We present a derivation and theoretical investigation of the Adams-Bashforth and Adams-Moulton family of linear multistep methods for solving ordinary differential equations, starting from a Gaussian process (GP) framework. In the limit, this formulation coincides with the classical deterministic methods, which have been used as higher-order initial value problem solvers for over a century. Furthermore, the natural probabilistic framework provided by the GP formulation allows us to derive probabilistic versions of these methods, in the spirit of a number of other probabilistic ODE solvers presented in the recent literature. In contrast to higher-order Runge-Kutta methods, which require multiple intermediate function evaluations per step, Adams family methods make use of previous function evaluations, so that increased accuracy arising from a higher-order multistep approach comes at very little additional computational cost. We show that through a careful choice of covariance function for the GP, the posterior mean and standard deviation over the numerical solution can be made to exactly coincide with the value given by the deterministic method and its local truncation error respectively. We provide a rigorous proof of the convergence of these new methods, as well as an empirical investigation (up to fifth order) demonstrating their convergence rates in practice.

Things to Do in L.A. Tap to enable a layout that focuses on the article. Supporters of the Green Dot charter at Locke High intently watch the debate over the school's future. On Tuesday, the board narrowly voted to close the school at the end of the year. This is read by an automated voice. Please report any issues or inconsistencies here .

Things to Do in L.A. Tap to enable a layout that focuses on the article. This is read by an automated voice. Please report any issues or inconsistencies here . Alberto Carvalho issued his first statement after an FBI raid on his home and office. The investigation is linked to AllHere, the company behind a failed multimillion-dollar chatbot initiative that the district launched with fanfare in 2024 but that failed within months.

Things to Do in L.A. Tap to enable a layout that focuses on the article. A pedestrian walks past Los Angeles Police Department headquarters along 1st Street downtown. This is read by an automated voice. Please report any issues or inconsistencies here . An LAPD official said an investigation is underway after video footage showed an officer from Transit Services Division enter a Tesla while responding to a call for service.

ROTC students at Old Dominion subdued and killed ISIS-linked gunman who left one dead, two wounded after shouting'Allahu Akbar' and opened fire Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' UFO-linked Air Force general's disappearance takes mysterious new twist amid'sighting' in New Mexico New details have emerged in the mysterious disappearance of a retired US Air Force general, including a possible sighting near a popular New Mexico trail. William Neil McCasland, 68, was last seen around 11am local time on Friday near Quail Run Court NE in Albuquerque, according to the Bernalillo County Sheriff's Office. Friends say McCasland left home without his watch or phone, unusual for the avid hiker and cyclist.

Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Alexander brothers' alleged HIGH SCHOOL gang rape video: Classmates speak out on sick'taking turns' footage... as creepy unseen photos are exposed Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting NFL superstar Xavier Worthy spills all on Travis Kelce, the Chiefs' struggles... and having Taylor Swift as his No 1 fan Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Nancy Mace throws herself into Iran warzone as she goes rogue on Middle East rescue mission: 'I AM that person' READ MORE: I hacked NASA's systems and finally found the truth about UFOs the cover-up is astonishing A retired US Air Force general who led laboratories linked to UFO research has been reported missing from New Mexico . William Neil McCasland, 68, was last seen around 11am Friday near Quail Run Court NE in Albuquerque, the Bernalillo County Sheriff's Office (BCSO) said. The sheriff's office issued a Silver Alert, a public notification system used to locate missing seniors or individuals with Alzheimer's, dementia, or other medical conditions. 'Due to his medical issues, law enforcement is concerned for his safety,' the office said. McCasland, a veteran of Kirtland Air Force Base in New Mexico, previously led both the Phillips Research Site and the Air Force Research Laboratory at Wright-Patterson in Ohio .

Trump rushed into Iran crisis meeting as insider warns'strike within hours' 'Deeply concerned' King Charles backs Andrew investigation after royal's arrest and says the'law must take its course' Model agency boss who'scouted' victims for Epstein was secretly planning to testify against him... only to suddenly change his mind before meeting chillingly similar fate to notorious pedophile The monarchy has survived wars and countless crises... but this is why it may not survive Andrew's arrest - and why the rift at the heart of the family is about to get so much worse: ROBERT JOBSON Widower whose wife set herself on fire after alleged affair with married congressman finally breaks silence to reveal their texts... and heartbreaking video of her death Whereabouts of Andrew's ex-wife and daughters remain unknown as former prince is arrested over public misconduct claims Andrew'pushed through' appointment of Jeffery Epstein's fixer to board of Windsor Castle trust despite opposition Virginia Giuffre's family hail Andrew's arrest and say'he was never a prince' But countless women (and some husbands) are secretly getting it for thrilling sex side effects... risking a truly putrid complication FBI'has names and photos of people who may be masked suspect caught on surveillance video outside Nancy Guthrie's home' How DID Beatrice afford her 20s jet-set lifestyle? US assembles the most aerial firepower since Iraq War as Trump prepares to strike Iran'in just DAYS'... and president is'choosing between two devastating options of attack' The side-effects were unbearable and I swore off the drug forever. This is the simple diet that helped me shed the pounds... and I'm not alone. The Prince and Princess of Wales express support for King Charles' statement after Andrew's arrest Jason Bateman says he quit cocaine and alcohol to ease'tension' in his marriage Humiliating real reason Mia Goth left Shia LaBeouf: What'friends and lovers' are all saying behind his back... after Mardi Gras brawl What happens now Andrew Mountbatten-Windsor has been arrested? I went into early menopause in my 30s.

Spain's Prime Minister Pedro Sanchez gives a speech during the World Governments Summit in Dubai on February 3, 2026. Spain's Prime Minister Pedro Sanchez gives a speech during the World Governments Summit in Dubai on February 3, 2026. The Spanish government has called for an investigation into social media giants X, Meta, and TikTok over their alleged role in producing and spreading AI-generated child sexual abuse material. The Council of Ministers will invoke Article 8 of the Organic Statute of the Public Prosecution Service to request that it investigate the crimes that X, Meta and TikTok may be committing through the creation and dissemination of child pornography by means of their AI, Spanish Prime Minister Pedro Sánchez wrote on X on Tuesday. Sánchez accused the platforms of "attacking the mental health, dignity and rights of our sons and daughters," saying that "the impunity of the giants must end."

Forum members discuss their discomfort with mass deportation efforts, debate how federal agents have interacted with civilians, and complain about their working conditions. Every day, people log in to an online forum for current and former Homeland Security Investigations (HSI) officers to share their thoughts on the news of the day and complain about their colleagues in Immigration and Customs Enforcement (ICE). "ERO is too busy dressing up as Black Ops Commandos with Tactical body armor, drop down thigh rigs, balaclavas, multiple M4 magazines, and Punisher patches, to do an Admin arrest of a non criminal, non-violent EWI that weighs 90 pounds and is 5 foot 2, inside a secure Federal building where everyone has been screened for weapons," wrote one user in July 2025. The forum describes itself as a space for current and prospective HSI agents, "designed for the seasoned HSI Special Agent as well as applicants for entry level Special Agent positions." HSI is the division within ICE whose agents are normally responsible for investigating crimes like drug smuggling, terrorism, and human trafficking.