{\Delta_i}.
   \end{eqnarray}
   For $j \neq i,$
   \begin{eqnarray*}
    \inp {\ell_{*,s} - \ell_{*,\zeta_q^{(i-1) p + \nu}} } {(I - Q_{n, j} ) v}_{\Delta_j}
     = (s - \zeta_q^{(i-1) p + \nu}) \inp {D^{(1, 0)} \ell_{*}  (\eta_q^{{(i-1) p + \nu}}, \cdot) } {(I - Q_{n, j} ) v}_{\Delta_j},
   \end{eqnarray*}
   for some $\eta_q^{{(i-1) p + \nu}} \in (t_{i - 1}, t_i).$
    Define the following constant function
   $$ g_q^{{(i-1) p + \nu}} (t) =  D^{(1, 0)} \ell_{*}  {\left (\eta_q^{{(i-1) p + \nu}}, \frac {t_{j-1} + t_j} {2} \right )} , \; \; \;
   t \in [t_{j - 1}, t_j].$$
   Then
   \begin{eqnarray*}
    && \inp {\ell_{*,s} - \ell_{*,\zeta_q^{(i-1) p + \nu}} } {(I - Q_{n, j} ) v}_{\Delta_j} \\
    &  & \hspace*{1 cm} = \left (s - \zeta_q^{(i-1) p + \nu}\right )  \inp { D^{(1, 0)} \ell_{*} (\eta_q^{(i-1) p + \nu}, \cdot) - g_q^{(i-1) p + \nu } } {(I - Q_{n, j} ) v}_{\Delta_j}.
   \end{eqnarray*}
   From (\ref{eq:2.11}) and (\ref{eq:2.14}),
   \begin{eqnarray*}
  &&  \left | \inp {\ell_{*,s} - \ell_{*,\zeta_q^{(i-1) p + \nu}} } {(I - Q_{n, j} ) v}_{\Delta_j} \right |
   \leq   C_5 \left ( \max_{s \neq t} \left |D^{(1, 1)} \ell_{*} (s, t)
   \right | \right ) \|v'\|_{ \infty} h^4.
  \end{eqnarray*}
    On the other hand,  from (\ref{eq:3.5}) with $r=0,$
    \begin{eqnarray*}
    \left | \inp {\ell_{*,s} - \ell_{*,\zeta_q^{(i-1) p + \nu}} } {(I - Q_{n, i} ) v}_{\Delta_i} \right |
    & \leq &  2  C_5 C_6  \|v'\|_\infty h^3, \;\;\; \nu = 1, \ldots, p.
    \end{eqnarray*}
   Thus, from (\ref{eq:4.5}) and the above two estimates,
\begin{eqnarray}\nonumber
 \|(I - Q_{n, i} ) \mathcal{K}_m'(\varphi) (I - Q_n ) v \|_{\Delta_i, \infty}
& \leq &  C_5 \max\left\{ 2 C_6, \max_{s \neq t} \left |D^{(1, 1)} \ell_{*} (s, t)
\right |  \right\}
   \|v'\|_\infty h^3.
\end{eqnarray}
  This completes the proof of (\ref{eq:4.3}).

  In order to prove (\ref{eq:4.4}), as before we consider two cases.
   If $s = t_i$ for some $i,$ then
  \begin{eqnarray*}
\mathcal{K}_m'(\varphi) (I - Q_n ) \mathcal{K}_m'(\varphi) (I - Q_n ) v (s)
& = & \sum_{j=1}^n \inp { (I - Q_{n,j}  ) \ell_{*,s} } {  (I - Q_{n,j}  )  \mathcal{K}_m'(\varphi) (I - Q_n )v}_{\Delta_j}.
\end{eqnarray*}
  If $s \in (t_{i-1}, t_i),$ then we write
  \begin{eqnarray}\nonumber
\mathcal{K}_m'(\varphi) (I - Q_n ) \mathcal{K}_m'(\varphi) (I - Q_n ) v (s)
& = & \sum_{\stackrel {j=1} {j \neq i}}^n \inp { (I - Q_{n,j}  ) \ell_{*,s} } {  (I - Q_{n,j}  )  \mathcal{K}_m'(\varphi) (I - Q_n )v}_{\Delta_j}\\\nonumber
& + & \inp {   \ell_{*,s} - g_i} {  (I - Q_{n,i}  )\mathcal{K}_m'(\varphi) (I - Q_n )v}_{\Delta_i}.
\end{eqnarray}
Proceeding as in the proof of Proposition 3.2 and using the estimate (\ref{eq:4.3}), we obtain the required result.
  \end{proof}

\begin{theorem}\label{thm:4.2}
Let   $ \kappa$ be of class $\mathcal{G}_2 (2, 0)$
and $ f \in C^{2} [0, 1].$
Let $\varphi$ be the unique solution of (\ref{eq:1.2}) and assume that $1$ is not an eigenvalue of $\mathcal{K}' (\varphi).$
Let $\mathcal{X}_n$ be the space of piecewise constant functions  with respect to the partition (\ref{eq:2.9})
and $Q_n: L^\infty [0, 1] \rightarrow \mathcal {X}_n$ be the discrete orthogonal projection defined by (\ref{eq:4.2}).
 Let $z_n^M $ be the discrete modified projection solution in $\mathcal{B} (\varphi, \delta_0 ).$ Then
\begin{equation}\label{eq:4.6}
\|z_n^M - \varphi \|_\infty = O (\max \{\tilde{h}^2, h^3 \}).
\end{equation}
\end{theorem}

\begin{proof}
Recall from (\ref{eq:3.10}) that
\begin{eqnarray}\nonumber
\| z_n^M - \varphi\|_\infty
& \leq & 6  \left \| \left (I - \mathcal{K}' (\varphi) \right )^{-1} \right \| \left ( \|\mathcal{K} (\varphi) -
\mathcal{K}_m (\varphi)\|_\infty + \|\mathcal{K}_m (\varphi) -\tilde{\mathcal{K}}_n^M (\varphi)\|_\infty \right ).
\end{eqnarray}
From (\ref{eq:2.5}) we have
\begin{eqnarray}\label{eq:4.7}
\|\mathcal{K} (\varphi) - \mathcal{K}_m (\varphi)\|_\infty = O (\tilde{h}^2).
\end{eqnarray}
On the other hand,
\begin{eqnarray}\nonumber
\|\mathcal{K}_m (\varphi) - \tilde{\mathcal{K}}_n^M (\varphi)  \|_\infty
& \leq & \| (I - Q_n) (\mathcal{K}_m (Q_n\varphi) - \mathcal{K}_m (\varphi) - \mathcal {K}_m'  (\varphi) (Q_n \varphi - \varphi ) )\|_\infty\\\label{new1}
&& + \| (I - Q_n) \mathcal {K}_m'  (\varphi) (Q_n \varphi - \varphi )\|_\infty.
\end{eqnarray}
Recall from (\ref{eq:2.7}) that
\begin{eqnarray}\nonumber
 \mathcal{K}_m (Q_n\varphi) - \mathcal{K}_m (\varphi) - \mathcal {K}_m'  (\varphi) (Q_n \varphi - \varphi )  =
R  ( Q_n \varphi - \varphi ),
\end{eqnarray}
where
\begin{align}\nonumber
R  ( Q_n \varphi - \varphi )(s) = \int_{0}^{1} \mathcal{K}_m''(\varphi + \theta(Q_n \varphi - \varphi )) (Q_n \varphi - \varphi )^2(s) (1-\theta) d\theta.
\end{align}
Note that
\begin{align*}
&\mathcal {K}_m''  (\varphi + \theta(Q_n \varphi - \varphi )) (Q_n \varphi - \varphi )^2 (s) \\ &\hspace{1cm} =  \tilde {h}  \sum_{j=1}^m  \sum_{q=1}^\rho  w_q \;  \frac {\partial^2 \kappa } {\partial u^2} (s,  \zeta_q^j, (\varphi + \theta(Q_n \varphi - \varphi )) (\zeta_q^j) (Q_{n, j} \varphi - \varphi )^2  (\zeta_q^j)
\end{align*}
Define
\begin{align*}
\sigma_n (s,t) = \frac {\partial^2 \kappa } {\partial u^2} (s,  t, (\varphi(t) + \theta(Q_n \varphi - \varphi )(t))
\end{align*}
and for a fixed $s \in [0, 1],$ let
\begin{equation*}
\sigma_{n, s} (t) = \sigma_n (s,t), \; t \in [0, 1].
\end{equation*}
Let
\begin{eqnarray*}
C_9 = \max\left\{ \sup_{ \stackrel {0\leq t<s\leq 1} {|u|\leq \|\varphi\|_\infty + \delta_0}}
\left |D^{(0,1,1) } \ell_1 (s, t, u) \right | , \sup_{ \stackrel {0\leq s<t\leq 1} {|u|\leq \|\varphi\|_\infty + \delta_0}}
\left |D^{(0,1,1) } \ell_2 (s, t, u) \right | \right\},
\end{eqnarray*}
Note that
\begin{eqnarray*}
\mathcal {K}_m''  (\varphi + \theta(Q_n \varphi - \varphi )) (Q_n \varphi - \varphi )^2 (s)
&  = &\sum_{j = 1}^n \left< (I-Q_{n, j})\sigma_{n, s}, ((I- Q_{n, j}) \varphi )^2 \right>_{\Delta_j}.
\end {eqnarray*}
If $s = t_i$ for some $i,$ then for all $j$ and if $s \in (t_{i-1}, t_i)$ for some $i,$ then for $j \neq i,$
\begin{eqnarray*}
\|(I-Q_{n, j})\sigma_{n, s}\|_{\Delta_j, \infty} \leq C_5 C_9 h.
\end{eqnarray*}
We then obtain $\;\;\; \left \|\mathcal {K}_m''  (\varphi + \theta(Q_n \varphi - \varphi )) (Q_n \varphi - \varphi )^2 \right \|_\infty =
O (h^3).$
It follows that
\begin{eqnarray}\label{eq:4.8}
\| (I - Q_n) (\mathcal{K}_m (Q_n\varphi) - \mathcal{K}_m (\varphi) - \mathcal {K}_m'  (\varphi) (Q_n \varphi - \varphi ) \|_\infty
& = & O (h^3).
\end{eqnarray}
Using the estimate (\ref{eq:4.3}) of Proposition 4.1 and \eqref{new1}, we thus obtain
\begin{eqnarray*}
\|\mathcal{K}_m (\varphi) - \tilde{\mathcal{K}}_n^M (\varphi)  \|_\infty   = O (h^3).
\end{eqnarray*}
The required result follows from (\ref{eq:4.7}) and the above estimate.
\end{proof}

\begin{theorem}\label{thm:4.3}
Let   $ \kappa$ be of class $\mathcal{G}_2 (2, 0)$
and $ f \in C^{2} [0, 1].$
Let $\varphi$ be the unique solution of (\ref{eq:1.2}) and assume that $1$ is not an eigenvalue of $\mathcal{K}' (\varphi).$
Let $\mathcal{X}_n$ be the space of piecewise constant functions with respect to the partition (\ref{eq:2.9})
and $Q_n: L^\infty [0, 1] \rightarrow \mathcal {X}_n$ be the discrete orthogonal projection defined by
(\ref{eq:4.2}). Let $\tilde{z}_n^M$ be the discrete iterated modified projection solution defined by (\ref{eq:2.18}). Then
\begin{eqnarray}\label{eq:4.9}
\|\tilde{z}_n^M - \varphi \|_\infty = O \left ( \max \left \{\tilde{h}^2, h^{4} \right \} \ \right).
\end{eqnarray}
\end{theorem}
\begin{proof}
Recall from Section 3.3 that
\begin{eqnarray}\nonumber
\tilde{z}_n^M - \varphi_m &=& \mathcal {K}_m' (\varphi_m) (z_n^M - \varphi_m) + O (\|z_n^M - \varphi \|_\infty^2)
\end{eqnarray}
Hence by Theorem 4.2,
\begin{eqnarray}\label{eq:4.10}
\tilde{z}_n^M - \varphi_m = \mathcal {K}_m' (\varphi_m) (z_n^M - \varphi_m) +
O \left (\max \{\tilde{h}^{2}, h^{ 3} \}^2 \right )
\end{eqnarray}
We now obtain estimates for the three terms in the expression for $\mathcal {K}_m' (\varphi_m) (z_n^M - \varphi_m)$ given in (\ref{eq:3.14}).
Note that
\begin{eqnarray}\label{eq:4.11}
\|\mathcal{K}_m (\varphi_m) - \tilde{\mathcal{K}}_n^M (\varphi_m)  \|_\infty
& \leq & \| (I - Q_n) (\mathcal{K}_m (Q_n\varphi_m) - \mathcal{K}_m (\varphi_m) - \mathcal {K}_m'  (\varphi_m) (Q_n \varphi_m - \varphi_m ) )\|_\infty \nonumber\\
&& + \| (I - Q_n) \mathcal {K}_m'  (\varphi_m) (Q_n \varphi_m - \varphi_m )\|_\infty.
\end{eqnarray}
Recall from (\ref{eq:2.7}) that
\begin{eqnarray}\nonumber
y_n = \mathcal{K}_m (Q_n\varphi_m) - \mathcal{K}_m (\varphi_m) - \mathcal {K}_m'  (\varphi_m) (Q_n \varphi_m - \varphi_m )  =
R  ( Q_n \varphi_m - \varphi_m ),
\end{eqnarray}
Now proceeding as in the proof of Theorem 4.2,  we obtain
\begin{align*}
\|y_n \|_\infty = \|R  ( Q_n \varphi_m - \varphi_m )\|_\infty = O\left( \max\left\{ \tilde{h}^2 , h^3 \right\} \right).
\end{align*}
Note that
\begin{align}\label{eq:4.12}
\|\mathcal{K}_m'(\varphi_m) (I - Q_n )y_n\|_\infty \leq C_7 \|y_n\|_\infty h = O\left(h \max\left\{ \tilde{h}^2, h^3 \right\} \right).
\end{align}
Using (\ref{eq:4.4}) it can be seen that
\begin{eqnarray*}
\left \|\mathcal {K}_m'  (\varphi_m)   (I - Q_n) \mathcal {K}_m'  (\varphi_m)   (I - Q_n) \varphi_m \right \|_{\infty} = O \left (h^4 \right ).
\end{eqnarray*}
Thus, from  \eqref{eq:4.11}, \eqref{eq:4.12} and the above estimate, we obtain
\begin{eqnarray}\label{eq:4.18}
 \|\mathcal{K}_m'   (\varphi_m)  (  \mathcal {K}_m (\varphi_m) -  \tilde{\mathcal{K}}_n^M (\varphi_m)  )\|_\infty = O\left(h \max\left\{ \tilde{h}^2, h^3 \right\} \right).
\end{eqnarray}
We recall the following result from Proposition 3.5:
\begin{eqnarray}\nonumber
\left \|\tilde{\mathcal{K}}_n^M (z_n^M) - \tilde{\mathcal{K}}_n^M (\varphi_m)  - \left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m) (z_n^M - \varphi_m) \right \|_\infty &\leq & C_8 \left \|z_n^M - \varphi_m \right \|_\infty^2 .\\\label{eq:4.19}
& = & O (\max \{\tilde{h}^{2}, h^{ 3} \}^2).
\end{eqnarray}
Note that
\begin{eqnarray}\nonumber
\left \| \mathcal{K}_m'   (\varphi_m)
\left (\left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m)  -  \mathcal{K}_m'   (\varphi_m) \right  ) \right \| = O (h).
\end{eqnarray}
Hence
\begin{equation}\label{eq:4.20}
\left \|\mathcal{K}_m'   (\varphi_m) \left (\left (  \tilde{\mathcal{K}}_n^M \right )' (\varphi_m)  -  \mathcal{K}_m'   (\varphi_m) \right  ) ( z_n^M - \varphi_m ) \right \|_\infty
 = O (h \max \{\tilde{h}^2, h^3 \}),
\end{equation}
We thus obtain the following estimate using (\ref{eq:3.14}), (\ref{eq:4.18}), (\ref{eq:4.19}) and (\ref{eq:4.20}):
\begin{eqnarray*}
\|\mathcal {K}_m' (\varphi_m) (z_n^M - \varphi_m) \|_\infty = O (h \max \{\tilde{h}^2, h^3 \}).
\end{eqnarray*}
From (\ref{eq:4.10}) it follows that
\begin{eqnarray*}
\|\tilde{z}_n^M - \varphi_m \|_\infty = O (h \max \{\tilde{h}^2, h^3 \}).
\end{eqnarray*}
Since
$ \|\varphi - \varphi_m \|_\infty = O (\tilde{h}^2),$
the required result  follows.
\end{proof}
\begin{remark}
It can be  shown that
\begin{equation}\label{**}
 \|z_n^G - \varphi \|_\infty = O \left (h \right ),
 \;\;\; \|z_n^S - \varphi \|_\infty = O \left ({h}^{2} \right ).
\end{equation}
\end{remark}

\setcounter{equation}{0}
\section{Numerical Results}

For the sake of illustration, we quote some  numerical results from Grammont et al \cite{Gram3}  for the following example considered
 in Atkinson-Potra \cite{AtkP1}.

Consider
\begin{equation}\label{eq:5.1}
 x  (s) - \int_0^1 \kappa (s, t) \left [ f (t, x (t) \right ] d t  = \int_0^1 \kappa (s, t) z (t) d t, \;\;\; 0 \leq s \leq 1,
 \end{equation}
where
$$
\kappa (s,t) = \left\{ {\begin{array}{ll}
(1 - s) t, &  0 \leq t \leq s \leq 1, \\
s ( 1 - t), &  0 \leq s \leq t \leq 1,
\end{array}}\right. \;\;\; \mbox{and} \;\;\; f (t, u) = \frac {1} {1 + t + u}
$$
with $z (t)$ so chosen that
$$ \varphi (t) = \frac { t (1 - t)} { t + 1}$$
is the solution of (\ref{eq:5.1}).
In this example,
 $r$ can be chosen as large as we want.

\subsection{ Piecewise Constant functions ($ r = 0$)}

Let $\mathcal{X}_n$ be the space of piecewise constant functions
with respect to the partition (\ref{eq:2.12}) and
$ Q_n: L^\infty [0, 1] \rightarrow \mathcal{X}_n$ be the
discrete orthogonal projection defined by (\ref{eq:4.4})-(\ref{eq:4.6}).
The numerical quadrature is chosen to be the  composite Gauss $2$ rule with respect to partition (\ref{eq:2.1}) with
$m = n^2$ subintervals. Then $\tilde{h} = h^2.$

In the following table, $\delta_G, \; \delta_S, \; \delta_M $ and $\delta_{IM}$ denote the computed orders of convergence in the
discrete Galerkin, discrete iterated Galerkin, discrete Modified Projection and the discrete iterated Modified Projection methods, respectively.  It can be seen that the computed values of order of convergence
match well with the theoretically predicted values in (\ref{eq:4.6}), (\ref{eq:4.9}) and (\ref{**}).

\begin{center}
Table 5.1

\bigskip

\begin{tabular} {|c|cc|cc|cc|cc|}\hline
$n$ & $\| \varphi  - z_n^G \|_\infty$ & $\delta_G$ & $\| \varphi - z_n^S \|$ & $\delta_S$
& $\| \varphi - z_n^M \|_\infty$ & $\delta_M$  & $\| \varphi - \tilde{z}_n^M \|_\infty$ & $\delta_{IM}$\\
\hline
2&  $  1.22 \times 10^{-1} $ &              &  $ 8.40 \times 10^{-3} $ &                &   $  4.34 \times 10^{-3} $  &            &    $  5.23 \times 10^{-3} $ & \\
4&  $  8.65 \times 10^{-2} $ & $ 0.49 $ & $  2.35 \times 10^{-3} $ & $ 1.84 $  &   $ 4.31 \times 10^{-4}   $ & $ 3.33 $  &    $  3.14 \times 10^{-4} $ &  $ 4.06$\\
8&  $  5.09 \times 10^{-2} $ & $ 0.77 $ & $ 6.22 \times 10^{- 4}  $ &  $ 1.92 $ &   $ 5.28 \times 10^{- 5 }  $ &   $ 3.03 $  &    $  1.89 \times 10^{-5} $ & $4.05$\\
16& $ 2.70 \times 10^{-2} $ & $ 0.91 $ & $ 1.59 \times 10^{-4}  $ & $ 1.96 $  &   $ 6.92 \times 10^{- 6}  $ & $ 2.93 $  &    $  1.36 \times 10^{-6} $ & $3.80$\\
32& $ 1.33 \times 10^{-2}  $ & $ 1.02 $ & $ 4.02 \times 10^{-5}  $ &  $ 1.98 $ &   $ 8.38 \times 10^{-  7}   $ & $ 3.05 $   &    $  4.55 \times 10^{-8} $ &  $4.90$\\\hline
\end{tabular}
\end{center}
 \subsection{Piecewise Linear Functions ($ r = 1$)}

Let $\mathcal{X}_n$ be the space of piecewise linear polynomials
with respect to the partition (\ref{eq:2.12}) and
$ Q_n: L^\infty [0, 1] \rightarrow \mathcal{X}_n$ be the
discrete orthogonal projection defined by (\ref{eq:2.15}).
The numerical quadrature is chosen to be the composite Gauss 2 point rule with $ n^2$ intervals for the Galerkin and the iterated Galerkin
method and the composite Gauss 2 point rule with $ n^3$ intervals for the modified projection and the iterated modified projection methods. In the latter case $\tilde{h}^2 = h^6.$ As a consequence,
it follows from   (\ref{eq:3.11}),  (\ref{*})  and (\ref{eq:3.19}) that the expected orders of convergence
in the discrete Galerkin, the discrete iterated Galerkin, the discrete modified projection and the discrete iterated modified projection methods are
 $2, 4, 4$ and $6,$ respectively. The computational results given below match well with these orders.

\begin{center}
Table 5.2

\bigskip

\begin{tabular} {|c|cc|cc|cc|cc|}\hline
$n$ & $\| \varphi  - \varphi_n^G \|_\infty$ & $\delta_G$ & $\| \varphi - \varphi_n^S \|$ & $\delta_S$
& $\| \varphi - \varphi_n^M \|_\infty$ & $\delta_M$  & $\| \varphi - \tilde{\varphi}_n^M \|_\infty$ & $\delta_{IM}$\\
\hline
2&  $  1.32 \times 10^{-1} $ &              &  $ 4.97 \times 10^{-3} $ &                &   $  1.54 \times 10^{-3} $  &            &    $  1.34 \times 10^{-3} $ & \\
4&  $  4.98 \times 10^{-2} $ & $ 1.41 $ & $  4.46 \times 10^{-4} $ & $ 3.48 $  &   $ 1.12 \times 10^{-4}   $ & $ 3.78 $  &    $  1.89 \times 10^{-5} $ &  $ 6.15 $\\
8&  $  1.58 \times 10^{-2} $ & $ 1.66 $ & $ 3.89 \times 10^{- 5}  $ &  $ 3.52 $ &   $ 1.06 \times 10^{- 5 }  $ &   $ 3.40 $  &    $  2.48\times 10^{-7} $ & $6.25$\\
16& $ 4.51 \times 10^{-3} $ & $ 1.81 $ & $ 3.15 \times 10^{- 6}  $ & $ 3.62 $  &   $ 9.10 \times 10^{- 7 } $ &  $ 3.54$&  $  2.92\times 10^{-9} $   &  $6.41$\\
\hline
\end{tabular}
\end{center}

redpajama_set_name: RedPajamaArXiv


text:
Editorial: Condolences To NBA Hall Of Famer Moses Malone From Vann Digital Networks
Sad news for the basketball community.
Pro Basketball Hall of Famer and three-time NBA MVP Moses Malone died Sunday, according to BET News. He was 60. The cause of his death is currently unknown.
Malone, a 12-time NBA All-Star, played for eight teams in over 20 seasons in the league. The 6-10 center led the Philadelphia 76ers to the league title in 1983 and was named Finals' MVP. After averaging 20.6 points and 12.3 rebounds per game over his career, Malone was named one of the NBA's 50 Greatest Players. He finished his career with 27,409 points and 16,212 rebounds.
In addition to two different stints with the 76ers, Malone also played for the Houston Rockets, Washington Bullets, Atlanta Hawks and the San Antonio Spurs, who he finished his career with during the 1994-95 season. He won his league MVPs in 1979, 1982 and 1983, having been inducted into the Hall of Fame in 2001.
My condolences to the family and friends of Moses Malone. You will truly be missed. Rest in peace Big Mo!!!
— Allen Iverson (@alleniverson) September 13, 2015
Previous PostVideo: @Karceno Speaks On Mayweather vs. Berto (Post Fight) [#MayweatherBerto]
Next PostVideo: @SuliBreaks Writes Open Letter Titled 'Dear Social Media'


text:
Q: When use EF stored procedure to handle with breezejs, how to return error from SP to client side? I call a raiserror in the SP, expect it will be returned to client side in some way, just like when I add raiserror to a trigger on a table, but now I don't see the message returned
"Message":"An error has occurred.","ExceptionMessage":"Store update, insert, or delete statement affected an unexpected number of rows (0). Entities may have been modified or deleted since entities were loaded. Refresh ObjectStateManager entries."
Any idea?

A: Try wrapping the exception in either a breeze EntityErrorsException or a standard .NET HttpResponseException. The EntityErrorsException will allow you to pass additional information about exactly which entities are causing the error(s), if you have this information available on the server.
The discussion here: http://www.breezejs.com/documentation/server-side-validation discusses the EntityErrorsException in more detail.

redpajama_set_name: RedPajamaStackExchange


Embarcadero developers Jim McKeeth and Malcolm Groves are literally on fire lately with their tutorials for building Delphi XE7 Firemonkey apps on diverse devices. Jim McKeeth has a extensive tutorial for using the new Delphi XE7 multi device designer to build an Android app for the Moto 360 watch from Motorola. In his tutorial he demonstrates how to create a custom View for the shape of the watch. Basically in XE7 you have a master View where your UI components are setup and then you have the individual device views where you can make changes for the specific device size that you are working with (like the Moto 360 watch). He also goes into detail on how to connect and deploy your app to the watch device. Malcolm Groves has a tutorial on building Delphi XE7 Firemonkey apps for the Beaglebone Black mini computer which is a small single board computer with an ARM Cortex A8 processor. The device also has a microHDMI port for connecting a monitor. Apparently the mini computer ships with Linux installed on it but Malcolm explains how to install Android on the device using a MicroSD card. Once you have the Beaglebone mini computer running Android he then shows how to get it connected to your Windows machine via USB. With the USB link setup the device should show up in the Delphi or AppMethod project target platform. One click later and the Android app is deployed from Delphi XE7 onto the Beaglebone Black. Both of these devices support the NEON instruction set which is required for the Firemonkey Android NDK support. The XE7 multi device designer really starts to shine once you start deploying apps to more than just one screen size.
Head over and check out the full tutorial from Jim on deploying Firemonkey apps to the Moto 360 Android watch.
Continue on over to Malcolm's blog and read the full write up on developing Delphi XE7 Firemonkey apps for the Beaglebone Black mini computer.


text:
Jakarta, for Eid celebration, no travel, prayers at home (but there are exceptions)
by Mathias Hariyadi
To curb the spread of the pandemic, travel is prohibited, but there are millions who have escaped control to visit relatives. No religious parade at the end of the fast. Everyone is urged to pray at home. But in Tegal, Kudus, Ciamis, Sidrap will be prayed in public.
Jakarta (AsiaNews) - To avoid the risks of a spread of the pandemic from Covid-19, the political and religious authorities of Indonesia have been busy for days advising, urging, obliging faithful to observe the Eid al-Fitr festival staying at home, without travel or by going to massive prayer gatherings in the mosque.
Eid Festival ends Ramadan. In Indonesia it falls tomorrow, May 24th. The celebration begins tonight with the family gathering that dissolves the last fast. The family gathering lasts at least two days and is called Silahturahmi. The authorities demand that Silahturahmi be celebrated only among close family members and in social distancing this year.
Traditionally, in the country with the largest number of Muslims in the world, millions of people leave the cities days before Eid to return to their villages and gather in the family. It is a real exodus which in the local language is called mudik.
This year, due to the pandemic, mudik is prohibited and the communication routes are controlled by the police. Despite this, there are many people - perhaps millions - who have managed to escape the controls, traveling at night or by secondary routes.
Tradition also has it that on the eve of the evening, after dinner, by motorbike or car, a religious parade will take place on the streets, which lasts until midnight. But even this is prohibited this year.
A large prayer at the mosque would also be planned for the following day. But for days, even with video messages, Prof. KH Nazaruddin Umar, head of the Istiqlal Grand Mosque, explains that this year "it is better to pray at home".
"Even on a rainy day in Medina - he explains - the Prophet advised his followers not to carry out prayer in the mosque, but in their homes. Today we are faced with a 'torrential rain of coronavirus' and therefore his appeal rings out louder: we must pray at home. "
In some cities, local authorities have set more stringent rules. Public prayers are likely to be held in the Grand Mosque of Tegal and in Kudus (Central Java), Ciamis (West Java) and Sidrap (South Sulawesi). For the occasion, masks and social distancing are mandatory.
COVID-19 hits Eid al-Fitr, the festival marking the end of Ramadan
On Eid Muslims prayed for the end of the COVID-19 pandemic
Eid al-Fitr: more solidarity, fewer trips for Muslims in Asia
As Indonesia stops for Eid al-Fitr, Catholics extend best wishes and show friendship
As Indonesian Muslims celebrate the end of Ramadan, the fear of attacks persists


This classic form of full body massage enhances circulation, improves muscle tone, relieves tension and soothes tired muscles. Swedish is a lighter form of massage; should you want targeted or deeper work, we suggest you schedule a Combo or Deep Tissue Massage. Aromatherapy Available. Please tell us which scent you would like upon scheduling your appointment.
The Graston Technique is a form of manual therapy known as soft-tissue instrument-assisted mobilization. It is one of a number of manual therapy approaches that use instruments with a specialized form of massage/scraping the skin gently.
The Graston Technique is a form of manual therapy known as soft-tissue instrument-assisted mobilization. It is one of a number of manual therapy approaches that use instruments with a specialized form of massage/scraping the skin gently. The therapy is designed to help the practitioner identify areas of restriction and attempt to break up scar tissue.
This massage incorporates Swedish massage with one or two targeted areas of deep pressure techniques.
From your Doctor to Us! Many Doctors will talk to you about needing to manage your pain. One of the ways your Doctor might suggest you do this is with Muscle Massage Therapy. This massage uses a variety of different modalities with assistance of Hydrotherapy depending on what your Doctor has suggested. Hydrotherapy aids to accelerate the body's healing process by manipulating the blood flow. The goal is to increase range of motion, open areas of tightness, reduce soreness and invigorate the body – wakes you up and stimulates healing.


Extremely delighted to be part of this wonderful Blog Tour today to share my thoughts on this stunning read! My thanks to the author, publisher and Anne of Random Things for letting me be part of it all!
I found this to be a enthralling and dark thriller! My mind was seriously messed with throughout!!
Lynn and Ed are married and 10 years down the line, Ed gets diagnosed with early onset dementia, and sinister things begin to happen - are the two things linked?! So all the dreams that Lynn had for their life together are dashed and she goes back to work to help ease the pressure at home.
What then follows is then a well written observation of dealing with a loved one and the effects that dementia has on that person, as well as those around them. There is a lot of anger and frustration on both sides as the illness takes hold and this really adds a realistic edge to the story, especially if you have personal dealings of a loved one and this illness.
The story also delves into the past of how they met, Ed and his life before she came along and the darker elements of life that begin to creep back in.
There was so much throughout this book that kept me guessing, and the twists were spectacular to say the least! They had me fooled!!
A top notch domestic noir tale that will leave you chilled to the core!!


Matthew West has some great songs that I have enjoyed for years, but this album is fresh and new and I love it! All In, Amen, The Sound of a Life Changing (SO GOOD), 1 Song (fun upbeat), are my favorites but really all good! If you need some fresh music to add to your collection, I HIGHLY recommend this album! I love hearing new Christian music that sounds different and that I want to listen to every day.