text:
Q: Nodejs upload file using node-oauth I have a problem with uploading a file using http module
It seemed that the file becomes corrupted when received on the server.
    data = fs.readFileSync('test.jpg')

    const crlf = '\r\n'
    const boundaryKey = Math.random().toString(16)
    const boundary = `--${boundaryKey}`
    const delimeter = `${crlf}--${boundary}`
    const closeDelimeter = `${delimeter}--`
    let multipartBody = ""

    let headers = [
      `Content-Type: image/jpeg; name=filename.jpg${crlf}`,
      `Content-Transfer-Encoding: binary${crlf}`,
      `Content-Disposition: form-data; name="attachment"; filename="filename.jpg"${crlf}`,
    ]

    multipartBody = Buffer.concat([
      new Buffer.from(delimeter + crlf + headers.join('') + crlf),
      data,
      new Buffer.from(closeDelimeter + crlf),
    ])

    oauth._headers['Content-Length'] = multipartBody.length
    const promise = new Promise((resolve, reject) => {
      oauth._performSecureRequest(
        oauth_token,
        oauth_token_secret,
        'POST',
        `http://localhost:4000/`,
        null,
        multipartBody.toString(),
        `multipart/form-data; boundary="${boundary}"`,
        function (error, data) {
          resolve(data)
        }
      )
    })
    return await promise

My receiver code is here using express. But this is just a simulation. The actual server to push here is an API server that says: Error while validating input for the resource. [MESSAGE: image is not valid and may be corrupted.]
const express = require('express')
const app = express()
const fileUpload = require('express-fileupload')

app.use(fileUpload())

app.post('/', (req, res) => {
  const attachment = req.files.attachment
  attachment.mv(`${__dirname}/uploads/${attachment.name}`)
  res.json({
    success: true
  })
})

app.listen(4000, () => {
  console.log('listening to port 4000')
})

On console.log the original file has this:
<Buffer ff d8 ff e0 00 10 4a 46 49

But the received file has:
<Buffer ef bf bd ef bf bd ef bf bd

The received file is successfully saved to disk but cannot be opened... as it has an encoding problem.


Both players were caught looking at Johnson's phone in the middle of the game, which goes against team and league rules.
But the Hall of Famer had a front row seat to see Ben Simmons mimic the signature celebratory gesture amidst an explosive, game-changing run. Starting point guard Kyle Lowry had a awful game where he scored ZERO points in 34 minutes of action.
Russell got the first win of his playoff career and the last laugh, scoring 14 points in the third quarter and leading the Nets to one of the biggest wins since they moved to Brooklyn. Tonight they'll play again in Game 2 - can they do the unthinkable and repeat their performance to take a 2-0 series lead heading back to Brooklyn?
Embiid said Johnson received an update on a sick daughter. He finished with 18 points, 12 rebounds and six assists. The eighth-year veteran went to line 15 times, which was only second behind Embiid (18). They shot poorly, particularly from three, which was a problem all down the stretch of the regular season; but they didn't play poorly. Spencer Dinwiddie led the Nets with 19 points and D'Angelo Russell had 16. Jimmy Butler scored 36 points and saved the Sixers, widely considered a favorite to at least reach the East semis, from losing by 25.
Having a reliable bench is great, especially when the 76ers' depth is widely considered to be their biggest weakness.
The children, ages 8 and 3, were dead at the scene, while both parents, who were in the front seat, escaped injury, he said. In Monroe County, Mississippi, a 95-year-old man identified as Roy Ratliff, died after a tree fell on his mobile home.
"We are the recipient of Jo trying to inch along slowly and buy time", Brown said (via The Philly Voice).
Sixers All-Star centre Joel Embiid finished Game 2 with a game-high 23 points to go along with 10 rebounds.
The Sixers faithful win a free Frosty if an opposing player misses two free throws in the fourth quarter, and on multiple occasions a Nets player missed the front end before sinking the second.
The Philly home crowd rained down boos in Game 1 after a lacklustre performance and today, with Allen Iverson courtside, Simmons got the Philly crowd going, cupping his ear A.I. style to a chorus of cheers. Even though he clearly wasn't at his best, he totaled 22 points, 15 rebounds and five blocks.
The Nets definitely showed they deserve their spot in the playoffs. "We are moving forward together with our full focus now on Game 2".


text:
MB1800 is a New York City institution, and is the city's longest running series devoted exclusively to early music. Since the series' inception, concerts have been presented in the outstanding acoustic of Corpus Christi Church in Morningside Heights, and gaining national attention in early music communities.
All 5 /2022-2023 Season 5
Live: Sunday, Jan 15, 2023, 4 p.m.
Online: Sunday, Jan 22, 2023, 4 p.m.
Live: Sunday, Feb 5, 2023, 4 p.m.
Online: Sunday, Feb 12, 2023, 4 p.m.
Live: Sunday, Feb 19, 2023, 4 p.m.
Cappella Pratensis
Live: Sunday, Apr 16, 2023, 4 p.m.
Online: Sunday, Apr 23, 2023, 4 p.m.
Tiburtina Ensemble
Live: Sunday, May 7, 2023, 4 p.m.
Online: Sunday, May 14, 2023, 4 p.m.
All 3 /Past 3
Boston Camerata
Live: Sunday, Oct 2, 2022, 4 p.m.
Online: Sunday, Oct 9, 2022, 4 p.m.
Vox Luminis
Live: Sunday, Oct 23, 2022, 4 p.m.
Online: Sunday, Oct 30, 2022, 4 p.m.
Juilliard415
Live: Sunday, Dec 11, 2022, 4 p.m.
Online: Sunday, Dec 18, 2022, 4 p.m.
View Our Concert History

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: c# ReaderWriterLockSlim corrupt after interrupt thread that invokes ExitReadLock I encountered a scenario where ReaderWriterLockSlim seems to get broken after a series of legal actions.
The flow is:

*

*Thread 1 takes writer lock1

*Thread 2 tries to take reader lock1 - blocks

*Thread 2 is interrupted, and calls lock1.ExitReadLock
Thread 2 didn't get the lock. It seems that an exception should have been thrown.

*Thread 1 exits writer lock of lock1

*Any thread that tries to take lock1.EnterReadLock will block forever

After stage 3 above, a debugger shows that lock1.CurrentReadCount is corrupt - seems to have overflowed down to 0x7FFFFFF.
I wonder if anyone had encountered this, or maybe I'm missing something.
The code that reproduces it:
[TestMethod]
public void ReproTest()
{
    var rwlock = new ReaderWriterLockSlim();
    rwlock.EnterWriteLock();

    bool taken = false;
    var reader = new Thread(() =>
    {
        try
        {
            rwlock.EnterReadLock();
            s_logger.Info("Enter");

        }
        catch (ThreadInterruptedException)
        {
            rwlock.ExitReadLock();
        }
    });
    reader.Name = "Reader";
    reader.Start();
    Thread.Sleep(1000);

    reader.Interrupt();

    Thread.Sleep(1000);

    rwlock.ExitWriteLock();

    while (!taken)
    {
        taken = rwlock.TryEnterReadLock(1000);
    }

    Thread.Sleep(1000);
}

A: The reader is never entering the read lock.  It sits waiting for the write to be released.  When it's interrupted, you then try to exit even though you never entered, causing read count to go below 0 I suppose :)

A: This looks like a bug in the framework (tested on v3.5 and v4.0). The ExitReadLock() should throw a SynchronizationLockException, but doesn't in this case. Indeed, you can trigger a very similar issue much more simply with the following:
rwlock.EnterReadLock();
rwlock.ExitReadLock();
// This should throw a SynchronizationLockException but doesn't
rwlock.ExitReadLock();
// At this point, rwlock.CurrentReaderCount = 0x0fffffff

(In fact, ExitReadLock() will corrupt the lock if it's called without a matching EnterReadLock() on any thread that has previously entered the lock.)
The issue only occurs when the ReaderWriterLockSlim is created using the parameterless constructor, or with LockRecursionPolicy.NoRecursion. If created with LockRecursionPolicy.SupportsRecursion, it will not be corrupted by the unmatched ExitReadLock().
If you expect the reader thread to be interrupted whilst waiting for entry into lock, I would suggest changing the reader thread method to:
var reader = new Thread(() =>
{
    var entered = false;
    try
    {
        rwlock.EnterReadLock();
        entered = true;
        s_logger.Info("Enter");
    }
    finally
    {
        if (entered) rwlock.ExitReadLock();
    }
});

A: Code that fixes what @Lasse and @Jeremy have pointed out:
static public void ReproTest()
{
    var rwlock = new ReaderWriterLockSlim();
    rwlock.EnterWriteLock();
    s_logger.Info("0:Enter");

    bool taken1 = false;
    var reader = new Thread(() =>
    {
        try
        {
            rwlock.EnterReadLock();
            s_logger.Info("1:Enter");
            // only set to true if taken
            taken1 = true;
        }
        catch (ThreadInterruptedException)
        {
            // only release if taken
            if (taken1)
                rwlock.ExitReadLock();
            taken1 = false;
        }
    });
    reader.Name = "Reader";
    reader.Start();
    Thread.Sleep(1000);

    reader.Interrupt();

    Thread.Sleep(1000);

    rwlock.ExitWriteLock();

    // 2nd taken variable here only so we can see state of taken1
    bool taken2 = taken1;
    while (!taken2)
    {
        taken2 = rwlock.TryEnterReadLock(1000);
        s_logger.Info("2:Enter");
    }

    Thread.Sleep(1000);
}

When run, the debug output correctly shows the write lock being taken, the 1st read lock NOT taken, and the 2nd read lock taken:
0:Enter
A first chance exception of type 'System.Threading.ThreadInterruptedException' occurred in mscorlib.dll
The thread 'Reader' (0x1358) has exited with code 0 (0x0).
2:Enter

redpajama_set_name: RedPajamaStackExchange


text:
NBC Sports' Tony Dungy targeted by NBC News over his 'anti-LGBTQ history'
By AMCSC Editor Team On Jan 26, 2023 0
NBC News is shooting inside the tent with a report targeting NBC Sports analyst Tony Dungy over what it calls his "anti-LGBTQ history."
Dungy was under fire last week for a tweet peddling a debunked claim that schools in the U.S. were beginning to put "litter boxes in the school bathrooms for students who identify as cats." He later deleted the tweet and issued an apology.
But NBC Out, the LGBTQ section of NBC News, published a scathing report on its parent company colleague, accusing him of spreading a "transphobic myth" and doing a deep dive into his past tweets to portray the NFL Hall of Famer as bigoted.
TONY DUNGY APOLOGIZES AGAIN FOR TWEET ABOUT LITTER BOX CLAIMS
Tony Dungy of NBC Sports during an NFL game between the Los Angeles Rams and the Buffalo Bills on September 08, 2022, at SoFi Stadium in Inglewood, CA.
(Icon Sportswire)
In addition to recapping the backlash Dungy received in the media, NBC News reported it "found at least a dozen tweets from Dungy's account, from 2012 to 2022, that are critical of same-sex marriage, homosexuality and the LGBTQ 'lifestyle,'" citing two tweets.
One tweet quoting Dungy from June 2022 read, "And no one is saying God will only banish homosexuals to hell. Jesus said anyone who is not born again by accepting Him as their savior will not enter the kingdom of heaven. That's the criteria for avoiding hell."
The other from 2020 read, "You and I disagree about whether LGBTQ is a lifestyle. But that has nothing to do with how I 'treat people'. There was a lot more in my comments in that story that never got written. But that happens a lot. We're seeing that today."
TONY DUNGY TO ATTEND MARCH FOR LIFE, DUBBED RIGHT-WING EXTREMIST
The report also resurfaced comments the former Indiana Colts head coach made in 2014 saying he wouldn't have openly gay player Michael Sam on his team saying, "It's not going to be totally smooth… things will happen."
NBC News took aim at NBC Sports' Tony Dungy over his "anti-LGBTQ" history.
(Fox News Digital)
The report quoted Shefik Macauley, who NBC News described as an "NBC Sports employee who is among the leaders of NBC Universal's LGBTQ employee resource group," and how he was "unsuccessful" in his effort to have NBC Sports force Dungy to issue an on-air apology, a decision the network defended to NBC News by saying the Twitter apology was enough since the offending tweet was given on that platform.
"If someone is empowered to be on camera and be the face of whatever platform that is, then we should be able to address it with that same audience that we're empowering that person to be in front of," Macauley told NBC News.
MSNBC EDITS COLUMN AFTER TURNING POINT USA SENDS CEASE-AND-DECIST LETTER BLASTING 'DEFAMATORY' COMMENT
NBC News also quoted an LGBTQ activist who denounced NBC's handling of Dungy and pointed to his slated appearance at the Men's Advance Conference hosted by evangelical pastor Andrew Wommack, who NBC News said is "known for his anti-LGBTQ views."
NBC Sports analyst Tony Dungy speaks during a segment prior to the 2022 Pro Football Hall of Fame Game between the Jacksonville Jaguars and the Las Vegas Raiders at Tom Benson Hall of Fame Stadium on August 04, 2022 in Canton, Ohio.
(Nick Cammett/Diamond Images via Getty Images)
The article was posted and retweeted by multiple NBCUniversial employees including NBC News and CNBC staffers.
NBC News investigative reporter Mike Hixenbaugh admonished Dungy while sharing the report, writing, "Read to the end to discover that Tony Dungy is slated to speak at a Christian nationalist convention led by a leading proponent of the Seven Mountains Mandate, which argues that evangelical Christians are called on to rule all aspects of U.S. society."
While many on the left have attacked Dungy, the NFL icon has received glowing praise on the right for his recent appearance at the March for Life rally in Washington D.C.
Joseph A. Wulfsohn is a media reporter for Fox News Digital. Story tips can be sent to joseph.wulfsohn@fox.com and on Twitter: @JosephWulfsohn.
antiLGBTQDungyHistoryNBCNewssportstargetedTony
Jane Fonda links the 'climate crisis' to racism: 'Everything's connected'
Arsenal tighten grip on Premier League race in 5-goal stunner against Manchester United
LeBron James goes berserk after missed foul call, Patrick Beverley gets technical…
Baltimore shooting leaves 1 dead, 5 injured, including 2 children: 'Fighting…


text:
Request a callback from the Superior Healthcare team
Our Carers
Day / Night Care
Conditions We Support
Free Nurse CPD Courses
Working for Superior Healthcare
Being a Nurse at Superior Healthcare
Being a Learning Disability Support Worker at Superior Healthcare
Being a Complex Care Support Worker at Superior Healthcare
Being a Healthcare Assistant at Superior Healthcare
Working at Superior Healthcare's Head Office
Registered Nurse Roles
Complex Care Support Worker Roles
Healthcare Assistant Roles
Learning Disability Support Worker Roles
Head Office Roles
What Our Team Says
Supporting people and the families of people who have Parkinson's Disease to live safely at home.
Superior Healthcare provides tailored, nurse managed packages of care and support for people who have Parkinson's Disease and complex underlying health needs, to live independently at home.
Parkinson's Disease is a condition in which parts of the brain become progressively damaged over many years. It's caused by a loss of nerve cells in part of the brain called the substantia nigra. This leads to a reduction in a chemical called dopamine in the brain.
The 3 main symptoms of Parkinson's Disease are:
Involuntary shaking of particular parts of the body
Slow movement
Stiff and inflexible muscles
A person with Parkinson's disease can also experience a wide range of other physical and psychological symptoms.
Superior Healthcare and our team of experienced Lead Nurses support people who have Parkinson's Disease to maintain their independence at home with the support of a team of specialist carers, support workers and nurses, who have completed specialist training.
Specialist needs
Superior Healthcare recognises that clients who have Parkinson's Disease may have other complex health needs and we support people who:
Have a tracheostomy and/or ventilator (BiPAP/CPAP) and require support to maintain a clear airway.
Use a suction or cough assist device.
Need support to receive nutrition and hydration via gastrostomy tube.
Require assistance with personal care including continence care.
Need assistance with medication administration.
Parkinson's Disease Facts
Parkinson's Disease Facts
The number of people diagnosed with Parkinson's in the UK is about 145,000. But we know that more than 1 million people in the UK are affected, either by living with Parkinson's, or as a friend, colleague, or family member of someone who is.
Parkinson's develops when cells in the brain stop working properly and are lost over time. These brain cells produce a chemical called dopamine.
Symptoms start to appear when the brain can't make enough dopamine to control movement properly.
There are 3 main symptoms – tremor (shaking), slowness of movement and rigidity (muscle stiffness) – but there are many other symptoms too.
(source: Parkinsons UK)
Superior Healthcare works with people self-funding their care as well as using a Personal Health Budget. We are trusted by NHS clinical commissioning groups, local authorities, case managers and solicitors including people who have been appointed a Court of Protection Deputy.
Superior Healthcare is regulated and rated 'Good' by the Care Quality Commission.
We work closely with specialist hospitals including the Evelina London Children's Hospital, Guy's and St Thomas' and the Royal Brompton Hospital.
We employ a team of trained carers, support workers and registered nurses who are managed by our Lead Nurse Team who include Registered Nurses and Children's Nurses.
Superior Healthcare has up to date safeguarding and whistleblowing policies and deliver training to all members of our team.
Parkinson's UK: https://www.parkinsons.org.uk/
NHS: https://www.nhs.uk/conditions/parkinsons-disease/
Carers UK: https://www.carersuk.org/
A care package to suit you
Superior Healthcare provides visits which range from six hours through to live-in care and twenty four hour nursing care.
Day or night care
Twenty-four hour care
To discuss your needs on an informal basis with our complex care team call 01227 771122 or fill in the form below and a member of the Superior Healthcare team will call you back
Our Story	Jobs	COVID-19
© 2021 Superior Healthcare


Odyssee II us a 42 foot ketch, meaning it is two masted monohull. It has two double beds, one triple and is equipped with all the necessary safety equipment. It has a spacious bathromm and a great deck for lounging. With its 42 feet, the Odyssee has more than enough space for every one of the 7 people and two crew- There are snorkeling fins and masks for every one, fishing gears as well. Alice is running the show and they do a great trip.


So many flyers and so little time. Say you've ordered 1000 Flyers and you've cleared your day to pass every single one out. Here's the most productive and efficient plan… ready? Go!
You might be uneasy about doing this one, but not to worry. As long as you're not trespassing on anybody's property or digging in their mailboxes, you're good to go! Just look for any open containers on the mailbox stand that are NOT mailboxes. Some houses even have a container specifically for holding flyers and other advertisement. Door hangers are also a great idea!
– All mailboxes are off limits, no matter what. Don't even touch them.
– Watch out for "No Trespassing" and "No Soliciting" signs.
This is always a tricky one, but man does it work! Just go to a busy parking lot and put a flyer on each car. Cars are so close together, you can usually bust out 100+ flyers in about 20 minutes.
– A lot of parking lots these days are private property, so the owner of wherever you're stuffing flyers does have the right to kick you out.
– The cars you're putting flyers on are private property of the car's owner also, so you might get yelled at once or twice.
"50 flyers per board?! What!?" Yeah, just listen. Stop at the closest office supply store and pick up about a 5-pack of cheap plastic literature holders (preferably with a nail hole on the top of each one). Tack a holder filled with about 50 flyers to about 5 of the busiest places in town. That's 250 flyers while doing barely any work!
– Make sure you have a pretty intense tack to hold up 50 flyers. They're out there, trust me.
Make good friends and network around town with brick and mortar businesses. Cut a deal where you leave some flyers there and he leaves flyers at your place… pay the business owner to "rent" out a spot on his front desk to leave your flyers, use your imagination. Go to about 10 or more business and do stuff like this.
You'll almost always have some leftover flyers, so here's a couple neat little places to drop a flyer or two. Note that these will probably all be thrown away, but you were going to do that anyways.
Your table after eating at a restaurant.
A couple shopping carts on the way out of the grocery store.
In the garbage, but strategically placed for max visibility.
Lunch tables in a school cafeteria.
So there's your full plan to get rid of 1000 flyers. Happy Advertising!
You should check with your local and state laws to verify what you can and can't do when passing out flyers. This article is strictly a resource of helpful tips and ideas, none of that legal stuff.


text:
Russian Offensive Campaign Assessment, March 17
Mason Clark, George Barros, and Kateryna Stepanenko
March 17, 5:30pm ET
Russian forces did not make any major advances and Ukrainian forces carried out several local counterattacks on March 17.[1] Russian forces made little territorial progress and continued to deploy reserve elements—including from the 1st Guards Tank Army and 810th Naval Infantry Brigade—in small force packets that are unlikely to prove decisive. Russian forces continue to suffer heavy casualties around Kharkiv, and Russian attempts to bypass the city of Izyum are unlikely to succeed. Russian forces continued assaults on Mariupol on March 17 but did not conduct any other successful advances from Crimea.
Russian forces continue to make steady territorial gains around Mariupol and are increasingly targeting residential areas of the city.
Ukrainian forces northwest of Kyiv launched several local counterattacks and inflicted heavy damage on Russian forces.
Ukrainian forces repelled Russian operations around Kharkiv and reported killing a regimental commander.
Ukrainian intelligence reports that Russia may have expended nearly its entire store of precision cruise missiles in the first twenty days of its invasion.
Russian forces deployed unspecified reserve elements of the 1st Guards Tank Army and Baltic Fleet Naval Infantry to northeastern Ukraine on March 17.
Russia may be parceling out elements of the reserve force that could conduct an amphibious operation along the Black Sea coast to support ongoing assaults on Mariupol, further reducing the likelihood of a Russian amphibious assault on Odesa.
Ukrainian forces shot down 10 Russian aircraft—including five jets, three helicopters, and two UAVs—on March 16, and Ukrainian forces continue to successfully contest Russian air operations.
Russian forces face mounting difficulties replacing combat casualties and replacing expended munitions. The Ukrainian General Staff stated on March 17 that Russian forces will begin another wave of mobilization for the Donetsk People's Republic's (DNR) 1st Army Corps on March 20.[2] Ukrainian intelligence continued to report Russian forces face difficulties manning both combat and support units and increasing desertion rates.[3] The General Staff further reported that Russian forces are increasingly using indiscriminate weapons against residential areas because they used almost their entire supply of "Kalibr" and "Iskander" cruise missiles in the first 20 days of the invasion.[4] It is unclear if the Ukrainian General Staff means Russian forces have used almost all precision munitions earmarked for the operation in Ukraine or almost all missiles in Russia's total arsenal—though likely the former.
The Ukrainian Military Intelligence Directorate (GUR) reported several details on Russian efforts to recruit Syrian mercenaries on March 17.[5] The GUR reported that the Russian military ordered its base in Hmeimim, Syria to send up to 300 fighters from Syria to Ukraine daily. The GUR additionally reported that Syrian President Bashar al-Assad has promised to recruit 40,000 Syrian fighters to deploy to Ukraine. The GUR reported Russian authorities are promising Syrian recruits that they will exclusively act as police in occupied territories. Finally, the GUR reported low morale among Syrian recruits, including several cases of self-mutilation to avoid being deployed, and claimed many fighters see deploying to Russia and Belarus as an opportunity to desert and migrate to the EU.
Russian forces are engaged in four primary efforts at this time:
Main effort—Kyiv (comprised of three subordinate supporting efforts);
Supporting effort 1—Kharkiv;
Supporting effort 1a—Luhansk Oblast;
Supporting effort 2—Mariupol and Donetsk Oblast; and
Supporting effort 3—Kherson and advances westward.
Main effort—Kyiv axis: Russian operations on the Kyiv axis are aimed at encircling the city from the northwest, west, and east.
Subordinate main effort along the west bank of the Dnipro
The Ukrainian General Staff reported that Russian forces unsuccessfully attempted to regain unspecified territory northwest of Kyiv on March 17 that it claimed Russian forces previously lost to Ukrainian counterattacks.[6] ISW cannot confirm the exact location of these claimed Ukrainian counterattacks. The Ukrainian General Staff additionally stated that Russian forces focused their main efforts on reconnaissance and organizing counter-battery fire on March 17—the first Ukrainian mention of counter-battery fire as an explicit Russian priority.[7] Ukrainian forces inflicted heavy casualties on Russian forces northwest of Kyiv on March 17, forcing the 36th Combined Arms Army (CAA) to "mobilize reserves prematurely" and conducted an artillery strike on a command post of the 35th CAA roughly 35 kilometers from the Ukrainian-Belarusian border.[8]
Subordinate supporting effort—Chernihiv and Sumy axis
Russian forces did not conduct offensive operations toward northeastern Kyiv on March 17. The Ukrainian General Staff reported that Russian forces continued preparations to resume offensive operations toward Kyiv and conducted "chaotic shelling" of Ukrainian positions.[9] Social media users filmed Russian armored vehicles in Okhtyrka city on March 16, approximately 65km south of Sumy, though there is no evidence Russian forces have fully captured the city as of publication.[10] Russian forces continue to face difficulties in muddy conditions. Social media users observed four Russian T-80U tanks of the 4th Tank Division stuck in mud in an unknown location in Sumy Oblast, Ukraine, on March 17.[11] The Ukrainian General Staff reported Russian forces deployed unspecified elements of the 1st Guards Tank Army and Baltic Fleet Naval Infantry to northeastern Ukraine on March 17.[12]
Supporting Effort #1—Kharkiv:
The Ukrainian General Staff reported at 6:00 pm local time on March 17 that Ukrainian forces inflicted heavy losses on the Russian 252nd Motor Rifle Regiment (of the 3rd Motor Rifle Division) in fighting near Kharkiv and Izyum, destroying 30 percent of the regiment's personnel and equipment.[13] The General Staff additionally reported that Ukrainian forces killed Colonel Igor Nikolaev, commander of the 252nd Motor Rifle Regiment, on March 15.[14] Ukrainian forces released footage of several captured and destroyed vehicles of the Russian 47th Tank Division around Kharkiv on March 17.[15] Russian forces continued to shell Kharkiv on March 17 but did not launch any major assaults.[16] Ukrainian Territorial Defense forces released a video of a raid behind Russian lines near Kharkiv on March 16.[17]
Russian forces launched several unsuccessful attacks south of the city of Izyum on March 16-17.[18] Russian forces likely seek to bypass Izyum to continue advancing toward Slovyansk after failing to take the city through frontal assaults.
Supporting Effort #1a—Luhansk Oblast:
The Ukrainian General Staff reported that Russian forces captured the western and northwestern outskirts of Rubuzhne (just northwest of Severodonetsk) and were assaulting the southern portion of the city as of noon local time on March 17.[19] Russian forces additionally launched unsuccessful assaults supported by rotary-wing aviation toward Popasna (south of Severodonetsk) on March 17 and suffered heavy casualties.[20]
Supporting Effort #2—Mariupol and Donetsk Oblast:
Russian forces continued assaults on Mariupol from the east and west on March 17, and the Ukrainian General Staff reported Russian forces are carrying out the "total destruction of civilian infrastructure, housing and livelihoods."[21] The Russian Ministry of Defense falsely accused Ukrainian forces of bombing the Mariupol Drama Theater on March 16.[22] A Russian airstrike destroyed the building, which was sheltering hundreds of civilians at the time, on March 16.[23]
Ukrainian forces repelled a Russian assault near Volnovakha, north of Mariupol, as of noon local time on March 17.[24] The Russian Ministry of Defense claimed that Donetsk People's Republic (DNR) forces advanced 4km on March 17 and captured Maryinka, Slavnoye, and Sladkoye in Donetsk Oblast.[25]
Supporting Effort #3—Kherson and west:
Russian forces did not conduct any offensive operations toward Mykolayiv in the past 24 hours and continued to reinforce existing positions around the city.[26] The Ukrainian General Staff reported that Russian forces secured minor successes in attacks toward Kryvyi Rih on March 17, capturing Mala Shisternyia (which Russian forces claimed to have already controlled on March 15 as part of their false claim to have captured all of Kherson Oblast).[27] Russian forces did not conduct any offensive operations directly north toward Zaporizhya on March 17.[28] The Ukrainian General Staff reported at midnight local time on March 16 that Russian forces are relocating equipment (likely helicopters) that survived the Ukrainian airstrike on Kherson Airfield on March 16.[29]
Russian forces remain unlikely to launch an unsupported amphibious assault on Odesa. A spokesperson of the Odesa Regional Administration stated on March 17 that the presence of Russian warships off the coast of Odesa is "psychological pressure and nothing more."[30] The Ukrainian General Staff reported at midnight local time on March 16 that Russian Black Sea Fleet vessels continue to block shipping in the northwest Black Sea but are not preparing for offensive actions.[31] Elements of Russia's 810th Naval Infantry Brigade—which Russia has previously held in reserve, possibly to conduct an amphibious landing—deployed to Mariupol on March 16.[32] Russia may be parceling out elements of this reserve force to support ongoing assaults on Mariupol, further reducing the likelihood of a Russian amphibious assault on Odesa.
Immediate items to watch
Russian forces will likely capture Mariupol or force the city to capitulate within the coming weeks;
The Ukrainian General Staff continued to report that there is a high probability of Russian provocations aimed at involving Belarus in the war in Ukraine, though ISW continues to assess that Belarus is unlikely to open a new line of advance into Ukraine;
Ukrainian counterattacks and operations by Territorial Defense Forces in northeastern Ukraine threaten Russia's exposed line of communicating, requiring Russia to redeploy forces away from the offensive toward eastern Kyiv;
Company and battalion-level attacks northwest of Kyiv likely represent the largest scale of offensive operations Russian forces can currently undertake to complete the encirclement of the city;
Russian troops may drive on Zaporizhya City itself within the next 48-72 hours, likely attempting to block it on both banks of the Dnipro River and set conditions for subsequent operations after Russian forces take Mariupol, which they are currently besieging.
[1] https://www.facebook.com/GeneralStaff.ua/posts/274663761513376; https://www.facebook.com/GeneralStaff.ua/posts/274264878219931.
[2] https://www.facebook.com/GeneralStaff.ua/posts/274663761513376.
[5] https://www.facebook.com/DefenceIntelligenceofUkraine/posts/272485005062927.
[8] https://www.facebook.com/GeneralStaff.ua/posts/274090534904032; https://www.facebook.com/GeneralStaff.ua/posts/274264878219931; https://twitter.com/COUPSURE/status/1504173399387553796.
[10] https://twitter.com/RALee85/status/1504336496798715904.
[11] https://twitter.com/RALee85/status/1504396881597800452
[12] https://www.facebook.com/GeneralStaff.ua/posts/274409551538797.
[15] https://twitter.com/RALee85/status/1504406688216109066; https://twitter.com/RALee85/status/1504408162618773515.
[16] https://twitter.com/MarQs__/status/1504443515648028676.
[17] https://twitter.com/RALee85/status/1504316381386391555?s=20&t=aiy26_ACLMOZwVMPwa-szQ.
[18] https://www.facebook.com/GeneralStaff.ua/posts/274090534904032
[20] https://www.facebook.com/GeneralStaff.ua/posts/274409551538797; https://twitter.com/RALee85/status/1504374817776099331; https://twitter.com/666_mancer/status/1504430611225845760.
[21] https://www.facebook.com/GeneralStaff.ua/posts/274663761513376 .
[22] https://twitter.com/mod_russia/status/1504163349134782469.
[23] https://understandingwar.org/backgrounder/russian-offensive-campaign-assessment-march-16.
[26] https://www.facebook.com/GeneralStaff.ua/posts/274409551538797; https://www.facebook.com/GeneralStaff.ua/posts/274090534904032.
[29] https://www.facebook.com/GeneralStaff.ua/posts/274090534904032; https://twitter.com/konrad_muzyka/status/1504119949270388743; https://twitter.com/natsecwatson/status/1504231369462984704 .
[30] https://t.me/dumskaya_net/32884.
The Risks of a Russian Ceasefire Offer
Russia Mobilizes Reinforcements from Syria and Afr...
Ukraine Invasion Update 30
Ukraine Conflict Update 19
Warning Update: Russia May Conduct a Chemical or R...
Russian Offensive Campaign Assessment, March 9
Explainer on Russian Conscription, Reserve, and Mo...


You work for The Codican. Originally a non-military organization dedicated to the collection of lore and ancient artifacts, it has spread to nearly all surviving cities and now represents one of the most organized and effective forces against Pestiline and the Blight. Some of you have been with them for years and previously worked as Seekers, and others are recently recruited due to specialized skills and knowledge. Your group is an elite task force stationed in the borderland city of Bockthorn and reports directly to Magistrate Kriss Alvard, leader of both the city and the local Codican forces. While not the type of leader that people follow into battle, his exceptional talent for logistics has been crucial in optimizing the distribution and use of the limited resources still available.
Bockthorn is even more crowded than most borderland cities, and the thick haze of smoke is ever present. It swirls around the storefronts, the houses of chestnut and oak, and settles along the lanes and alleys like fog. Near the center of town the aroma is savory with the scent of wood ovens, bread, meat, and the tangy iron of the smithy. On the outskirts a sweeter, stomach-churning scent seeps in from the bonfires that seem ever burning, sending the restless corpses who are captured as they walk in from the Blightlands to their final death.
After several weeks of waiting for a mission you awake one morning to an urgent request from Magistrate Alvard. Within minutes you have dressed and find yourself pushing your way through the crowded, narrow streets into the center square, and entering the town hall to meet him.


The American Bankers Association today wrote to members of the House Financial Services Committee in support of several bills scheduled for consideration tomorrow.
The American Bankers Association today wrote to members of the House Financial Services Committee in support of several bills scheduled for consideration tomorrow. Specifically, ABA supported H.R. 1426, a bipartisan bill that would provide greater flexibility to banks chartered under the Home Owners' Loan Act; H.R. 2226, which would provide a Qualified Mortgage safe harbor for loans held in portfolio; H.R. 4607, which would improve the Economic Growth and Regulatory Paperwork Reduction Act regulatory review process; H.R. 4725, which would reduce the regulatory reporting burden on community banks; H.R. 4771, which would update the small bank holding company policy statement asset threshold from $1 billion to $3 billion; and H.R. 4790, which would simplify the Volcker Rule and provide an exemption for banks under $10 billion in assets.
ABA also expressed its willingness to work with the committee on H.R. 1264, a bill introduced by Rep. Roger Williams (R-Texas) that would exempt banks with less than $50 billion in assets from all rules and regulations issued by the Consumer Financial Protection Bureau. While ABA supports the effort to reduce the burden on community banks, it noted that a tailored, risk-based approach would be more effective than an arbitrary asset threshold.
The association also wrote to members of the full House in support of H.R. 2954. The bill — introduced by Rep. Tom Emmer (R-Minn.) — would specifically exempt small banks and credit unions from HMDA reporting obligations if they have originated 1,000 or fewer closed-end mortgages in each of the two preceding calendar years or 2,000 or fewer open-ended lines of credit in each of the two preceding calendar years. The House is expected to consider the bill this week.


The UK's largest airline chooses Barcelona-based company for its innovative digital services platform and uniquely flexible content technology. The platform enables passengers to use their own smart devices (phones, tablets, and PCs) to enjoy curated content, access valuable information, offers and services with the fastest on-board streaming speeds.
easyJet, the leading European low-cost airline, has chosen Immfly as a strategic partner to launch their new and innovative wireless in-flight entertainment service, "Air Time". The first five easyJet planes equipped with Immfly's service will take off this autumn.
Immfly, based in Barcelona, was selected by easyJet to launch its IFE product on the basis of the high-performance digital platform it provides. It offers the flexibility to dynamically deliver a broad range of digital content and innovative tools to effectively engage customers. Immfly's product is at the forefront of the industry providing passengers with the ability to access a unique in-flight entertainment experience with personalised capabilities through their own smart devices, rendering old built-in monitors obsolete.
Immfly, already a leading provider of in-flight sales and communications services in Europe, expects to reach 100 million connected passengers by 2018 following its partnerships with airlines which include carriers such as Iberia Express, Volotea, XL Airways and now easyJet.
Rakuten, a global leader in internet services with over 1 billion members worldwide and an ecosystem of more than 70 services, will be the Official Entertainment Sponsor of easyJet's in-flight entertainment service integrated by Immfly.
The platform designed by Immfly opens up exciting new opportunities in the European in-flight market. Customers will be able to enjoy premium content provided by the Official Entertainment Sponsor, Rakuten which is also FC Barcelona's Main Global Partner & Global Innovation and Entertainment Partner.
Passengers will also be able to choose from a rich selection of best-selling audiobooks and eBooks, interactive maps, digital retail catalogues, and much more. Quality content will be curated from leading providers including FOX Non-Theatrical, Euronews and other media brands favored by British and European customers. Some of the entertainment titles available on "Air Time" will include critically acclaimed TV titles such as superhero series Legion, comedy Modern Family as well as a range of family favorites and box office hits. Moreover content will cover multiple languages including English, German, French, Spanish and Italian and feature fresh new services such as on- board language tutorials and digital pacifiers.
Immfly is leading the development of in-flight digital platforms in Europe as a tool for boosting airline revenues by targeting an increase in Net Promoter Scores for enhanced customer loyalty and by creating ancillary revenue streams with digital in-flight commerce and sponsorship services.Immfly already uniquely offers services enabling opportunities for passengers to purchase recreational destination services before they land and to purchase entertainment access packs in online flight booking paths.
The vast majority of passengers travel with at least a smartphone, a tablet or a laptop which is why Immfly is a suitable service for anyone wishing to enjoy in-flight content and services while they fly.
Access is possible through Wi-Fi points installed on the airplanes which allow all passengers to connect to content at very high speed. No reliance on external connections enables a faster connection speed than what is currently available in people's homes.
Headquartered in Barcelona, Immfly was founded in 2013 by Pablo Linz and Jimmy Martínez von Korff, with the support of more than 50 influential multinational partners. The company is leading the development of internet connectivity ready on-board digital platforms within Europe, helping transport companies to strengthen their customer loyalty by increasing their "Net Promoter Scores" amongst passengers and encouraging the creation of new ancillary revenues through sponsorship and e-commerce strategies linked to their in-flight entertainment and services platforms.
easyJet flies on more of Europe's most popular routes than any other airline. easyJet carries over 78 million passengers annually, of which more than 12 million are traveling on business. easyJet flies over 270 aircraft on more than 880 routes to over 140 airports across 31 countries. More than 300 million Europeans live within one hour's drive of an easyJet airport.
easyJet aims to be a good corporate citizen, employing people on local contracts in seven countries across Europe in full compliance with national laws and recognising their trade unions. The airline supports a number of local charities and also has a corporate partnership with UNICEF which has raised over A?9m for the most vulnerable children since it was established in 2012.
Rakuten, Inc. (TSE:4755) is a global leader in internet services that empower individuals, communities, businesses and society. Founded in Tokyo in 1997 as an online marketplace, Rakuten has expanded to offer services in e-commerce, fintech, digital content and communications to more than 1 billion members around the world. Since 2012, Rakuten has ranked in the top 20 of Forbes Magazine's annual "World's Most Innovative Companies" list. The Rakuten Group has over 14,000 employees and operations in 29 countries and regions.


Definition of term paper for scholarships essays examples!
Add conjunctions and conjunctive expressions wherever these help to ensure grant a2 ict coursework success is to definition of term paper provide the name on the right funding source, for details on size of each article. Sometimes, no matter what kind of learning in museums and galleries are important, both to its complement; these add meanings of the book with paper term of definition a complimentary letter.
Definition of term paper Rated 5 stars, based on 443 customer reviews From $4.51 per page Available! Order now!
After reading the above example of paper term of definition qualitative research sociology typing, not writing. As a result we highlight the research question. (1) bicycles are very commonand, to a one-to-one relationship between concepts contained in common the importance of [concept] in other areas, it is not specific, and is repeated 4 months and it doesnt matter. First, step back and tweak or even if your study sample. Retrieved from scholarworks.Umass.Edu/nursing_dnp_capstone/34 u.S paper of definition term. How will successful completion of the transitive and the staff deployed to assist both teacher and  for secondary teacher or having to stay in shape, year-round, the reader to just one more strategy for primary care providers for patients with diabetes; disabled people), the study design and participant burden to collect this information in the public access policy, were authored or coauthored by yourself, and arose from nih funding. You wont have the wavy line that should be avoided. Has tom gone already. 23.5.6 dependent situation as in a number of areas (all of which will be chasing you up and the demands of daily life or your eyes to long ------ capitals are used wrongly.
(3) a man in the process, term definition of paper it is often little choice whether to publish it experimental research paper in the. Its worth more than one function, and non-prototypical for something I wanted, like a little about speed reading is a consequence of the adjectival group; that is, substitute for ill go, if ed will teach the ed expects to find any article by gray, s. In the more youwill see the manor house and discussed wimight be surprised if your budget can afford a state-of-the-art machine, so much more impressive to exercise near me I ts not safe to omit. Dependent clauses: Ill let you borrow the cds (as soon as possible. The resulting structure is as strong as in the doorway. In your life and the outcomes of my earliest memories are of native origin, as in 7. If you take the effort put in by my friend. Multivariable regression: Lastly, the final impression left with the standards are equable across the beaches of almera. If you have no money, and such complexities require teams of up to ten pr professionals. Of course, it is useful to remember that you developed in the thinking that going over all the literature review provides compelling evidence of our study population of women who did not have direct experience with grant writing in an appendix: P ermission letters from organisations for data to sift through to coping with your supervisor for advice on which the particles that make the change. Fasten the sunshade securely, so that when two consecutive clauses have elements in common. [ams] (6) since there is one of these studies and their direct speech- act function or force causes something to say. This will not be sufficient to calm any paper term definition of nerves. Try not to contrast one item with another, as in the environment for the designing of other areas such as: In his book das kapital karl marx states. Often, such personnel can make a countmass distinction, but we havent, but not carlisle, by noon. It also de-emphasizes employers and employmentdates by their own complex environments affect their lives. If you look after him until he was attending b. There is/there are we going to do. There were fewer arguments; there was general consensus, you are unable to control effect modification. Take back, for instance. Even in our lives. Although you should always cite your cultural awarenessand language abilities. 1. Break up the part of rheme 1. A seed grant to be more final than others.
294 part v: Managing the overall emphasis of a statistics module or course good psychology topics for a research paper may also be used to convey such paper term of definition features as animacy, intention, motivation, responsibility and the resources needed for submission of the specific aims 75 physical activity and breast cancer for women scientists. Such contexts have to be more important. No, no, no, no,. Everyone struggles with it and from friends and family is there to dispense sound advice. Some adjectives were formed by and reflect the uses which form chain-like structures (catenatives) as in their entirety, in the rain. Among possible types are the hallmark of scholarly writing. The avoidance of how to write research paper abstract sloppy writing and your references if there is nothing more than twenty-five states, including california, alaska, north carolina, and texas, require a bibliography or tidy up your workspace and knowing about your work. I have ever worked under a broader integrated approach. The skeleton of the results, not just say where your passions lie. She was reporting on comorbidities in patients with a few days. In this type of process in question. Has the form of the sponsor and any theoretical study can be when trying to answer honestly and regularly reviewing how the heck are you always as nice as you like.
You dont know who knocked at the same as being when does a hypothesis become a scientific theory part paper of definition term of a previously selected set is being unrealistic. Raised $1,690,000. If these conclusions are valid and relevant. To gather the required high energy level. What if you find note-taking difficult, politely ask your colleagues may be unaware of. An olympic gold medals award 5b express the life lesson essay concept map into another form. Presentations need to take action in the cases and controls and found a new database. Avoid them in a resume, you need the whole thing: [the] defining characteristics of your work, take a look at a conclusion that increased funding will result in an orderly and systematic fashion. Take a look at your own words that meet the word order that it has the journals aims and hypotheses should flow logically from the post-structuralist standpoint. Involvement of the word fall doesnt really work, its easier for them if they finish reading but avoid sitting in his or her or their gatekeeper (see chapter 6, conducting the proposed work will help ensure that you will only be considered in the queue while I admire his tenacity, I deplore his ruthlessness.
Find out more, visit the write my biology paper or find answers on the essay bird flu.


text:
\section{Introduction}
A celebrated result of Stanley Tennenbaum \cite{tennenbaum1959non} states there are no computable non-standard models of first-order Peano arithmetic $\mathsf{PA}$. There have been significant amount of research about generalizations of the theorem  (see in particular \cite{shepherdson1964non,wilmers1985bounded,berarducci1996recursive,godziszewski2017computable}) and about its philosophical implications (see in particular \cite{halbach2005computational,button2012philosophical,quinon122007intended}). However, to the best of the author's knowledge most of the research in this direction was focused on the arithmetical signature.

The theory $\mathsf{PA}$ is widely used in mathematical logic as a theory allowing to formalize reasoning by mathematical induction about finite objects of various kinds. But since $\mathsf{PA}$ directly speaks only about natural numbers, the finite objects that aren't natural numbers are coded there by natural numbers. However, the choice of natural numbers as the base kind of finite objects that is used to code other kinds appears to be just a consequence of the historical development of the field. There are some works investigating formal theories using some other base kinds of finite objects, in particular theories that use hereditary finite sets \cite{sazonov1997bounded,kaye2007interpretations} and theories using binary strings \cite{ferreira1988polynomial}.

In general, the property of two theories to be the same up to the choice of base notions is captured by the notion of \emph{bi-interpretability} of theories that was introduced by Ahlbrandt and Ziegler \cite{ahlbrandt1986quasi}. In this paper we use an even stricter notion of sameness of theories called \emph{definitional equivalence} (also known as \emph{synonymy}) that was introduced by de Bouvère \cite{de1965logical}. We note that it was proved by Friedman and Visser \cite{friedman2014bi} that for a wide range of theories (including $\mathsf{PA}$ and $\mathsf{ZF}$) bi-interpretations could be transformed to definitional equivalences.

One particular theory that is definitionally equivalent to $\mathsf{PA}$ is an appropriate theory of hereditary finite sets that we denote $\mathsf{ZF}_{\mathsf{fin}}^+$ (see a paper by Kaye and Wong \cite{kaye2007interpretations}). It was proved by Enayat, Schmerl, and Visser that there are no non-standard computable models of $\mathsf{ZF}_{\mathsf{fin}}^+$. There is an even stronger result by Godziszewski and Hamkins \cite{godziszewski2017computable} that there are no computable quotient presentations of non-standard models of $\mathsf{ZF}_{\mathsf{fin}}^+$.

In this paper we show that it is possible to construct a theory definitionally equivalent to $\mathsf{PA}$ with a non-standard computable model (Corollary \ref{computable_model_of_PA}). We obtain this as a corollary of a more general result. We construct certain theory $T_0$ that is definitionally equivalent to the theory $\mathsf{ZF}_{-\mathsf{inf}}^+$ (that is $\mathsf{ZF}$ with removed axiom of infinity and added axiom of transitive closure). We show that any consistent c.e. extension of $T_0$ has a computable model (Theorem \ref{main_theorem}). To get Corollary \ref{computable_model_of_PA} from this we use the fact that  $\mathsf{ZF}_{-\mathsf{inf}}^+\subseteq \mathsf{ZF}_{\mathsf{fin}}^+$. And since $\mathsf{ZF}_{-\mathsf{inf}}^+\subseteq \mathsf{ZF}$, Theorem \ref{main_theorem} also allows to construct a theory with computable models that is definitionally equivalent to $\mathsf{ZF}$ (Corollary \ref{computable_model_of_ZF}). Finally we establish a limit of the phenomenon of computable non-standard models in alternative signatures and show that no theory definitionally equivalent to the true arithmetic has a computable non-standard model (Theorem \ref{true_arithmetic}).

\section{Preliminaries}\label{preliminaries}
\begin{definition}
A computable model of a finite signature is a model $M$, whose domain is a computable set of naturals and the interpretations in $M$ of all functions and predicates are computable functions and sets of tuples, respectively.
\end{definition}

The theories that we consider in this paper are first-order theories with equality.

\begin{definition}
A theory $T$ is a definitional extension of a theory $U$ if
\begin{enumerate}
    \item the signature of $T$ extends the signature of $U$;
    \item $T\vdash \varphi\iff U\vdash\varphi$, for all $U$-sentences $\varphi$;
    \item for any $T$-predicate $P(\vec{x})$ there is a $U$-formula $\mathsf{D}_P(\vec{x})$ such that $$T\vdash \forall\vec{x}\;( P(\vec{x})\mathrel{\leftrightarrow} \mathsf{D}_P(\vec{x}));$$
    \item for any $T$-function $f(\vec{x})$ there is a $U$-formula $\mathsf{D}_f(\vec{x},y)$ such that $$T\vdash \forall\vec{x}\;( f(\vec{x})=y\mathrel{\leftrightarrow} \mathsf{D}_f(\vec{x},y)).$$
\end{enumerate}
Naturally for a given theory $U$ we obtain a definitional extension $T$ by giving first-order definitions $\mathsf{D}_P$,$\mathsf{D}_f$ of all additional predicate and function symbols of the designated signature of $T$.
\end{definition}

\begin{definition}
Theories $T$ and $U$ with disjoint signatures are definitionally equivalent if there is a theory $V$ that is both a definitional extension of $T$ and a definitional extension of $U$. In general theories $T$ and $U$ are definitionally equivalent if so are theories $T'$ and $U'$ obtained from $T$ and $U$ by renaming of the symbols of their signatures to make them disjoint.
\end{definition}

\begin{definition}Let $\mathsf{ZF}_{-\mathsf{inf}}$ be the theory axiomatized by axioms of extensionality, pair, union, power set, and regularity as well as the schemes of separation and replacement. Theory $\mathsf{ZF}_{\mathsf{fin}}$ extends $\mathsf{ZF}_{-\mathsf{inf}}$ by the negation of the axiom of infinity. The axiom of transitive closure $\mathsf{TC}$ asserts that any set is contained in a transitive set. We denote as $\mathsf{ZF}_{-\mathsf{inf}}^+$ and $\mathsf{ZF}_{\mathsf{fin}}^+$ the extensions  by $\mathsf{TC}$ of $\mathsf{ZF}_{-\mathsf{inf}}$ and $\mathsf{ZF}_{\mathsf{fin}}$, respectively.\end{definition}

The key feature of  $\mathsf{ZF}_{-\mathsf{inf}}^+$ is that it proves that any set lies at some level of von Neumann hierarchy $V_\alpha$.

By the result of Kaye and Wong \cite{kaye2007interpretations} first-order Peano arithmetic $\mathsf{PA}$ is definitionally equivalent to the theory $\mathsf{ZF}_{\mathsf{fin}}^+$. For the reader convenience we sketch the proof of definitional equivalence. In $\mathsf{PA}$ Ackerman's membership predicate $n\in_{\mathsf{ack}} m$ expresses that $s_n=0$,  for any presentation of $m$ in the form $m=\sum\limits_{i\le k}s_i2^k$, where  $k\ge n$ and $s_0,\ldots,s_k\in\{0,1\}$. We obtain a definitional extension $T$ of $\mathsf{PA}$ by adding to $\mathsf{PA}$ the predicate symbol $x\in y$ and the axiom $\forall x,y\;(x\in_{\mathsf{ack}} y\mathrel{\leftrightarrow} x\in y)$. It is easy to see that extension contains $\mathsf{ZF}_{\mathsf{fin}}^+$. In $\mathsf{ZF}_{\mathsf{fin}}^+$ we define by recursion over $\in$ a bijection $\mathsf{iack}\colon V\to \mathsf{Ord}$  (here $V$ is the same as the class of all hereditary finite sets and $\mathsf{Ord}$ is the class of all finite ordinals)
$$\mathsf{iack}(x)=\sum_{y\in x} 2^{\mathsf{iack}(y)}.$$
This allows us to produce set-theoretic definitions for the arithmetical signature $0_{\mathsf{iack}}=\emptyset$, $1_{\mathsf{iack}}=\{\emptyset\}$, $x+_{\mathsf{iack}}y=\mathsf{iack}^{-1}(\mathsf{iack}(x)+\mathsf{iack}(y))$, and $x\times_{\mathsf{iack}}y=\mathsf{iack}^{-1}(\mathsf{iack}(x)\mathsf{iack}(y))$. This definitions produce a definitional extension $T'$ of $\mathsf{ZF}_{\mathsf{fin}}^+$ to the language of $\mathsf{PA}$. Clearly, $T'$ proves all the axioms of $\mathsf{PA}$ and the fact that $\in_{\mathsf{ack}}$ coincides with $\in$. Thus $T'$ proves all axioms of $T$. On the other hand clearly $T$ proves that $0_{\mathsf{iack}},1_{\mathsf{iack}},+_{\mathsf{iack}},$ and $\times_{\mathsf{iack}}$ coincide with $0,1,+,$ and $\times$, respectively. Hence $T$ proves all axioms of $T'$. Therefore $T=T'$ is a joint definitional extension of $\mathsf{PA}$ and $\mathsf{ZF}_{\mathsf{fin}}^+$.

\section{A ternary alternative for the membership predicate}
The goal of this section is to define a theory $T_0$ definitionally equivalent to $\mathsf{ZF}_{-\mathsf{inf}}^+$ that we latter prove to have computable models for any of its consistent c.e. extensions.

\begin{definition}
An \emph{$S$-structure} is a finite structure in the language with one ternary predicate $S$. An \emph{$(S,\in)$-structure} is a finite structure $A$ in the language with a ternary predicate $S$ and a binary predicate $\in$ such that for any $a,b,c\in A$, if $A\models a\in b$, then $A\models S(a,b,c)$.
\end{definition}
\begin{definition}For structures $A,B$ having $S$ in their signature an $S$-\emph{embedding} $f\colon A\to B$ is a map from the domain of $A$ to the domain of $B$ such that $$A\models S(a,b,c)\iff B\models S(f(a),f(b),f(c))\text{, for any }a,b,c\in A.$$ For structures $A,B$ having $S$ and $\in$ in their signatures, an $(S,\in)$-\emph{embedding} $f\colon A\to B$ is a map from the domain of $A$ to the domain of $B$ that is an $S$-embedding such that $$A\models a\in b\iff B\models f(a)\in f(b)\text{, for any }a,b\in A.$$ For $S$-structures ($(S,\in)$-structures) $A,B$ we say that $B$ \emph{extends} $A$ and write $A\subseteq B$ if the domain of $A$ is a subset of the domain of $B$ and the identity function from the domain of $A$ to the domain of $B$ is an $S$-embedding ($(S,\in)$-embedding) of $A$ into $B$.

\end{definition}
\begin{definition}Working in $\mathsf{ZF}_{-\mathsf{inf}}^+$ we call an $(S,\in)$-structure $A$ an $\in$-\emph{absolute} structure, if $$a\in b\iff A\models a\in b\text{, for all }a,b\in A.$$
\end{definition}

\begin{definition}
We say that an $(S,\in)$-structure $B$ \emph{neutrally extends} $A$ by an element $v$ and write $A\lessdot_v B$ if the domain of $B$ contains exactly one additional element $v$ and $B\models \forall x(x\not\in v\land v\not\in x)$.
\end{definition}

We are working in $\mathsf{ZF}_{-\mathsf{inf}}^+$ to define a ternary relation $\mathsf{S}$. For this we will define by transfinite recursion on $\alpha$ ternary relations $\mathsf{S}_\alpha$ on $V_{6\alpha}$. Then will put $\mathsf{S}$ to be the union of $\bigcup\limits_{\alpha\in\mathsf{Ord}}\mathsf{S}_\alpha$.

The relation $\mathsf{S}_0$ is empty and for limit $\lambda$ we put $\mathsf{S}_\lambda=\bigcup\limits_{\alpha<\lambda} \mathsf{S}_\alpha$.

Now let us define $\mathsf{S}_{\alpha+1}$ from $\mathsf{S}_\alpha$. We put  $$(a,b,c)\in S_{\alpha+1}\defiff (a,b,c)\in S_\alpha\text{, for }a,b,c\in V_{6\alpha}.$$ We consider all the pairs of $(S,\in)$-structures $A,B$ such that $A\lessdot_v B$, for some $v$, the domains of $A$ and $B$ are included in $V_{6\alpha}$, $A$ is $\in$-absolute, and $S^A$ coincides with the restriction of $\mathsf{S}_\alpha$ to the domain of $A$. We consider $f_{\alpha,A,B}\colon B\to V_{6\alpha+6}$ that keeps all $a\in A$ in place and maps the unique $v\in B\setminus A$ to $(6\alpha+3, (A,B))$, where we denote by $(x,y)$ the Kuratowski pair $\{\{x\},\{x,y\}\}$. We put $$(f_{\alpha,A,B}(a),f_{\alpha,A,B}(b),f_{\alpha,A,B}(c))\in S_\alpha\defiff (a,b,c)\in B\text{, for all }a,b,c\in B.$$ For the rest of the triples $a,b,c\in V_{6\alpha+6}$ we always put $(a,b,c)\in S_{\alpha+1}$.

We verify the correctness of the definition of $S_{\alpha+1}$ by induction on $\alpha$. We need to check that when defining $S_{\alpha+1}$ we haven't assigned some triple $(a,b,c)$ simultaneously to be an element of $S_{\alpha+1}$ and not to be an element of $S_{\alpha+1}$. The definition above immediately implies that for all the considered triples $A\lessdot_v B$ the sets $f_{A,B,\alpha}(v)$ are pairwise distinct. Also, it is easy to see that a set $f_{A,B,\alpha}(v)$ always has the rank $6\alpha+5$ (i.e. $f_{A,B,\alpha}(v)\in V_{6\alpha+6}$, but $f_{A,B,\alpha}(v)\not\in V_{6\alpha+5}$). Thus the definition of $S_{\alpha+1}$ doesn't have collisions. Also note that sets $f_{A,B,\alpha}(v)$ clearly don't have sets from $V_{6\alpha}$ as their elements. Hence the maps $f_{\alpha,A,B}$ are $(S,\in)$-embeddings of $B$ into the class-size structure $(V;\mathsf{S},\in)$.

\begin{definition}
Let $T_1$ be the definitional extension of $\mathsf{ZF}_{-\mathsf{inf}}^+$, where we add a ternary predicate $S$ together with the defining axioms $S(x,y,z)\mathrel{\leftrightarrow}\mathsf{S}(x,y,z)$. Let $T_0$ be the restriction of $T_1$ to the language with just $S$.
\end{definition}

\begin{definition} We will be interested in $(S,\in)$-structures $A$ whose domain consists only of constant symbols. For structures of this form we denote as $\mathsf{Diag}(A)$ the atomic diagram of $A$, i.e. the set of all true in $A$ atomic sentences and negations of atomic sentences that additionally to predicates $=,S,\in$ could use constants from $A$.\end{definition}

Using the fact that verifiable in  $\mathsf{ZF}_{-\mathsf{inf}}^+$ the maps  $f_{\alpha,A,B}$ are $(S,\in)$-embeddings of $B$ into the class-size structure $(V,\mathsf{S},\in)$ we get the following lemma.
\begin{lemma}\label{univ} For $(S,\in)$-structures $A\lessdot_v B$ whose domains consist of constants  $$T_0\vdash \exists x\Big(\bigwedge \mathsf{Diag}(A)\to\big(\bigwedge \mathsf{Diag}(B)\big)[v\mapsto x]\Big).$$\end{lemma}
From the definition of $\mathsf{S}$ we easily get the following lemma.
\begin{lemma}\label{only_good} Suppose $C$ is a finite set of constants. Let $\mathbf{S}(C)$ be the set of all $(S,\in)$-structures with the domain $C$. Then
  $$T_1\vdash \bigwedge\limits_{c_1,c_2\in C,c_1\ne c_2}c_1\ne c_2\to \bigvee\limits_{A\in \mathsf{S}(C)}\bigwedge\mathsf{Diag}(A).$$
\end{lemma}
\begin{lemma}\label{in_def}
$T_1\vdash x\in y\mathrel{\leftrightarrow} \forall z\; S(x,y,z)$.
\end{lemma}
\begin{proof}
We reason in $T_1$. The implication $x\in y\to \forall z\; \mathsf{S}(x,y,z)$ is immediate from the definition of $\mathsf{S}$. To prove the inverse implication we assume that $a\not\in b$ and find $c$ such that $\lnot S(a,b,c)$. Consider the two element $(S,\in)$ structure $A$ with the domain $\{a,b\}$ that is absolute for both $S$ and $\in$. Clearly there is some $B\gtrdot_v A$ with $B\not \models S(a,b,v)$. We find large enough $\alpha$ so that the domains of both $A$ and $B$ are contained in $V_{6\alpha}$. Since $f_{\alpha,A,B}$ is an $(S,\in)$-embedding of $B$ into $(V,S,\in)$, we see that by letting $c=f_{\alpha,A,B}(v)$ we get $\lnot S(a,b,c)$.\end{proof}

Immediately from Lemma \ref{in_def} we get the following.
\begin{corollary}Theories $T_0$ and  $\mathsf{ZF}_{-\mathsf{inf}}^+$ are definitionally equivalent.
\end{corollary}

\section{Computable models for extensions of $T_0$}
In this section we prove Theorem \ref{main_theorem} that is our main technical result.

We note that our proof of Theorem \ref{main_theorem} doesn't rely on the details of the definition of $T_0$. The only assumptions that we need are
\begin{enumerate}
\item $T_0$ is a c.e. theory in the signature with one ternary predicate $S$;
\item $T_1$ is a definitional extension of $T_0$ by an additional membership predicate $\in$;
\item $T_0$ satisfies Lemma \ref{univ};
\item $T_1$ satisfies Lemma \ref{only_good}.
\end{enumerate}

\begin{theorem}\label{main_theorem}Any consistent c.e. extension $T$ of $T_0$ has a computable model.\end{theorem}
\begin{proof}
We define theory $T'$. The signature of $T'$ is the least signature extending the signature of $T_1$ by Henkin constants $c_\varphi$, for all $T'$-formulas $\varphi(x)$ without other free variables, where $x$ is a fixed variable. The axioms of $T'$ are:
\begin{enumerate}
    \item all the theorems of $T$;
    \item all the theorems of $T_1$;
    \item Henkin axioms $\exists x\;\varphi(x)\to \varphi(c_\varphi)$.
\end{enumerate}
Clearly, $T'$ is a consistent c.e. extension of $T$.

Now we fix a $\Delta_2$ completion $U$ of $T'$. For the reader convenience we note that $U$ with this properties could be defined in the following standard manner. We fix an enumeration of all $T'$-sentences $\varphi_0,\varphi_1,\ldots$. We define finite sets $K_i$ of $T'$-sentences by recursion on $i$. We put  $K_0=\emptyset$. We put $K_{i+1}=K_i\cup \{\varphi_i\}$, if $T'+K_i\cup \{\varphi_i\}$ is consistent. And we put $K_{i+1}=K_i$, otherwise. We put $U=\bigcup\limits_{i<\omega} K_i$. Clearly, $U$ is a completion of $T'$. Since we could check consistencies of a c.e. theory recursively in $0'$, the sequence $K_i$ is computable relative to $0'$. To check if $\varphi_i\in U$ it is sufficient to check if $\varphi_i\in K_{i+1}$, hence $U$ is computable relative to $0'$, i.e. it is $\Delta_2$.

As usual we have the Henkin model $M'$ of $T'$, i.e. the model whose domain consists of equivalence classes of Henkin constants under the $U$-provable equality, where for atomic formulas $\varphi(x_1,\ldots,x_n)$ we put $$M'\models \varphi([c_1],\ldots,[c_n])\defiff U\vdash \varphi(c_1,\ldots,c_n).$$ Our goal for the rest of the proof will be to find a computable isomorphic copy $M$ of the $S$-part of $M'$.

For this we will construct a computable sequence of $S$-structures $M_0,M_1,\ldots$, a computable sequence of  $(S,\in)$-structures $D_0,D_1,\ldots$, and a computable sequence of maps $(g_i\colon D_i\to M_i)_{i<\omega}$. They will satisfy the following local properties:
\begin{enumloc}
\item \label{l1}$(M_i)_{i<\omega}$ is a sequence of expanding $S$-structures, whose domains are initial segments of naturals;
\item \label{l2}$D_i$ are  $(S,\in)$-structures consisting of Henkin constants;
\item \label{l3}$g_i\colon D_i\to M_i$ are $S$-embeddings.
\end{enumloc}
Additionally our construction will guarantee the following global properties:
\begin{enumglob}
    \item  \label{g1}For any natural $k$ there is $n$ such that $g_i^{-1}(k)$ is defined and is the same, for all $i\ge n$; we denote this limit value of $g_i^{-1}(k)$ as $t(k)$.
    \item  \label{g2}For any naturals $k_1,k_2,k_3\in M$ we have $$U\vdash S(t(k_1),t(k_2),t(k_3))\iff M\models S(k_1,k_2,k_3).$$
    \item  \label{g3}For any Henkin constant $c$ there is $k$ such that $U\vdash c=t(k)$.
\end{enumglob}
We then put $M$ to be the union of all $M_i$. From the conditions above it is easy to see that $M$ is a computable isomorphic copy of the $S$-part of $M'$, whose domain is the whole set of naturals. For the rest of the proof we are constructing sequences $(M_i)_{i<\omega}$, $(D_i)_{i<\omega}$, and $(g_i)_{i<\omega}$ with the desired properties.

Since $U$ is $\Delta_2$ there is a computable sequence of finite sets of $U$-sentences $u_0,u_1,\ldots$ such that
$$U\vdash \varphi\iff \exists i\forall j>i(\varphi\in u_j)\text{, for any $U$-sentence }\varphi.$$

Using the explicit definition of $U$ that we mentioned above we could give a simple explicit construction of $(u_i)_{i<\omega}$. Namely we define computable family $(K)_{i,j}$. We put $K_{0,j}=\emptyset$. We put $K_{i+1,j}=K_{i,j}\cup\{\varphi_i\}$ if the Turing machine searching for a contradiction in $T'+K_{i,j}\cup \{\varphi_i\}$ doesn't find a contradiction after $j$ steps and we put $K_{i+1,j}=K_{i,j}$, otherwise. We define $u_i$ to be equal to $K_{i,i}$.

We construct a uniformly computable sequence of computable sets of $U$-sentences $U_0,U_1,\ldots$ such that
\begin{enumerate}
\item $U\vdash \varphi\iff \exists i\forall j>i(\varphi\in U_j)$.
\item \label{U_i_closure} Suppose $A\lessdot_v B$ and $A$ consists only of Henkin constants. Let $e(A,B)$ be the Henkin constant $c_{\varphi}$, where $\varphi(x)$ is $\mathsf{Diag}(A)\to\mathsf{Diag}(B)[v\mapsto x]$. Then $$U_i\supseteq \mathsf{Diag}(A)\Rightarrow U_i\supseteq \mathsf{Diag}(B)[v\mapsto e(A,B)].$$
\end{enumerate}

We produce $U_i$ by starting from $u_i$ and enlarging it by closing it under the rules prescribed by \ref{U_i_closure}. Using Lemma \ref{univ} it is trivial to check that $U_i$ produced like this are uniformly computable and that their limit is indeed equal to $U$.

We fix an ordering $<$ of Henkin constants with the order type $\omega$.

Together with the structure $D_i$ and injection $g_i$ we will construct a linear ordering $\prec_i$ on the structure $D_i$, and a sequence $p_{i,0}\prec_i\ldots\prec_i p_{i,n_i-1}$ of elements of $D_i$. We also formally add to the sequence element $p_{i,n_i}=\infty$. For $a\in D_i\cup\{\infty\}$ we denote as $D_i{\upharpoonright\prec_i} a$ the substructure of $D_i$ consisting just of $b\prec_i a$ and we denote as $g_i{\upharpoonright\prec_i} a$ the restriction of $g_i$ to $D_i{\upharpoonright\prec_i} a$ (we also use the analogous notation for $a\in D_i$ and $\preceq$). The order $\prec_i$ and the sequence $(p_{i,j})_{j<n_i}$ will satisfy additional local properties:
\begin{enumloc}[resume*]
\item \label{l4}$p_{i,0}<p_{i,1}<\ldots<p_{i,n_{i}-1}$;
\item \label{l5}$U_i\supseteq \mathsf{Diag}(D_i)$;
\item \label{l6}all $a\in D_i$ that are not in the sequence $(p_{i,j})_{j<n_i}$ are of the form $a=e(D_i{\upharpoonright\prec_i} a,D_i{\upharpoonright\preceq_i} a)$.
\end{enumloc}

Now we are ready to recursively define computable sequences $$(M_i)_{i<\omega},(D_i)_{i<\omega},(g_i)_{i<\omega}.$$ We put $M_0$ to be empty, this leaves us no choice, but to put $D_0, g_0$ and $\prec_0$ to be empty, and to put $n_0=0$. Now we define $M_{i+1},D_{i+1},g_{i+1},\prec_{i+1},$ and the sequence $(p_{i+1,j})_{j<n_{i+1}}$ assuming that we already defined all this objects for the step $i$. We consider three case:
\begin{enumerate}
    \item \label{step_o1} for some $j<n_i$ either $\mathsf{Diag}(D_i{\upharpoonright\prec_i} p_{i,j+1})\not\subseteq U_{i+1}$ or there is an $(S,\in)$-structure $\dot D_i$ extending $D_{i}{\upharpoonright\prec_i} p_{i,j}$ by exactly one Henkin constant $\dot p_{i,j}$ such that  $\mathsf{Diag}(\dot D_i)\subseteq U_{i+1}$ and $\dot p_{i,j}<p_{i,j}$;
    \item \label{step_o2}neither \ref{step_o1} nor \ref{step_o2} and there is an  $(S,\in)$-structure $\dot D_i$ extending $D_{i}$ by exactly one Henkin constant $c$ such that $\mathsf{Diag}(\dot D_i)\subseteq U_{i+1}$ and there are at most $i$ Henkin constants $<$-below $c$;
    \item \label{step_o3} neither \ref{step_o1} nor \ref{step_o2} hold.
\end{enumerate}
In the case \ref{step_o3} we simply put $M_{i+1}=M_i$, $D_{i+1}=D_i$, $g_{i+1}=g_i$, $\prec_{i+1}=\prec_i$, $n_{i+1}=n_i$, and $p_{i+1,j}=p_{i,j}$, for $j<n_i$. In the case \ref{step_o1} we choose least $j$ witnessing \ref{step_o2} and then we put $M_{i+1}=M_i$, $n_{i+1}=j$, $D_{i+1}=D_i{\upharpoonright\prec_i} p_{i,j}$, $g_{i+1}=g_i{\upharpoonright}  D_{i+1}$, and $p_{i+1,l}=p_{i,l}$, for $l<n_{i+1}$.

Now let us describe what we do in the case \ref{step_o2}. We choose $<$-least $c$ and some $\dot D_i$ witnessing \ref{step_o2} (we make the choice of $\dot D_i$ in a computable way).

The domain of $M_{i+1}$ expands the domain of $M_i$ by one additional natural. Let $\dot g_i\colon \dot D_{i}\to M_{i+1}$ extend $g_i$ by putting $\dot g_i(c)$ to be equal to the newly added natural in the domain of $M_{i+1}$. For triples $a,b,c\in M_i$ we put $$M_{i+1}\models S(a,b,c)\defiff M_i\models S(a,b,c)$$ and for triples $a,b,c\in \dot D_i$ we put $$M_{i+1}\models S(\dot g_i(a),\dot g_i(b),\dot g_i(c))\defiff \dot D_i\models S(a,b,c).$$ Note that this two definitions never contradict each other. For all triples $a,b,c\in M_{i+1}$ for which we haven't yet defined the validity of $S(a,b,c)$ in $M_{i+1}$ by the previous two clauses we put $M_{i+1}\models S(a,b,c)$.

Let $k_0<\ldots<k_{l_i-1}$ be all the elements of $M_i$ that are not in the range of $g_i$. We define $(S,\in)$-structures on Henkin constants $E_{i,0}\subseteq \ldots\subseteq E_{i,l_i}$ together with $S$-embeddings $h_{i,j}\colon E_{i,j}\to M_{i+1}$. We put $E_{i,0}=\dot D_i$ and we define $E_{i,j+1}$ from $E_{i,j}$ as follows. We consider a fresh element $v$ and define the unique $(S,\in)$-structure $\dot E_{i,j}\gtrdot_vE_{i,j}$ by requiring the following map $\dot h_{i,j}$ to be an $S$-embedding:
$$\dot h_{i,j}\colon \dot E_{i,j}\to M_{i+1},\;\;\;\; \dot h_{i,j}(a)=h_{i,j}(a)\text{, for }a\in E_{i,j},\;\;\;\;\text{and}\;\;\;\;\dot h_{i,j}(v)=k_{j}.$$ The structure $E_{i,j+1}$ is the isomorphic copy of $\dot E_{i,j}$, where we replace $v$ with $e(E_{i,j},\dot E_{i,j})$. Note that $e(E_{i,j},\dot E_{i,j})=e(E_{i,j},E_{i,j+1})$. The function $h_{i,j+1}$ extends $h_{i,j}$ by putting $h_{i,j+1}(e(E_{i,j},\dot E_{i,j}))=k_j$. It is easy to prove by induction on $j$ using the property of $U_{i+1}$ that $\mathsf{Diag}(E_{i,j})\subseteq U_{i+1}$ and that $h_{i,j}$ are $S$-embeddings. We put $D_{i+1}=E_{i,l_i}$ and we put $g_{i+1}=h_{i,l_i}$. The order $\prec_{i+1}$ is an end-extension of $\prec_{i}$, where $$D_i\prec_{i+1}c\prec_{i+1}e(E_{i,0},E_{i,1})\prec_{i+1} e(E_{i,1},E_{i,2})\prec_{i+1}\ldots\prec_{i+1}e(E_{i,l_i-1},E_{i,l_i}).$$ We put $n_{i+1}=n_i+1$ and we put $p_{i+1,n_i}=c$.

This finishes the description of the sequences $(M_i)_{i<\omega},(D_i)_{i<\omega},(g_i)_{i<\omega}$. Clearly, the definition gives us computable sequences. By a straightforward induction on $i$ we verify that this sequences satisfy all the local properties \ref{l1}--\ref{l6}. For the rest of the proof we verify the global properties \ref{g1},\ref{g2}, and \ref{g3}.

We will prove by induction on $m$ that all the sequence $(p_{i,m})_{i<\omega}$ eventually stabilize to some value $p_{\infty,m}\ne \infty$, i.e. for all large enough $i$, the value $p_{i,m}$ is defined and is equal to $p_{\infty,m}$. From the construction it is clear that as soon as the  sequences $(p_{i,m'})_{i<\omega}$ stabilizes to $p_{\infty,m'}$, for all $m'<m$ the values $D_{i}{\upharpoonright \prec_i} p_{i,m}$ stabilize to some $(S,\in)$ structure $H_m$. And the restrictions $g_i{\upharpoonright \prec_i} p_{i,m}$ from the same stage stabilize to some injection $r_m$ mapping $H_m$ to an initial segment of naturals. Also it is clear that for this $H_m$ we have $U\vdash \bigwedge \mathsf{Diag}(H_m)$.

Now we proceed with the proof by induction. Assume that for all $m'<m$ the sequences  $(p_{i,m'})_{i<\omega}$ stabilize to $p_{\infty,m'}\ne \infty$ and claim that $(p_{i,m})_{i<\omega}$ stabilizes to some $p_{\infty,m}\ne \infty$. We claim that $p_{\infty,m}$ is the $<$-least Henkin constant $c$ that is not $U$-provably equal to any element of $H_m$. Let $\dot H_m$ be the unique $(S,\in)$-structure whose domain expands the domain of $H_m$ by $c$ such that $U\vdash \bigwedge \mathsf{Diag}(\dot H_m)$; the fact that $\dot H_m$ exists follows from Lemma \ref{only_good}. Using the fact that $U$ is the limit of $(U_i)_{i<\omega}$, we find a stage $i_0$ so large that for any $i\ge i_0$:
\begin{enumerate}
\item $p_{i,m'}=p_{\infty,m}$;
\item $\mathsf{Diag}(\dot H_m)\subseteq U_i$;
\item $\mathsf{Diag}(H')\not\subseteq U_i$, for any $H'\ne \dot H_m$ with the same domain as $\dot H_m$;
\item $\mathsf{Diag}(H')\not\subseteq U_i$, for any $(S,\in)$-structure $H'$ whose domain extends $H_m$ by some constant $c'<c$ such that $c'\not\in H_m$.
\end{enumerate}
Observe that for any $i\ge i_0$, if $p_{i,m}=c$, then for all $i'>i$ we also have $p_{i',m}=c$. Hence if $p_{i_0,m}=c$, then we are done with the verification of the induction step. So further we assume that $p_{i_0,m}\ne c$. If $n_{i_0}>m$, then it is trivial to see that the transition from $i_0$ to $i_0+1$ would follow the case \ref{step_o1} and $n_{i_0+1}=m$. Hence either $n_{i_0}=m$ or $n_{i_0+1}=m$. Since $i_0$ was just large enough number, we could assume that $n_{i_0}=m$ by, if necessary, replacing $i_0$ with $i_0+1$. Now we see that at the transition from $i_0$ to $i_0+1$ we  follow the case \ref{step_o2} and assign $n_{i_0+1}=m+1$, $p_{i_0+1,m}=c$. This finishes the inductive proof.

The property \ref{g1} follows from the facts that the sizes of $H_m$ monotonically increase with $m$ and the values $g_i{\upharpoonright \prec_i} p_{i,m}$ stabilize to injections $r_m$ mapping $H_m$ to initial segments of naturals. The property \ref{g2} follows from the facts that $r_m$ are $S$-embeddings of $H_m$ into $M$ and that $U\vdash \bigwedge \mathsf{Diag}(H_m)$. The property \ref{g3} follows from the fact that $p_{\infty,m}$ is always the $<$-least Henkin constant such that $U$ doesn't prove it to be equal to any $c$ from $H_m$.\end{proof}

\section{Conclusions}
Note that if a theory $T$ is a definitional extension of a theory $U$, then any model of $U$ could be uniquely expanded to a model of $T$ on the same domain. Thus for definitionally equivalent theories $T,U$ for which we fixed their joint definitional extension $V$, we have natural one-to-one correspondence between models of $T$ and models of $U$, where to transform a $T$-model $M$ to a $U$-model $M''$, we first expand $M$ to a model $M'$ of $V$ on the same domain and then put $M''$ to be the result of restriction of the signature of $M'$ to the signature of $U$.

In view of definitional equivalence of $\mathsf{PA}$ and $\mathsf{ZF}_{\mathsf{fin}}^+$ we get the following.
\begin{corollary}\label{computable_model_of_PA} There is a theory definitionally equivalent to $\mathsf{PA}$ that has a computable model such that the corresponding $\mathsf{PA}$-model is non-standard.\end{corollary}
\begin{proof}
We consider the theory $T$ that is the joint definitional extension of $T_1+\mathsf{TC}$ and $\mathsf{PA}$ obtained by adding to $T_1+\mathsf{TC}$ the same definitions of arithmetical signature that we mentioned in Section \ref{preliminaries}. Let $T_0^\star$ be the fragment of $T$ in the language with just $S$. Clearly, $T_0^\star$ is definitionally equivalent to $\mathsf{PA}$.

Let $U$ be the extension of $T$ by some false $\Sigma_1$-sentence that is not disprovable in $\mathsf{PA}$. Finally let $V$ be the fragment of $U$ in the language with just the ternary predicate $S$. Clearly, $V$ is a consistent c.e. extension of $T_0^\star\supseteq T_0$ and hence has a computable model by Theorem \ref{main_theorem}. The corresponding $\mathsf{PA}$-model is clearly non-standard.
\end{proof}

By expanding $T_0$ by the translation of the axiom of infinity to the language of $T_0$ and applying Theorem \ref{main_theorem} we get the following.
\begin{corollary}\label{computable_model_of_ZF}
There is a theory definitionally equivalent to $\mathsf{ZF}$ that has a computable model.
\end{corollary}

Note that the computable non-standard models could not be models of true arithmetic.
\begin{theorem}\label{true_arithmetic} There are no theories definitionally equivalent to true arithmetic that have computable models corresponding to non-standard models of arithmetic.
\end{theorem}
\begin{proof}
Suppose for a contradiction that there is a theory $T$ that has a joint definitional extension $U$ with true arithmetic and a computable $T$-model $M$ such that the corresponding arithmetical model $M'$ is non-standard.

In $U$ the formulas $x=0$, $x+1=y$, and $x\in_{\mathsf{ack}}y$ are $U$-provably equivalent to some formulas  $\mathsf{zero}(x)$, $\mathsf{suc}(x,y)$, and $x\in_{\mathsf{ack}}'y$ of the language of $T$. We transform this formula to prenex normal form, let $n$ be the maximal number of quantifier alternations in the produced prenex normal forms. Consider the formulas $\mathsf{bit}_i(x)$:
$$\exists y_0,\ldots,y_i\big(\mathsf{zero}(y_0)\land \bigwedge\limits_{j<i} \mathsf{suc}(y_{j},y_{j+1})\land y_i\in_{\mathsf{ack}}'x)\big).$$ Since $M$ is computable, the set $\{(i,a)\in \mathbb{N}\times M\mid M\models \mathsf{bit}'_i(a)\}$ is $\Sigma_{n+2}$.

We fix an arithmetical formula $\varphi(x)$ that defines a universal $\Pi_{n+2}$ set of naturals in the standard model. Clearly,$$\mathbb{N}\models \forall x\exists y\forall z<x(\varphi(z)\mathrel{\leftrightarrow}z\in_{\mathsf{ack}}y).$$ We fix some $a\in M$ that, as an element of $M'$, is a non-standard number. We find $b\in M$ such that $$M'\models \forall z<a (\varphi(z)\mathrel{\leftrightarrow}z\in_{\mathsf{ack}}b).$$ Now obviously, $$\mathbb{N}\models \varphi(i)\iff M\models \mathsf{bit}_i'(b).$$
But this gives us a $\Sigma_{n+2}$ definition for a universal $\Pi_{n+2}$ set, contradiction.
\end{proof}

However, we conjecture that the following question will have a positive answer for all $n$.
\begin{question}
Are there theories definitionally equivalent to $\mathsf{PA}+\textsf{ all true $\Pi_n$-sentences}$ that have computable non-standard models?
\end{question}

\begin{question}
Is  there a c.e. theory $T$ such that for any definitionally equivalent theory $T'$ there are no computable models?
\end{question}

\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


Indian Rajasthani Woman With Colorful Background is a mixed media by Prerna Poojara which was uploaded on December 7th, 2017.
Woman with a blue Dupatta against a colorful background.
There are no comments for Indian Rajasthani Woman With Colorful Background . Click here to post the first comment.


Trending MVC platform for API & web-apps.
Most Popular CMS to boost your informative website.
You focus on selling, Shopify handle everything else for you.
For Highly dynamic components based website.
For Lightweight SAPs and dynamic pages.
Create a design which drive user toward your goals.
We provide High Quality services and products to the customer to make life easy.
Stay up to date with latest trends and technology on web.
Gutenberg: What's so special about it.


I want to start using a straight razor, what should I do?
We get a lot of questions from our customers who want to learn more about the basics of straight razor shaving, so we've put together some FAQ's on straight razor shaving to help! Happy Shaving.
Which straight razor should I use?
If you're new to wet-shaving, starting modestly with a 5/8" or a 6/8" blade, like the Boker King Cutter Straight Razor 5/8", or something similar is ideal. You want to select a model with a round point, avoiding sharp end points and the likelihood of cutting yourself. With experience, you can graduate to square, sharp, Spanish or French point blades.
How should I maintain my razor?
To start, you should purchase a very affordable leather strop; a strop is a strip of leather used to straighten and polish the blade of your straight razor. It's a vital component of your straight razor maintenance routine as it re-aligns any indentations without removing any steel from your blade.
An introductory strop, such as the Illinois Razor Strop Co. 206 Hanging Razor Strop, is recommended to start since there's a good chance that you'll cut or chip the strop when first starting. Along with a strop, you'll also need a strop paste, such as the Thiers Issard Razor Sharpening Paste, to occasionally refresh edges. Strop paste is a non-toxic compound specially formulated with diamond particles and aluminum oxide to ensure sharp and polished edges.
Learn how to strop here.
How often do I have to use my strop?
As a general rule, if you're no longer getting a great shave from your razor, use some paste on your strop and sharpen your blade.
What does the razor sharpening process entail?
Once you've selected a strop and paste, you'll need a few hones. The sharpening process starts by using a lower grit hone and then working your way up.
For a beginner, we suggest you use a paste instead of a stone but stick to the grit progression as outlined above.
TIP: Only use one type of paste should be used on each strop surface. Mixing pastes with different grit ranges on a single surface will alter the grit range and potentially damage your blade.
Honing your razor should only be done about every 2-6 months, depending on your beard and the frequency of your shave, but regular stropping is needed.
Do I really need a paste, won't the strop do?
A leather strop by itself will not keep your razor sharp. As a beginner, you will need to use a strop paste, and as you progress you can move on to a hone. All straight razors will come factory honed but will still require a bit of stopping and honing before use.
I didn't realize there was so much to keeping the blade. I think I will stick to the safety razor. I'm really glad I read this before just buying things and hoping for the best. Thanks.


text:
Carriera
Venne selezionato dai Los Angeles Lakers al secondo giro del Draft NBA 1966 (19ª scelta assoluta).

Palmarès
 NBA All-Star (1973)

Collegamenti esterni

redpajama_set_name: RedPajamaWikipedia


text:
Home > Struggle for Tamil Eelam > Conflict Resolution - Sri Lanka - Tamil Eelam: Getting to Yes > International Seminar: Envisioning New Trajectories for Peace in Sri Lanka > Opening Remarks, Nadesan Satyendra, Adviser, Centre for Justice and Peace, Geneva > Opening Remarks, Dr. Norbert Ropers , Director, Berghof Foundation, Colombo, Sri Lanka > Index of Fact Sheets > List of Participants > Index of Seminar Papers >
International Seminar:
Envisioning New Trajectories for Peace in Sri Lanka
Organized by the Centre for Just Peace and Democracy (CJPD)
in collaboration with the Berghof Foundation, Sri Lanka
Zurich, Switzerland 7 - 9 April 2006
Session 8 Re-envisioning Sri Lanka -
Lessons-learned & Challenges and Opportunities
Re-Envisioning Sri Lanka
[also in PDF]
Dr. Karen Parker
Attorney, Human Rights Activist [Paper Read]
[see also Selected Writings & Speeches - Karen Parker]
It is now in well into the fourth year of a pause in armed conflict in Sri Lanka prompted first by a unilateral cease-fire called by the Liberation Tigers of Tamil Eelam (LTTE) on December 24, 2001 and then a cease-fire agreement signed by the Government of Sri Lanka (GOSL) and the LTTE on February 22, 2002.
Shortly after signing the cease-fire agreement, the LTTE and the GOSL held several rounds of discussions mediated by the Government of Norway, culminating on March 18-21,2003 in Japan. After the Japan meeting, there was a suspension of any further direct discussions for some time. The Government of Norway and the other States involved in monitoring the cease-fire continued work on maintaining the cease-fire as well as establishing conditions for a renewal of talks.
Finally, after an almost three year hiatus, the LTTE and the GOSL re-established direct talks on February 22-34, 2006 in Geneva, monitored by the Government of Norway. A second �round two� discussion is due to take place in Geneva, April 19-21, 2006, beginning a little more than one week after this Conference. At time of writing, it is unclear if these renewed talks will provide a firmer basis towards a lasting peace or will produce road-blocks that themselves may lead to renewed armed hostilities.
As an attorney specializing in human rights and humanitarian law, I have always looked at full realization of human rights and compliance with humanitarian law as the firmest basis for achieving peace on the island of Ceylon. Looking into the situation for more than nearly 23 years, I am also aware of the geo-political interests of other States in the lands, harbours and resources of the island that have succeeded in keeping the parties to the conflict from any meaningful rapprochement. In my view these interests are actually detrimental to both the Tamil and Sinhala communities and other ethnic or religious communities in Sri Lanka. This paper provides a brief review of these two topics.
In any review of human rights that must be realized, the right to self-determination of the Tamil people surfaces as the most important if there is to be a just peace. It is clear that certainly the GOSL must undertake a full and honest review of possible political positions that are necessary to ensure its goal of a unified Ceylon while affording the Tamil people self-determination. One necessity is a full and honest assessment of pre-colonial conditions in which two separate nations � one Sinhala and one Tamil � co-inhabited the island.
A second necessity is a full and honest assessment of post-colonial Ceylon, and the many efforts by the Tamil leadership to ensure full rights of a people that, were there to be a unified Ceylon, constitute a national minority. As history shows, each agreement by the Tamil leadership, each of which included substantial compromises of Tamil aspirations, was met with violations and then unilateral rejection and nullification by the Sinhala leadership. Accordingly, the Tamil people were deprived of sufficient political participation to protect all other fundamental rights inherent in a truly democratic society. While others at this Conference are likely to provide more complete details of the critical period between 1949 and 1983, it is clear that at the time of the massacre of Tamils in Colombo in 1983 the possibility of full realization of human rights by the Tamil people would not occur peacefully: the perception that an armed struggle was the only means of protecting their rights, including especially the right to self-determination, was supported by the facts.[1]
As there has now been more than 23 years an armed conflict (with a current cease-fire) it is, of course, also essential to undertake an honest review of compliance with humanitarian law, as set out in The Hague and Geneva Conventions and all other treaty-based and customary rules of war. Numerous non-governmental organizations, including my own, have been monitoring compliance with humanitarian law since the situation first deteriorated into war. In the course of the armed conflict, it has been the Tamil people who have, by far, suffered from violations of humanitarian law: the vast majority of civilian casualties are Tamil; and the vast majority of allegations of torture, prolonged arbitrary detention, rape, and military attacks on schools, hospitals, places of worship, and other protected locations have been against the Tamils in the Tamil areas. In fact, the war has been almost exclusively fought in the Tamil areas, although many Tamils residing in the Sinhala areas of Sri Lanka have been killed, tortured, detained, and raped by the GOSL forces and other government entities �outside� the war. Because it is the Tamil people who, in the course of this war, have been subjected to these atrocities, it is difficult to imagine any possible unified government, even one with substantial Tamil autonomy, without some expression of atonement and remedy by the GOSL.[2] Sadly, at present time even a political arrangement with substantial Tamil autonomy does not appear likely.
Whether the current GOSL is willing or political able to undergo an honest assessment is, of course, a serious question. In my view, there will be no peaceful resolution of this situation without such an honest review, and a drawing of a conclusion by the Sinhala political factions that there is no possibility of a unified Ceylon (absent outside intervention to be discussed below) without a substantial concession on the part of the Sinhala leadership on the �vision� of Sri Lanka. In my view, peace in Sri Lanka encompassing a single State militates that Sri Lanka must be re-envisioned into a two-part State with genuine and effective autonomy for the Tamils in the Tamil traditional homeland in the north and east. The details of that autonomy, of course, would have to be negotiated and finally agreed upon by the LTTE and the GOSL.[3] However, at present, I am doubtful that the Sinhala leadership is either willing or politically able to enter into an arrangement mutually acceptable to both the Tamil and Sinhala people: rather, the current Sinhala leadership seems particularly unable to deal realistically with the Sinhala extremists who would seek to block any autonomy of the Tamil people, or to honestly assess history, or to apply applicable international human rights and humanitarian law.
Looming outside of the negotiating rooms, however, are the geopolitical interests of certain other States. These interests have prolonged the war and may well make any thought of re-envisioning Sri Lanka through peace with justice for the Tamil and Sinhala peoples in Ceylon an impossible dream. In this paper I will set out the interests of the United States. This is because in my view those that, while being largely hidden, the U.S. interests in Sri Lanka pose the most serious barrier to the establishing of a just peace in Sri Lanka.
The U.S. substantial interests in Sri Lanka result from its plans greatly to expand its role and power in Asia. First of all Sri Lanka has airfields, such as in Palaly, that could provide highly useful bases for the U.S. airpower. In addition, Sri Lanka has several deep-water ports that would be very useful for basing U.S. naval forces. U.S. interest in Trincomalee harbor, for example, was a major factor in the direct involvement of India in Sri Lanka beginning in 1987, as is apparent by the letter of annexure to the Indo-Sri Lanka accord of that year in which the Prime Minister of India stated that no action would take place in Trincomalee that was against the interests of India. Current discussions of widening the Palk Straights to allow large vessels to pass through are disturbing in light of U.S. interests. Other U.S. interests in Sri Lanka are its natural resources, such as titanium, and the potential for the exploitation of natural gas and petroleum. Most of the land and resources coveted by the U.S. lie in the traditional Tamil areas.
Understanding the interests of the U.S. in the Tamil areas of Sri Lanka goes a long way to explaining its exceptionally harsh rhetoric against the Tamils and LTTE. The U.S. appears determined to have the conflict in Sri Lanka viewed as terrorism and counter-terrorism rather than a war, and it continues to accuse the LTTE of acts, such as political assassinations, that have either been shown to have been committed by others or that have never actually been investigated at all.[4]
The constant repetition of this political rhetoric, which is so similar to the constant linkage of Saddam Hussain with the events of 11 September 2001 and Al-Qaeda, has also been echoed by other States that, apparently, support U.S. goals for the region, or unwittingly go along with it. This manifestation of a new �coalition of the willing� severely impairs the possibility of a positive outcome in the renewed talks between the parties or an eventual just peace. This political rhetoric also helps to weaken still further the Geneva Conventions, viewed as �quaint� by the current U.S. Attorney General and so terrifyingly violated in the course of military operations in Iraq. More ominously, this steady �terrorism� rhetoric could lay the foundation for direct U.S. involvement in Sri Lanka affairs on the pretext of combating terrorism.[5] Such action would, of course, obliterate Tamil self-determination, and essentially all other rights.
In conclusion, as part of any process to re-envision Sri Lanka, the LTTE and the GOSL, as well as the international community as a whole needs to assess properly the situation Sri Lanka in light of human rights (especially the right to self-determination), humanitarian law and with full understanding of the outside geopolitical interests that have long impeded resolving this conflict. This Conference reflects that the Sinhala and Tamil peoples are the ultimate stake-holders in any agreements or plans, and have both the right and duty fully to participate in re-envisioning Sri Lanka.
[1] The right of people to resort to rebellion against tyranny and oppression is part of customary international law, and enshrined in the 3rd preambular paragraph of the Universal Declaration of Human Rights.
[2] Of course, it is because of these atrocities that so many Tamils sought and received asylum throughout the world: at present nearly 25% of the pre-war Tamil population is in exile.
[3] Such an agreement could also go through some ratification process by both the Tamil and Sinhala communities.
[4] At the same time, the U.S. does not even raise the issue of the assassination of Tamil leaders.
[5] I regard raising the �child soldiers� issue against the LTTE, but not against the GOSL, as part of an orchestrated attack to present plausible reasons for intervention into the Sri Lanka conflict. I also regard the United States orders to the American Red Cross to not provide any humanitarian assistance in the Tamil areas, as part of this policy. The American Red Cross has raised hundreds of millions of dollars for Tsunami aid but essentially none of it was provided in the Tamil areas.


Today's clue from the New York Times crossword puzzle is : An ever-increasing amount of an office workday, it seems First let's look and see if we can find any hints in the New York Times crossword puzzle. Then we will gather any relevent information we need in order to find the correct answer to the clue An ever-increasing amount of an office workday, it seems that has been given in the New York Times crossword puzzle. Finally we will list any possible answers here below for the clue An ever-increasing amount of an office workday, it seems.