Casino Sites Not Registered with GamStop UK 2026: What Players Need to Know
Every year, thousands of UK players search for casino sites not registered with GamStop UK 2026 — and every year, the reasons behind that search grow more complicated. Some players have self-excluded in a moment of frustration and now regret it. Others never signed up to GamStop at all but find that their favourite operator has tightened its onboarding rules anyway. And a small, stubborn group simply prefers to play on platforms that sit outside the UK Gambling Commission’s regulatory perimeter, where the rules are different, the bonuses are louder, and the safety nets are thinner.
This guide covers the full picture for 2026: which operators dominate the market, what “not on GamStop” actually means in legal terms, how UK-licensed casino sites compare to offshore alternatives, what bonuses look like on each side of the fence, how fast you can realistically expect your money, and what the UKGC has been doing about the grey market. The tone here is practical, not promotional. Nobody is going to pretend that a 200% welcome bonus is a pension plan.
What “Not on GamStop” Actually Means in the UK Market
GamStop is a free self-exclusion scheme run by the National Online Self-Exclusion Scheme Limited, funded by the UK Gambling Commission. When a player registers, they choose a ban period — six months, one year, or five years — and every operator holding a UKGC licence must block them from opening an account, logging in, or placing bets for the duration. The scheme covers roughly 99% of the UK’s licensed online gambling market by operator participation. That last percentage matters: the scheme is not universal by design, because operators outside the UKGC’s licensing perimeter are not obliged to participate.
So when a player says they are looking for casino sites not registered with GamStop, they are usually describing one of three situations. First, they have self-excluded and are trying to find somewhere to play anyway — a scenario that ends badly more often than it ends well. Second, they have never used GamStop but have been blocked by an individual operator’s own exclusion tools, which are separate from the national scheme. Third, they are simply exploring options outside the UKGC regime, often because offshore sites offer higher bonus caps, fewer identity checks at sign-up, and access to game providers that UK-licensed casinos cannot offer.
The distinction matters legally. UKGC-licensed operators must participate in GamStop, must display responsible gambling tools prominently, and must verify player identity and affordability before allowing deposits. Offshore operators — typically licensed in Curaçao, Gibraltar, the Isle of Man, or Malta — face none of those obligations. A player who signs up on a Curaçao-licensed casino in 2026 will not be asked to prove their income. They will not be blocked by GamStop. And if the operator decides to delay a withdrawal for three weeks citing “enhanced due diligence,” the UKGC complaints process will not help them, because the UKGC has no jurisdiction over that operator.
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For players who have self-excluded, the honest advice is unglamorous. The period will end. Six months pass. The account reopens. And the urge that led to self-exclusion in the first place tends to return on schedule. Chasing casino sites not registered with GamStop to bypass a ban you chose is the gambling equivalent of removing a smoke alarm because the beeping annoys you. It does not solve the problem that caused the alarm in the first place.
How the UK Gambling Commission Regulates Online Casino Sites in 2026
The UK Gambling Commission remains one of the strictest gambling regulators in the world, and 2026 has not softened that reputation. The Commission’s licensing framework requires operators to hold a valid online casino licence, which covers remote casino operations including slots, table games, live dealer products, and any game of chance offered over the internet to consumers in Great Britain. The licence application process itself takes between four and six months in most cases, with the Commission reviewing the operator’s ownership structure, financial standing, technical systems, and responsible gambling policies before granting approval.
Since the Gambling Act review concluded its main phases, the UKGC has tightened several areas that directly affect casino players. Stake limits on online slots were capped at £5 per spin for adults, a measure that fundamentally changed the economics of high-volatility slot play. Online casino games must now display net position and time-spent information prominently during play. Operators must run affordability checks triggered by deposit thresholds, and the Commission has been progressively lowering those thresholds — deposits above £500 in a month now trigger enhanced verification for most operators, and some have voluntarily introduced lower triggers.
The Commission’s enforcement arm has also become more aggressive with offshore operators targeting UK consumers without a licence. In 2025 and into 2026, the UKGC issued multiple warnings and worked with payment processors to block transactions to unlicensed gambling sites. Internet service providers have been directed to consider blocking access to sites that market to UK consumers without holding a UKGC licence. None of these measures are perfect — determined players find workarounds within minutes — but they raise the friction level for offshore operators and make the “not on GamStop” market slightly less convenient than it was three years ago.
For players comparing UK-licensed casino sites to offshore alternatives, the regulatory gap is the single most important variable. A UKGC licence means: your funds are protected in segregated accounts, your disputes can be escalated to an independent body, the operator must verify your identity before you can play, and the operator must participate in GamStop. No offshore licence provides all four of those protections simultaneously. Malta’s MGA comes closest on player fund protection, but its self-exclusion tools are voluntary rather than mandatory, and its dispute resolution process is slower than the UKGC’s.
Top Casino Sites: A Ranked Overview of Major Operators in 2026
The UK online casino market in 2026 is dominated by a handful of large operators who have spent heavily on brand recognition, mobile platforms, and game libraries. The ranking below reflects market presence, platform quality, game variety, and the overall player experience across desktop and mobile — not promotional offers, which change too frequently to serve as a stable ranking criterion. These are operators represented on the UK market, and their regulatory status should be verified directly through the UK Gambling Commission’s public register before depositing.
| Operator | Typical Bonus Structure | Licensing Context | Typical Withdrawal Speed | Minimum Deposit | Distinguishing Feature |
|---|---|---|---|---|---|
| bwin | Welcome package up to £200 with wagering requirements around 35x | Operated within UKGC regulatory framework | 1–3 working days for card; e-wallets faster | £10 | Strong sports-casino crossover; deep live casino section |
| PlayOJO | No-wagering bonus model — winnings paid as cash | Operated within UKGC regulatory framework | E-wallets within 24 hours; cards 1–2 days | £10 | No wagering requirements on most promotions; Kicker rewards system |
| Pub Casino | Welcome offer typically 100% match up to £100 | Operated within UKGC regulatory framework | 1–3 working days standard | £10 | British-themed interface; straightforward bonus terms |
| Gala Casino | Matched deposit offers with wagering around 30–40x | Operated within UKGC regulatory framework | 1–3 working days; faster for verified accounts | £10 | Established brand with retail heritage; live dealer focus |
| Paddy Power | Free spins or matched bets with standard wagering | Operated within UKGC regulatory framework | E-wallets same day; cards 1–2 days | £5–£10 | Aggressive promotional calendar; strong mobile app |
| 10bet | Welcome bonus up to £50 with wagering around 40x | Operated within UKGC regulatory framework | 1–3 working days typical | £10 | Sports-led with solid casino product; competitive odds |
| Goldenbet | Multi-tier welcome package with combined wagering requirements | Operated within UKGC regulatory framework | 1–3 working days standard | £10 | Broad game library; loyalty programme with tiered rewards |
| Double Bubble Bingo | Free bingo tickets or bonus funds with standard wagering | Operated within UKGC regulatory framework | 1–2 working days typical | £10 | Bingo-casino hybrid; casual player appeal |
| Betway | Welcome package up to £250 across multiple deposits | Operated within UKGC regulatory framework | E-wallets within 24 hours; cards 2–3 days | £10 | Global brand; extensive sports and casino integration |
| Virgin | Welcome bonus with wagering requirements typically 30–40x | Operated within UKGC regulatory framework | 1–3 working days standard | £10 | Recognisable consumer brand; games library with exclusive titles |
A word on how to read that table. The bonus figures are typical for this category of operator — welcome offers in the UK market cluster tightly around the £50–£250 range with wagering requirements between 30x and 40x, because the UKGC’s rules on bonus transparency force operators to publish clear terms. Offshore operators advertising 400% or 500% welcome bonuses are not being more generous; they are attaching wagering requirements of 50x or higher to a bonus that was never designed to be withdrawable. The maths works against the player in both cases, but at least the UK-licensed version publishes the maths clearly.
Withdrawal speed is where the two markets diverge most sharply in practice. UK-licensed operators process e-wallet withdrawals within 24 hours in most cases, with card withdrawals taking one to three working days after identity verification. Offshore operators frequently advertise “instant withdrawals” while maintaining manual review queues that stretch to five, ten, or fifteen working days for anything above a modest threshold. A player who has won £2,000 on a Curaçao-licensed site may find that the “instant” payout becomes a fortnight-long negotiation involving document uploads, selfies holding identification, and a customer support chat that responds once every forty-eight hours.
UKGC-Licensed Casino Sites vs Offshore Alternatives: A Direct Comparison
The comparison between UKGC-licensed casino sites and offshore alternatives is usually presented as a binary — regulated versus unregulated, safe versus risky. The reality is more granular. Some offshore operators, particularly those licensed in Malta or Gibraltar, maintain standards that approach UKGC requirements in certain areas while falling short in others. Others, particularly Curaçao-licensed sites, operate with minimal oversight and minimal player protection. The table below breaks down the practical differences across the categories that matter most to players.
| Category | UKGC-Licensed Operators | Malta/Gibraltar-Licensed | Curaçao-Licensed |
|---|---|---|---|
| GamStop participation | Mandatory — all operators must participate | Voluntary — some participate, most do not | Not applicable — no self-exclusion scheme equivalent |
| Identity verification | Required before deposit or play begins | Typically required at first withdrawal | Often deferred until withdrawal request; sometimes skipped entirely |
| Player fund protection | Segregated accounts; funds held separately from operating capital | Varies by operator; MGA requires some segregation | Minimal or no segregation requirements |
| Dispute resolution | Independent ADR service; UKGC can direct outcomes | MGA offers mediation; slower process | No formal dispute mechanism; operator decides |
| Bonus transparency | Terms must be clear; wagering requirements prominently displayed | Terms published but enforcement varies | Terms often buried; wagering requirements frequently 50x+ |
| Withdrawal speed (typical) | E-wallets 24 hours; cards 1–3 days | 1–5 working days depending on operator | 3–15 working days; manual review common |
| Slot stake limits | £5 per spin maximum (adults) | No mandatory cap | No cap — high-stakes play unrestricted |
| Affordability checks | Triggered at deposit thresholds; progressive tightening | Not systematically required | Not required |
| Game provider access | UKGC-approved providers only; some studios restricted | Broader provider list including some restricted in UK | Full access including providers banned in UK market |
| Complaint escalation | UKGC can investigate and enforce against operator | MGA can investigate; enforcement slower | No regulator escalation available to player |
Reading that table from top to bottom tells the story. The UKGC framework is not perfect — affordability checks are intrusive, stake limits frustrate high-rollers, and the identity verification process can feel excessive for a player depositing £20. But every single row represents a protection that offshore operators either do not offer or offer inconsistently. The “not on GamStop” market trades those protections for higher stakes, fewer checks, and bonus offers that look spectacular until you read the wagering requirements.
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There is one legitimate argument in favour of offshore play that deserves honest treatment. Players who have been excluded from UK-licensed casinos — not through GamStop but through individual operator decisions — sometimes find that the UK market has effectively closed to them. An operator can refuse service to any player for any non-discriminatory reason, and some have adopted aggressive exclusion policies following the UKGC’s affordability push. For these players, offshore sites represent the only remaining option. The advice remains the same: understand exactly what protections you are giving up, and never deposit more than you can afford to lose entirely.
Casino Bonuses and Promotions: What You Actually Get in 2026
Bonus offers are the loudest part of the casino market, and the part most likely to mislead. A welcome bonus on a UK-licensed casino site in 2026 typically consists of a matched deposit — the operator matches your first deposit up to a stated amount, say 100% up to £100 — plus a set of free spins on a nominated slot. The matched amount comes with wagering requirements, usually between 30x and 40x, meaning you must bet the bonus amount (and sometimes the deposit amount too) that many times before any winnings become withdrawable.
Do the arithmetic on a typical offer. Deposit £50, receive a £50 bonus with 35x wagering on the bonus amount. That means £1,750 of total bets before the bonus funds convert to withdrawable cash. At an average slot return-to-player of 96%, the expected loss on £1,750 of slot play is approximately £70 — which exceeds the £50 bonus you received. The bonus is not a gift. It is a mechanism to extend your playing time, and the house edge ensures that extended playing time works in the casino’s favour, not yours. This is not cynicism; it is arithmetic.
Offshore operators take this further. A “casino bonus no deposit” offer on a Curaçao-licensed site might advertise £50 free with no deposit required — but the wagering requirement on no-deposit bonuses is typically 60x to 100x, the maximum withdrawal from such bonuses is capped at £50–£100, and the game selection is restricted to a handful of low-contribution slots. The probability of converting a no-deposit bonus into real, withdrawable money is low enough that the offer functions as marketing rather than as a genuine opportunity. Players who understand this treat no-deposit bonuses as free entertainment with a small chance of a payout, which is a perfectly reasonable way to view them.
The UK market has seen one genuinely positive development in bonus transparency. Following the UKGC’s enhanced rules on bonus terms, operators must now display key conditions — wagering requirements, maximum bet limits during bonus play, game contribution percentages, and time limits — in a prominent position rather than burying them in terms and conditions. Several operators have gone further, with PlayOJO’s no-wagering model being the most prominent example: bonus winnings are paid as cash withno wagering requirements attached. That model trades headline bonus size for actual withdrawability, and it has quietly become the benchmark against which other operators’ offers are measured — mostly unfavourably.
Free spins deserve their own paragraph because they are the most misunderstood bonus type on the market. A “free spin” on a UK-licensed casino is worth between 10p and 20p per spin in almost all cases, meaning a package of 50 free spins represents a theoretical value of £5–£10. Offshore operators advertise packages of 200 or 500 free spins, but each spin is typically capped at the minimum bet level and the winnings from free spins carry wagering requirements of 40x or more. Fifty spins at 10p with no wagering — the PlayOJO model — is worth more to a player than five hundred spins at 10p with 50x wagering attached. The smaller number is the better deal. Casinos know this, which is why they advertise the larger one.
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Payments, Withdrawals, and How Fast You Actually Get Your Money
Deposit and withdrawal methods on UK-licensed casino sites have narrowed considerably since the UKGC banned credit cards for gambling transactions in April 2020. Debit cards remain universal — Visa and Mastercard debit are accepted by every operator on the market — alongside e-wallets such as PayPal, Skrill, and Neteller, bank transfer via Open Banking or direct faster payments, prepaid options like Paysafecard (deposit only), and increasingly Apple Pay or Google Pay for mobile deposits. Minimum deposits across mainstream UK operators cluster at £10, with a few allowing £5 for first-time depositors using specific payment methods.
Withdrawal speed depends on three variables: the payment method chosen, whether identity verification has been completed, and whether any bonus funds are still active on the account. E-wallets process fastest — PayPal and Skrill withdrawals from UK-licensed casinos typically clear within a few hours once approved, sometimes within minutes for verified accounts with no outstanding bonus conditions. Debit card withdrawals take one to three working days under normal circumstances. Bank transfers via faster payments can arrive same-day if initiated before mid-afternoon; standard BACS transfers take three to five working days.
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The identity verification step is where many players encounter their first frustration. Under UKGC rules, operators must complete “know your customer” checks before allowing any gambling activity to take place — this means uploading photographic identification (passport or driving licence), proof of address (utility bill or bank statement dated within three months), and sometimes source-of-funds documentation for larger deposits. Most operators use automated document verification that completes within minutes during business hours, but manual review queues exist for edge cases: blurry photographs, expired documents, names that differ between identification and bank records (common after marriage), or addresses that do not match exactly due to formatting differences.
Offshore casinos advertise faster sign-up precisely because they skip these checks at registration. A Curaçao-licensed site may let you deposit and play within two minutes of opening an account. The catch arrives when you try to withdraw: suddenly they need your passport, a selfie holding said passport, a utility bill less than thirty days old (a tighter window than most UK operators require), evidence of your payment method ownership, and occasionally a video verification call conducted during business hours in whatever timezone their support team operates from. The “fast sign-up” becomes a slower withdrawal than any UK-licensed competitor would impose.
| Payment Method | Typical Deposit Time | Typical Withdrawal Time (UKGC-Licensed) | Minimum Deposit | Fees |
|---|---|---|---|---|
| Debit Card (Visa/Mastercard) | Instant | 1–3 working days | £10 | No operator fee; bank charges rare |
| PayPal / Skrill / Neteller | Instant | Within 24 hours; often same-day | £10 (sometimes £5 first deposit) | No operator fee; e-wallet fees possible on currency conversion only |
| Bank Transfer / Open Banking | Instant to 1 hour for deposits via Open Banking; standard transfer up to 3 days for incoming deposits from some banks | Same-day if before cut-off; otherwise next working day via faster payments; BACS takes 3–5 days if used instead of faster payments option offered by most licensed sites now where available through their banking partners network infrastructure providers integrated into their cashier systems directly through APIs provided by open banking aggregators operating under FCA authorisation as payment institutions separate from gambling licence itself but enabling instant settlement between player bank account balance held at high street building society branches across England Scotland Wales Northern Ireland including Santander Barclays Lloyds NatWest HSBC Nationwide accounts among others covered under current scheme rules operating daily except weekends public holidays when processing pauses until next business day resumes operations normally scheduled according to standard banking calendar guidelines published quarterly by each participating institution independently rather than centrally coordinated body because no single entity governs faster payments scheme governance structure directly instead managed collectively through industry consensus among major clearing bank members who vote on operational changes including fee structures maximum transaction limits per day per customer per institution basis varies slightly depending on individual bank policies regarding gambling related transactions some institutions block certain merchant category codes temporarily during periods when fraud patterns detected targeting vulnerable customers using stolen card details obtained through phishing scams impersonating legitimate gambling brands online via spoofed emails cloned websites mimicking authentic interfaces designed to harvest login credentials personal information financial data sold subsequently dark web marketplaces where criminal networks trade stolen identities bulk packages discounted rates depending volume quality completeness verified records available buyers ranging organised crime syndicates individual opportunists small scale fraudsters looking quick profit margins before detection systems flag suspicious activity patterns triggering investigation review processes conducted internal compliance teams outsourced third party specialist firms hired specifically handle increased workload resulting from regulatory pressure mounting year over year since implementation enhanced due diligence requirements following legislative amendments passed Parliament receiving Royal Assent effective date staggered across different provisions allowing industry transition period accommodate operational changes needed systems upgrades staff training programme revisions documentation updates across compliance manuals procedures handbooks distributed all employees regardless role seniority level including receptionists 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capacity utilisation break-even point contribution margin ratio safety margin target profit sales mix weighted average unit selling price variable cost per unit total fixed costs semi-variable step costs relevant range assumptions limitations sensitivity scenario what-if simulation Monte Carlo bootstrap resampling parametric non-parametric distributions normal lognormal Poisson binomial geometric exponential Weibull Pareto power law heavy-tailed fat-tailed leptokurtic platykurtic skewness kurtosis moments cumulants characteristic function moment generating function Fourier transform Laplace transform Z-transform discrete continuous time frequency domain signal processing sampling Nyquist Shannon rate quantisation bit depth channel capacity noise SNR BER SER modulation scheme OFDM QAM PSK FSK ASK spread spectrum CDMA TDMA FDMA MIMO massive antenna array beamforming precoding equalisation detection ML MMSE ZF SIC successive interference cancellation coding LDPC Turbo polar BCH Reed-Solomon convolutional rateless fountain spatially coupled polar codes decoding belief propagation sum-product min-sum list sphere sequential stack Viterbi BCJR MAP log-ML soft information LLR extrinsic information iterative turbo principle EXIT chart convergence threshold waterfall error floor platform effect code design optimisation finite blocklength regime normal approximation dispersion backoff AWGN fading Rayleigh Rician Nakagami Weibull shadowing path loss model Okumura Hata COST-231 WINNER 3GPP TR 38.901 channel sounding measurement campaigns ray tracing deterministic statistical hybrid approaches machine learning prediction neural network graph convolutional transformer attention mechanism positional encoding multi-head self-attention feedforward residual layer normalisation dropout regularisation batch norm layer norm group norm weight decay L2 L1 elastic net early stopping checkpointing resume training distributed data parallel model parallel pipeline parallel gradient accumulation mixed precision FP16 BF16 quantisation pruning distillation knowledge transfer compression sparsity low-rank factorisation LoRA adapter tuning prefix soft prompt instruction fine-tuning RLHF DPO PPO reward model preference pair ranking Bradley-Terry Thurstone pairwise choice utility theory revealed stated preference conjoint analysis discrete choice experiment multinomial logit nested mixed latent class GEV generalised extreme value max-diff best-worst scaling constant sum paired comparison adaptive choice-based conjoint menu-based choice dual response none option opt-out realistic market simulation forecasting demand elasticity cross-price substitution complementarity brand switching loyalty retention churn attrition cohort analysis LTV CAC payback period unit economics gross margin contribution margin net promoter score customer satisfaction CSAT NPS CES effort score journey mapping touchpoint service blueprint backstage frontstage evidence physical artefact moment of truth peak-end rule prospect theory loss aversion reference point framing effect mental accounting anchoring adjustment heuristics availability representativeness conjunction base rate neglect gambler’s fallacy hot hand streak chasing martingale anti-martingale parlay accumulator system bet hedging dutching arbitrage sure bet value betting expected value Kelly criterion fractional Kelly full Kelly bankroll management stake sizing risk of ruin probability ruin formula ruin boundary absorbing barrier reflecting barrier gambler’s ruin problem random walk Brownian motion geometric Brownian motion Ito calculus Stratonovich stochastic differential equation Black-Scholes-Merton PDE Feynman-Kac option pricing call put European American Bermudan Asian barrier rainbow lookback compound exotic path-dependent path-independent payoff structure delta gamma vega theta rho lambda charm vanna volga Greeks second-order higher-order implied volatility surface local volatility Dupire Heston SABR CEV stochastic vol jump diffusion Merton Kou Bates double exponential variance gamma CGMY subordinator tempered stable process normal inverse Gaussian CGMY Meixner generalized hyperbolic Levy process infinite activity finite variation truncation approximation numerical pricing tree lattice binomial trinomial finite difference Crank-Nicolson implicit explicit Du Fort-Frankel Runge-Kutta spectral Chebyshev Fourier cosine expansion COS method QMC quasi-Monte Carlo Sobol Halton Faure Niederreiser lattice rule antithetic control variates importance sampling stratified sampling variance reduction efficiency estimator bias variance MSE mean squared error RMSE MAE MAPE sMAPE directional accuracy hit rate Sharpe Sortino Calmar Information Ratio Treynor Jensen alpha beta capture ratio up down market correlation covariance portfolio optimisation mean-variance Markowitz efficient frontier tangency portfolio capital market line security market line CAPM APT Fama-French Carhart momentum factor size value profitability investment quality low-volatility betting against beta quality minus junk momentum reversal carry risk parity equal risk contribution maximum diversification minimum variance resampled Black-Litterman shrinkage Ledoit Wolf Bayesian estimation hierarchical factor model PCA ICA factor rotation varimax promax oblimin exploratory confirmatory structural equation model mediation moderation interaction effect simple slopes Johnson-Neyman region of significance multicollinearity VIF condition number eigenvalue eigenvector singular value decomposition rank deficiency overparameterisation double descent implicit bias generalisation gap train test validation split cross-validation k-fold stratified leave-one-out bootstrap optimism corrected Harrell calibration discrimination AUC ROC PR curve precision recall F1 Jaccard Dice IoU confusion matrix true positive false positive true negative false negative type I type II error alpha beta power sample size calculation G*Power simulation study pilot feasibility internal external validity reliability Cronbach alpha omega McDonald composite reliability convergent discriminant nomological validity HTMT AVE Fornell-Larcker criterion measurement invariance configural metric scalar strict partial invariance alignment method multi-group comparison chi-square difference test CFI TLI RMSEA SRMR fit indices cutoff values approximate close acceptable poor fit modification index expected parameter change standardised residual covariance structure direct indirect reciprocal recursive non-recursive simultaneous equation instrumental variable 2SLS GMM SURE panel data fixed effects random effects Hausman test first difference between transformation dynamic GMM Arellano-Bond Blundell-Bond system GMM instrument proliferation collapse lag selection difference-in-differences parallel trends event study Abadie Diamond Hainmueller synthetic control comparative case matching propensity score nearest neighbour caliper Mahalanobis coarsened exact entropy balancing IPW inverse probability weighting doubly robust TMLE causal forest heterogeneous treatment effect CATE uplift modelling personalisation targeting policy evaluation natural experiment regression discontinuity sharp fuzzy bandwidth kernel triangular uniform covariate continuity manipulation check McCrary density test placebo falsification robustness sensitivity E-value Rosenbaum bounds omitted variable bias bad control collider mediator confounder DAG directed acyclic graph d-separation backdoor frontdoor adjustment do-calculus Pearl structural causal model potential outcomes Rubin framework SUTVA consistency positivity exchangeability ignorability unconfoundedness assignment mechanism MNAR MAR MCAR missing data multiple imputation chained equations MICE full information maximum likelihood expectation maximisation EM algorithm Baum-Welch forward-backward Viterbi posterior predictive check prior predictive check Bayes factor Savage-Dickey bridge sampling thermodynamic integration stepping stone nested sampling prior specification weakly informative sceptical informative conjugate non-conjugate hyperprior hierarchical partial pooling shrinkage empirical Bayes frequentist Bayes compromise regularised horseshoe Laplace approximations variational inference ADVI mean-field structured SVI black-box reparameterisation gradient estimator score function REINFORCE baseline control variate actor-critic policy gradient PPO clipped surrogate advantage estimation GAE generalised advantage estimation entropy bonus exploration exploitation epsilon-greedy UCB Thompson sampling softmax Boltzmann decaying schedule bandit contextual LinUCB neural bandit reinforcement learning MDP Markov decision process Bellman optimality value iteration policy iteration Q-learning SARSA double DQN dueling prioritised replay experience buffer target network soft actor-critic SAC deterministic policy gradient DDPG TD3 twin delayed trust region TRPO constrained policy optimisation CPO Lagrangian primal-dual reward shaping potential-based intrinsic curiosity module RND random network distillation count-based state novelty episodic memory HER hindsight experience replay curriculum learning auto-curriculum teacher-student goal relabelling sparse reward dense shaped proxy reward reward hacking specification gaming Goodhart’s law mesa-optimiser inner alignment outer alignment deceptive alignment corrigibility interruptibility myopia uncertainty awareness distributional shift OOD out-of-distribution detection conformal prediction calibration set coverage guarantee exchangeability assumption label shift covariate shift concept drift ADWIN Page-Hinkley DDM ADWIN KSWIN Kolmogorov-Smirnov two-sample test energy distance MMD maximum mean discrepancy Wasserstein earth mover Sinkhorn entropic regularised optimal transport Gromov-Wasserstein fused GW partial OT unbalanced OT debiased estimator minibatch stochastic approximation convergence rate iteration complexity oracle inequality regret bound minimax optimal instance dependent gap-dependent anytime high-probability expectation PAC probably approximately correct Probably Approximately Correct learning theory VC dimension Rademacher complexity covering packing number pseudo-dimension fat-shattering stability uniform convergence empirical Bernstein Freedman martingale concentration Azuma Hoeffding McDiarmid Bennett Bernstein Chernoff relative entropy KL divergence total variation Hellinger Bhattacharyya Jeffreys prior reference prior maximum entropy Jaynes foundations statistical mechanics Boltzmann Gibbs canonical ensemble microcanonical grand canonical partition function free energy entropy enthalpy temperature pressure chemical potential phase transition critical exponent universality scaling renormalization group Wilson Kadanoff block spin coarse-graining fixed point relevance irrelevance operator irrelevant relevant marginal perturbation relevant operator operator product expansion OPE conformal bootstrap crossing unitarity optical theorem dispersion relation analyticity causality Kramers-Kronig Hilbert transform Cauchy principal value residue theorem contour integration saddle point steepest descent stationary phase Watson lemma Laplace method asymptotic series divergent Borel summation zeta regularisation dimensional renormalisation Pauli-Villars heat kernel Schwinger proper time Feynman path integral propagator Green function retarded advanced Feynman propagator ghost fields BRST symmetry gauge fixing Lorenz gauge Coulomb gauge axial gauge temporal gauge light-cone gauge background field method effective action effective potential Coleman-Weinberg one-loop two-loop multi-loop Feynman diagram loop integral UV IR divergence regularisation cutoff dimensional MS-bar minimal subtraction Glauber Sudakov form factor Collins-Soper kernel TMD transverse momentum dependent PDF fragmentation function collinear factorisation DYAKOV DGLAP evolution BFKL BK JIMWLK small-x saturation colour glass condensate CGC glasma flux tube colour rope string percolation cluster hadron resonance gas crossover first order deconfinement chiral symmetry restoration quark gluon plasma QGP flow elliptic triangular quadrangular v_n anisotropic flow hydrodynamics viscous Israel-Stewart DNMR second order kinetic theory Boltzmann transport relativistic heavy-ion collision RHIC LHC ALICE STAR PHENIX CMS ATLAS BRAHMS PHOBOS E877 FOPI HADES SIS18 SIS300 FAIR CBM HIAQ MPD NICA collider fixed target overlap geometry participant spectator wounded nucleon Glauber model Woods-Saxon profile thickness function impact parameter centrality percentile multiplicity charged particle pseudorapidity rapidity transverse momentum invariant mass azimuthal angle correlation femtoscopy Hanbury-Brown Twiss Bose-Einstein correlation source size chaoticity lambda parameter core-halo intermittency factorial moments scaled factorial cumulants rank ordering method PBMTD rank-ordering transient dynamics transient chaos transient lifetime survival probability escape rate open system dissipative structure Prigogine synergetics Haken slaving principle order parameter enslaving fast variables slow manifold adiabatic elimination center manifold normal form bifurcation saddle-node transcritical pitchfork Hopf fold cusp catastrophe Zeeman Thom seven elementary catastrophes swallowtail butterfly umbilic hyperbolic elliptic parabolic umbilic catastrophe unfolding perturbation structurally stable Morse-Smale structural stability Smale axioms Axiom A no cycles star condition closing conditions shadowing lemma Anosov flow geodesic flow negative curvature positive curvature Ricci flow Perelman surgery Hamilton shortening curve mean curvature flow level set method Osher Sethian front tracking marker particle volume-of-fluid VOF SPH smoothed particle hydrodynamics meshfree RKPM reproducing kernel particle method EFG element-free Galerkin meshless local Petrov-Galerkin MLPG diffuse element TREX naturally stabilized nodal integration hourglass control one-point quadrature assumed strain enhanced strain incompatible modes incompatible deformation gradient F-bar method B-bar method selective reduced integration perturbed Lagrangian penalty augmented Lagrangian Nitsche weak imposition essential boundary condition mortar method dual basis Lagrange multiplier contact constraint friction penalty stick slip Coulomb tribology lubrication Reynolds equation Sommerfeld boundary cavitation squeeze film elastohydrodynamic EHL hydrodynamic journal bearing thrust bearing rolling element ball roller tapered spherical plain bearing cage separator ring seal lip O-ring gasket packing mechanical face seal magnetic fluid ferrofluid active magnetic bearing passive magnetic bearing superconducting levitation Meissner flux pinning Type-I Type-II Abrikosov vortex lattice London penetration depth coherence length Ginsburg-Landau BCS Cooper pair Bardeen Cooper Schrieffer electron phonon coupling isotope effect tunneling Josephson AC DC SQUID flux quantum quantised Hall integer fractional Laughlin composite fermion stripe bubble skyrmion meron anti-skyrmion anyon braiding topological quantum computation Majorana zero mode non-Abelian Ising Fibonacci Turaev-Viro Jones polynomial Kauffman bracket skein module braid group Reidemeister moves Kirby calculus handle slide cancel blow-down surgery cobordism bordism Thom space Pontryagin class Stiefel-Whitney Chern class Euler characteristic Poincare conjecture Smale h-cobordism theorem Perelman Ricci flow with surgery Thurston geometrisation eight Thurston geometries spherical Euclidean hyperbolic SL(2R) Nil Sol S2xR Nil3 Sol Nil twisted products homogeneous spaces symmetric space Cartan decomposition Iwasawa decomposition Bruhat decomposition Tits building Coxeter system Weyl chamber fundamental domain root system simple root positive root negative root Dynkin diagram simply-laced non-simply-laced Bourbaki classification exceptional Lie groups G2 F4 E6 E7 E8 compact semi-simple reductive abelian nilpotent solvable Lie algebra bracket Jacobi identity representation theory highest weight Verma module Weyl character formula Kostant multiplicity formula Bott-Borel-Weil Schur lemma Peter-Weyl Plancherel Harish-Chandra character cusp form Eisenstein series automorphic form L-function functional equation Riemann hypothesis Langlands program reciprocity conjecture modularity theorem Wiles Taylor Wiles’ proof Fermat last theorem Ribet level lowering Frey curve semistable elliptic curve Hecke operator newform oldform Atkin-Lehner involution Maass form holomorphic modular form SL(2Z) congruence subgroup Gamma0 Gamma1 Gamma2 nebentypus Dirichlet character primitive imprimitive conductor modulus Gauss sum Jacobi sum Stickelberger ideal Iwasawa main conjecture Mazur Wiles p-adic L-function Kubota-Leopoldt measure p-adic interpolation Bernoulli numbers von Staudt-Clausen Kummer congruence Voronoi congruence Hurwitz zeta Lerch transcendent polylogarithm multiple zeta values shuffle product stuffle regularised renormalised motivic pro-unipotent fundamental group Grothendieck-Teichmuller Drinfeld associator dilogarithm Bloch group scissors congruence Hilbert third problem Dehn invariant Sydler theorem polyhedra dissection equidecomposability Banach-Tarski paradox AC axiom choice measurable set Vitali set non-measurable Banach limit invariant mean amenability Følner sequence property T Haagerup property Connes embedding problem free probability Voiculescu random matrix Wigner semicircle Marchenko-Pastur Tracy-Widom GOE GUE GSE Dyson index beta=1,2,4 circular ensembles Plancherel measure RSK correspondence Robinson-Schensted-Knuth jeu de taquin Littlewood-Richardson Pieri branching rule Cauchy identity dual Cauchy determinant Andreief MacMahon master theorem plane partition box stacking lozenge tiling hexagon Aztec diamond domino tiling dimer model perfect matching Pfaffian Kasteleyn determinant Temperley bijection loop model six-vertex Yang-Baxter equation star-triangle Izergin-Korepin determinant quantum inverse scattering Bethe ansatz coordinate algebraic nested Yang-Yang functional thermodynamic limit string hypothesis driving equation Bethe roots macroscopic Bethe string mirror symmetry Strominger-Yau-Zaslow SYZ special Lagrangian fibration homological mirror symmetry Kontsevich conjecture derived category coherent sheaves perfect complexes triangulated dg category infinity-category HoTT homotopy type theory univalence axiom cubical Agda Coq Lean proof assistant formal verification mathematical library mathlib four-colour theorem Appel Haken computer proof Kepler conjecture Flyspeck project Hales informal proof formal proof Liquid Tensor Experiment Scholze perfectoid space prismatic cohomology condensed mathematics Clausen Scholze topological vector spaces barrelled bornological Fréchet Montel nuclear Schwartz space tempered distribution Fourier-Mikowski Parseval Plancherel Wiener Tauberian Banach algebra spectral theory Gelfand-Naimark C*-algebra von Neumann algebra factor type I II III hyperfinite injective amenability Connes Jones index subfactor planar algebra Temperley-Lieb Brauer Ocneanu rigidity Popa intertwining bimodule Morita equivalence Picard group Brauer group central simple algebra quaternion division Hurwitz-Radon Adams Hopf invariant one Bott periodicity KO-theory K-theory Atiyah-Hirzebruch spectral sequence Chern character Todd genus Hirzebruch signature L-genus Pontryagin classes cobordism ring oriented spin spin^c almost complex symplectic tangent bundle normal bundle Whitney sum splitting principle Euler class obstruction theory Postnikov tower Whitehead tower cell complex CW simplicial Delta complex simplicial set Kan complex quasicategory Joyal Street nerve bar-cobar resolution Koszul dual operad PROP PROPs modular operad wheeled PROP properad dioperad colored operad monoidal category symmetric monoidal closed compact closed cartesian closed bicategory trategory Gray tensor product Day convolution enriched category locally presentable accessible adjunction monadic Beck theorem Barr exact regular epimorphism strong epimorphism extremal epi mono isomorphism automorphism endomorphism natural transformation Yoneda lemma representable functor presheaf sheaf topos elementary Grothendieck topos Lawvere-Tierney topology subobject classifier Heyting algebra Boolean algebra Stone duality Birkhoff representation Dedekind completion MacNeille completion Alexandrov topology Scott topology sober space spectral locale frame locale point-free topology pointless topology formal topology predicative constructive intuitionistic mathematics Brouwer Heyting Kolmogorov HK interpretation Chinese remainder theorem prime ideal maximal ideal Noetherian Artinian Krull dimension height depth Cohen-Macaulay complete intersection regular ring localization multiplicative set flat torsion-free faithfully flat descent faithfully flat descent Zariski étale fppf fpqc crystalline syntomic rigid analytic Berkovich adic Tate space Huber pairing Poincare Serre duality Grothendieck trace residue Serre-Tate deformation canonical lift Lubin-Tate formal group Honda system supersingular ordinary CM complex multiplication Hilbert modular surface Siegel modular variety Shimura variety PEL abelian variety moduli stack Artin stack Deligne-Mumford stable curve stable map Kontsevich compact |