Watermarking promises a statistical trace of large language model (LLM) use, but real documents, after editing or paraphrasing, rarely arrive as purely human-written or purely machine-generated. This motivates a quantitative question beyond detection: what proportion of a document is generated from a pre-specified watermarked LLM? We study this watermark proportion estimation problem under the Gumbel--max watermarking mechanism, treating the next-token prediction (NTP) distributions as unknown and arbitrary nuisance parameters subject to a non-degeneracy condition. We compare two observation regimes: in the full observation regime, the estimator observes the pseudorandom vector and the selected token at each position; under the more popular setting of pivotal reduction, it observes only a scalar pivot, which follows a one-dimensional Uniform--Beta mixture distribution. Under pivotal reduction, we develop a Laguerre-polynomial estimator and establish a matching information-theoretic lower bound for the sample complexity. For full observation, we introduce an event-counting estimator and show a matching lower bound, yielding a substantially smaller sample complexity. As our results imply, although reducing to pivotal statistics is an elegant and widely used procedure, it is not always sample-efficient for estimating the proportion of watermarks.

We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-$T$ solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier--Stokes, Allen--Cahn, and Cahn--Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.

Online ride-hailing platforms aim to deliver efficient mobility-on-demand services, often facing challenges in balancing dynamic and spatially heterogeneous supply and demand. Existing methods typically fall into two categories: reinforcement learning (RL) approaches, which suffer from data inefficiency, oversimplified modeling of real-world dynamics, and difficulty enforcing operational constraints; or decomposed online optimization methods, which rely on manually designed highlevel objectives that lack awareness of low-level routing dynamics. To address this issue, we propose a novel hybrid framework that integrates large language model (LLM) with mathematical optimization in a dynamic hierarchical system: (1) it is training-free, removing the need for large-scale interaction data as in RL, and (2) it leverages LLM to bridge cognitive limitations caused by problem decomposition by adaptively generating high-level objectives. Within this framework, LLM serves as a meta-optimizer, producing semantic heuristics that guide a low-level optimizer responsible for constraint enforcement and real-time decision execution. These heuristics are refined through a closed-loop evolutionary process, driven by harmony search, which iteratively adapts the LLM prompts based on feasibility and performance feedback from the optimization layer. Extensive experiments based on scenarios derived from both the New York and Chicago taxi datasets demonstrate the effectiveness of our approach, achieving an average improvement of 16% compared to state-of-the-art baselines.

Terrifying stomach cancer explosion sweeps the US: After fitness influencer's shock death, experts reveal subtle early signs that are too often ignored... and lifestyle tweaks that can PREVENT it Actress, 43, announces she is expecting with sweet video after detailing'complicated' journey to motherhood and hope of having third child Trump foe Rosie O'Donnell to replace Jimmy Kimmel as he steps back from his show Deadly secrets of gorgeous California enclave where college girls were killed by a'sneaker'... now experts say they could have been SAVED The other women left devastated by Jelly Roll's divorce: Why his daughter is now'disgusted'... as Bunnie's baby bombshell rocks Nashville The shaming of America's original mommy influencer after tragedy that divided the nation: Bode Miller's wife Morgan breaks cover to reveal agonizing regret that still haunts her since daughter's drowning Trump boasts there's'no limits' to his power and posts bizarre memo by fake historian comparing him to Hitler More young Americans are living with their parents than ever before... and there is a shocking reason behind the boomerang trend I was mortified when my husband always said no to sex. Then I realised the mistake I was making. This is the change that's completely transformed marital love-making in middle age: ALICE SNAPE Revealed: Hero, 24, who saved man's LIFE in dramatic rescue during New York Knicks victory parade after defying cops' orders: 'I'm just another New Yorker' REVEALED: Gavin Newsom steered millions of dollars of donations to nonprofits connected to his wife... as Trump's DOJ probes couple The shingles vaccine could lower dementia risk'by up to a quarter' - but scientists are still puzzled why Farce of Obama's $850m'monstrosity': As clucking liberal elite cheer Barack's grand opening, outraged Chicago locals tell HARRIET ALEXANDER awkward truth about library Why turnips MUST be in your grocery cart if you're trying to lose weight Taco Bell's finally fixes a glaring menu gap - and brings back a fan favorite after years Mom thought popular'natural' health supplement was safer than Xanax. She took it... then never woke up. Don't make the same mistake Mother and child in critical condition after being swallowed into ocean by ANOTHER monstrous California wave... just days after college students were killed by breaker'I was taken from school and trained to fly UFOs with my mind,' claims child genius A former gifted child has come forward with claims that he was removed from public school and secretly trained to develop psychic abilities for military and UFO-related applications.

Gilgo Beach serial killer Rex Heuermann's ex-wife reacts to his sentencing as monster who killed eight women is transferred to new prison to begin life behind bars Boy, three, 'attacked by at least one crocodile' after being'thrown into zoo pit by man with learning difficulties who broke away from carers' - as suspect'not fit for interview' is bailed Jelly Roll stops concert to respond to wife Bunnie XO's bombshell podcast on their divorce Hegseth puts NATO on notice as he launches review of US troops in Europe and blasts allies for'shameful' behavior I was mortified when my husband always said no to sex. Then I realised the mistake I was making. This is the change that's completely transformed marital love-making in middle age: ALICE SNAPE Mom thought popular'natural' health supplement was safer than Xanax. She took it... then never woke up. Don't make the same mistake JD Vance turns on Israeli allies who are criticizing Trump's Iran deal: 'Wake up and smell reality' The other women left devastated by Jelly Roll's divorce... why his daughter is now'disgusted'... and Bunnie XO's one red-line demand before she would agree to the split Joe Biden mumbles to himself and requires stage direction as he aimlessly wanders off at Obama's library debut Tourists run for their lives as gunfire erupts in New York's Times Square as terrified parents drag children to safety Heartbroken family of college girls who drowned dispute account of their final moments before they were swept out to sea as they mourn'responsible and kind' students Oscar-winning director's daughter and her husband's deaths'medically related' as cops give grim update after couple were found in SUV on California highway Furious woke woman storms out of restaurant because customers were singing National Anthem ...and vows never to return A bold new experiment to streamline how Americans buy new cars... and auto dealerships are already scared Secret White House blacklist leaked by insider: 'Worst' influencers named and shamed... as foul-mouthed backstabbing erupts Watch horrifying drone video that follows woman's plunge to death after bungee team threw her from bridge without rope Bill Clinton's VERY cozy moment with Michelle while Hillary looks the other way... and the best UNSEEN moments from Obama public library opening Farce of Obama's $850m'monstrosity': As clucking liberal elite cheer Barack's grand opening, outraged Chicago locals tell HARRIET ALEXANDER awkward truth about library Humiliating new joke about Trump that's the talk of Washington... as White House moles tell me there's more to this story than meets the eye: MARK HALPERIN Humanoid robot is spotted BEGGING on a street in China - claiming it has'no money to recharge' READ MORE: China unveils the world's first self-driving TOILET While many people worry that robots are coming to take their jobs, one unlucky bot seems to have fallen on hard times.

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'We had to get out of the way': The backlash over delivery robots The first time Chicago resident John Roberts saw a delivery robot trundling down the sidewalk on his street he was impressed. I actually thought they were kind of neat - it felt futuristic, he says. But his attitude started to change when, soon after, he was out for a walk with his family. As another robot approached, they found themselves having to dodge it. To us it felt a little off - the fact that we were on the one strip reserved for walking, and we were having to get out of the way, says Roberts.

Large language models (LLMs) are increasingly capable of processing long inputs and locating specific information within them, as evidenced by their performance on the Needle in a Haystack (NIAH) test. However, while models excel at recalling surprising information, they still struggle to identify clearly omitted information. We introduce AbsenceBench to assesses LLMs' capacity to detect missing information across three domains: numerical sequences, poetry, and GitHub pull requests. AbsenceBenchasks models to identify which pieces of a document were deliberately removed, given access to both the original and edited contexts. Despite the apparent straightforwardness of these tasks, our experiments reveal that even state-of-the-art models like Claude-3.7-Sonnet

We study the statistical and computational limits of learning bounded linear operators between Sobolev spaces from noisy input-output data. In wavelet coordinates, the problem is recast as an infinite-dimensional matrix regression problem with a heterogeneous two-sided multiscale structure. We establish minimax rates under Sobolev operator-norm loss and construct a finite-resolution blockwise least-squares estimator attaining these rates. The analysis reveals a nonuniform local estimation difficulty across scales, which can be exploited algorithmically: by assigning scale-adaptive sample sizes, the estimator achieves the optimal computational cost among dense least-squares implementations.

Online ride-hailing platforms aim to deliver efficient mobility-on-demand services, often facing challenges in balancing dynamic and spatially heterogeneous supply and demand. Existing methods typically fall into two categories: reinforcement learning (RL) approaches, which suffer from data inefficiency, oversimplified modeling of real-world dynamics, and difficulty enforcing operational constraints; or decomposed online optimization methods, which rely on manually designed high-level objectives that lack awareness of low-level routing dynamics. To address this issue, we propose a novel hybrid framework that integrates large language model (LLM) with mathematical optimization in a dynamic hierarchical system: (1) it is training-free, removing the need for large-scale interaction data as in RL, and (2) it leverages LLM to bridge cognitive limitations caused by problem decomposition by adaptively generating high-level objectives. Within this framework, LLM serves as a meta-optimizer, producing semantic heuristics that guide a low-level optimizer responsible for constraint enforcement and real-time decision execution. These heuristics are refined through a closed-loop evolutionary process, driven by harmony search, which iteratively adapts the LLM prompts based on feasibility and performance feedback from the optimization layer. Extensive experiments based on scenarios derived from both the New York and Chicago taxi datasets demonstrate the effectiveness of our approach, achieving an average improvement of 16% compared to state-of-the-art baselines.