Police probe Andrew Mountbatten-Windsor over'sex offences': Stunning update on investigation of former prince as officers appeal for potential'victim survivors' to come forward Trump celebrates Stephen Colbert's final show with brutal'no talent' swipe as bitter host takes one last jab at CBS on way out door Trump warns of possible military action in Cuba and says'I'd be happy to do it' as Marco Rubio declares the nation a'US national security threat' Dangerous truth about melatonin side effects... the astonishing dose you SHOULD be taking... and a new natural grocery store alternative hailed by doctors CIA Nostradamus warned Trump about Iran... now he's calling the President's doctors. Dirty secret Hollywood's Cool Girls don't want you to know. Mom-of-two abandons home in Pennsylvania to live on board CRUISE SHIP year-round - and her kids have'zero concept' their life isn't normal This quiet announcement from Prince William was missed by most... but this is why royal insiders tell me it spells disaster for Harry and Meghan's future: RICHARD EDEN White man charged after he was filmed screaming at black female neighbor and using the phrase'You people' Shock moment'slurring' Britney Spears is arrested for DUI after failing sobriety test Astonishing secret list of elite Hollywood liberals conspiring to elect Spencer Pratt revealed to KENNEDY by her LA moles. Suspected Somali fraudster filmed leaping off Minnesota balcony and driving away in luxury Genesis sedan after feds announced they were charging him with alleged $3.3m scam Inside Pizza Hut restaurant that's still EXACTLY like it was in the 90s... complete with checkered tablecloths, arcade and famous buffet Stephen Colbert's final Late Show episode leaves fans unimpressed as Ryan Reynolds leads series of surprise celebrity cameos How Meryl Streep's husband really feels about her secret relationship with Martin Short: Their years of agony... his hard red line... and why she won't divorce him Look away now, Carrie Bradshaw! Fears for Ariana Grande: Insiders lift the lid on Ethan Slater's costly sacrifice... her private nightmares... and what's really keeping them apart Inside the UK's first AI-powered fertility clinic using state-of-the-art technologies to help women get pregnant - as one couple says'artificial intelligence allowed us to hold our baby in our arms' If you were asked to think about artificial intelligence ( AI), visions of killer robots, dodgy chatbots, or deepfakes might spring to mind.

Tune into live broadcasts from your Windows desktop with Trdo, a free and open-source application. I may be old-fashioned, but I prefer actual radio stations to Spotify's algorithms. The best human DJs find music I'd never seek out, and that even the best recommendation system would never point out to me. Even better: If you're good at finding community and public radio stations that appeal to your tastes, there are no commercials. I've found several of my favorite bands in the past few years listening to radio stations like KEXP, Indie XFM, and the various stations offered by SomaFM . It's simple to listen to such stations in your browser, but leaving a tab open just for the radio annoys me.

The delay reportedly came after pressure from big tech leaders. President Trump has postponed the signing an executive order around government AI oversight, CNN reported, saying he didn't like certain aspects of it. Originally, the order would have compelled AI companies to share advanced models with the government ahead of launch to ensure their safety, but it was later watered down to make tech company participation voluntary. The delay reportedly came about due to last-minute pressure from AI industry leaders, including Elon Musk and Mark Zuckerberg, along with former US AI and crypto czar David Sacks. They told Trump that the new system could slow development of AI tech that has become integral to the US economy, anonymous insiders told .

When you purchase through links in our articles, we may earn a small commission. Get lifetime access to AcePDF Converter & Editor for $23.99 with promo code PDF20 PDF tools have a way of turning simple tasks into unnecessary extra steps -- especially when basic features are locked behind subscriptions or usage limits. Right now, though, AcePDF Converter & Editor is available for a one-time payment of $23.99 with promo code PDF20 Whether you're merging pages, converting files to Word or Excel, or extracting text from scanned documents, this software gives you all the PDF power you need in one place. It's fast, reliable, and trusted by individuals and businesses to make daily document workflows way smoother. If PDFs are part of your daily grind, this is an easy one-time investment that will pay for itself the next time you need to edit or convert a document.

Fans react to Stephen Colbert's final show Fans lined up outside the Ed Sullivan Theater one last time for "The Late Show" as Stephen Colbert prepared to host his final episode, marking the end of the franchise's 33-year run on CBS. The network announced last summer that the show would end, calling it "purely a financial decision against a challenging backdrop in late night. Colbert took on the show on September 2015 after David Letterman retired from the role he'd held for 22 years. Rescuers removed the vehicle from near the shoreline and police arrested the driver, who was attempting to use the vehicle's wade mode. As part of the ongoing investigation into the crash that killed 14 people, officials released footage showing the engine detaching during takeoff.

Global affairs chief Chris Lehane wants to tone down the debate over AI's societal impacts--and get states to pass laws that won't derail OpenAI's meteoric rise. Three months ago, OpenAI cofounder Greg Brockman told me his concerns about a mounting public relations crisis facing artificial intelligence companies: Despite the popularity of tools like ChatGPT, an increasingly large share of the population said they viewed AI negatively. Since then, the backlash has only intensified. College commencement speakers are now getting booed for talking about AI in optimistic terms. Last month, someone threw a Molotov cocktail at OpenAI CEO Sam Altman's San Francisco home and wrote a manifesto advocating for crimes against AI executives.

Recently, Deng et al. (2026) proposed Generative Modeling via Drifting (GMD), a novel framework for generative tasks. This note presents an analysis of GMD through the lens of Wasserstein Gradient Flows (WGF), i.e., the path of steepest descent for a functional in the space of probability measures, equipped with the geometry of optimal transport. Unlike previous WGF-based contributions, GMD can be thought of as directly targeting a fixed point of a specific WGF flow. We demonstrate three main results: first, that one algorithm proposed by Deng et al. (2026) corresponds to finding the limiting point of a WGF on the KL divergence, with Parzen smoothing on the densities. Second, that the algorithm actually implemented by Deng et al. (2026) corresponds to a different procedure, which bears some resemblance to the fixed point of a WGF on the Sinkhorn divergence, but lacks certain desirable properties of the latter. Third, the same same idea can be extended to the limiting point of other WGFs, including the Maximum Mean Discrepancy (MMD), the sliced Wasserstein distance, and GAN critic functions.

We study the evolution of hidden-weight spectra in wide neural networks trained by (stochastic) gradient descent. We develop a two-level dynamical mean-field theory (DMFT) that jointly tracks bulk and outlier spectral dynamics for spiked ensembles whose spike directions remain statistically dependent on the random bulk. We apply this framework to two settings: (1) infinite-width nonlinear networks in mean-field/$μ$P scaling and (2) deep linear networks in the proportional high-dimensional limit, where width, input dimension, and sample size diverge with fixed ratios. Our theory predicts how outliers evolve with training time, width, output scale, and initialization variance. In deep linear networks, $μ$P yields width-consistent outlier dynamics and hyperparameter transfer, including width-stable growth of the leading NTK mode toward the edge of stability (EoS). In contrast, NTK parameterization exhibits strongly width-dependent outlier dynamics, despite converging to a stable large-width limit. We show that this bulk+outlier picture is descriptive of simple tasks with small output channels, but that tasks involving large numbers of outputs (ImageNet classification or GPT language modeling) are better described by a restructuring of the spectral bulk. We develop a toy model with extensive output channels that recapitulates this phenomenon and show that edge of the spectrum still converges for sufficiently wide networks.

No feature ranking can be simultaneously faithful, stable, and complete when features are collinear. For collinear pairs, ranking reduces to a coin flip. We prove this impossibility, quantify it for four model classes, resolve it via ensemble averaging (DASH), and machine-verify it with 305 Lean 4 theorems. We characterize the complete attribution design space: exactly two families of methods exist -- faithful-complete methods (unstable, with rankings that flip up to 50% of the time) and ensemble methods like DASH (stable, reporting ties for symmetric features) -- and no method lies outside this dichotomy. The impossibility is quantitative: the attribution ratio diverges as 1/(1-rho^2) for gradient boosting, is infinite for Lasso, and converges for random forests. DASH (Diversified Aggregation of SHAP) is provably Pareto-optimal among unbiased aggregations, achieving the Cramer-Rao variance bound with a tight ensemble size formula. In a survey of 77 public datasets, 68% exhibit attribution instability. Switching to conditional SHAP does not escape the impossibility when features have equal causal effects. The framework includes practical diagnostics -- a Z-test workflow and single-model screening tool -- and has direct consequences for fairness auditing: SHAP-based proxy discrimination audits are provably unreliable under collinearity. The design space theorem, diagnostics, and impossibility are mechanically verified in Lean 4 (305 theorems from 16 axioms, 0 sorry) -- to our knowledge, the first formally verified impossibility in explainable AI.

Protein-protein interactions (PPIs) govern nearly all cellular processes, yet computational methods for identifying binding partners typically produce ranked predictions without mechanistic justification. This creates a fundamental barrier to adoption because biologists cannot assess whether predictions reflect genuine biochemical insight or spurious correlations. We present \textbf{Protein Thoughts}, a framework that reformulates PPI discovery as an interpretable search problem with explicit reasoning. The system decomposes binding evidence into four biologically meaningful signals: sequence similarity reflecting evolutionary relationships, structural complementarity capturing geometric fit, interface balance, and chemical compatibility encoding residue-level interactions. Rather than collapsing these signals into an opaque score, we preserve their individual contributions through a transparent value function that enables both ranking and auditing. To navigate large candidate spaces efficiently, we introduce hypothesis-guided entropy-regularized Tree-of-Thoughts search. A fine-tuned language model generates search directives from embedding-derived features, classifying candidates as high-priority, exploratory, or skippable. These directives condition a Boltzmann policy that balances exploitation with entropy-driven exploration, while hypothesis-aware pruning prevents premature abandonment of promising candidates. For candidates exhibiting score disagreement, hypothesis-conditioned embedding-space flow matching transports protein embeddings toward the binder manifold. On the SHS148k benchmark, Protein Thoughts achieves mean best-binder rank of 11.2 versus 47.7 for an entropic tree search baseline, a 76% improvement, and for binding prediction the trained value function achieves $91.08 \pm 0.19$ Micro-F1, outperforming existing PPI methods on the same dataset.