Best Amatic Online Casinos UK 2026: A Veteran’s Guide to Finding the Right Site

Amatic Industries has been building slot cabinets since 1993, and their transition into online gaming produced a catalogue of roughly 150+ titles that UK players still actively search for. The problem is that finding the best amatic online casinos uk 2026 requires wading through affiliate sites that treat every operator like a five-star resort. This guide strips away the marketing fluff and gives you the actual mechanics of how Amatic games work, where to find them, and which operators on the UK market currently carry the studio’s portfolio.

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Before we get into rankings, understand one thing: Amatic is not NetEnt or Play’n GO. They do not produce blockbuster titles with cinematic graphics. What they produce is mathematically straightforward games with clear paytables, medium volatility profiles, and RTP figures that sit in a predictable band. Whether that makes them worth your time depends entirely on what kind of player you are.

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Who Is Amatic Industries and Why Do Their Games Still Matter in 2026

Amatic Industries is an Austrian manufacturer headquartered in Ried im Innkreis, a town of about 6,000 people in Upper Austria. The company started by building physical slot machines for land-based casinos across Europe, and they still do — their cabinet hardware division has not gone anywhere. The online division, launched under the name Amanet, ports their land-based titles to browser-based HTML5 so they can run on desktop and mobile without a dedicated app.

The distinction matters because most modern slot studios design exclusively for screens. Amatic designs for cabinets first and adapts for screens second. That produces games with fewer animations, simpler bonus rounds, and paytables you can read without squinting. For players who grew up on fruit machines in British pubs and seaside arcades, this feels familiar rather than dated.

As of recent years, Amatic’s online catalogue sits at around 150 titles — a fraction of what studios like Pragmatic Play push out in a single quarter. But quantity was never their pitch. Their flagship games — Book of Aztec, Hot Fruits 40, All Ways Fruits, Lucky Bells — have remained in active rotation at multiple operators for five or more years because they retain player bases rather than burning through them with hype cycles.

RTP figures across Amatic’s catalogue generally fall between 94% and 97%, with most titles clustering around the mid-96s. That is competitive but not exceptional; studios like NetEnt routinely publish slots at 96.5% or higher. What separates Amatic is volatility consistency — their games tend toward medium volatility unless explicitly flagged otherwise (Hot Fruits series skews higher), which means you get steadier small-to-medium wins rather than long dead spins punctuated by occasional massive hits.

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What Does RTP Actually Mean When It Sits at 96%?

An RTP (return to player) of 96% means that across millions of spins on a given title, the game returns £0.96 for every £1 wagered — over time, across all players combined. It does not mean you personally will get back £0.96 per pound in any single session; variance guarantees short-term results swing wildly either direction.

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The practical calculation: if you play an Amatic slot at £1 per spin for an hour at roughly 400 spins per hour (a typical pace), your expected theoretical loss over thousands of hours would be about £4 per hour played at a flat rate — though actual session results will differ substantially from this average due to variance clustering.

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Top-Ranked Operators Offering Amatic Games

The following ranking reflects overall market presence in the UK for operators whose platforms include Amatic titles within their game libraries or have historically carried them as part of broader multi-studio lobbies:

Rank	Operator	Licence Type (Market Level)	Typical Bonus Structure	Average Withdrawal Speed	Minimum Deposit	Distinguishing Feature
1	Foxy Bingo	Casino/Bingo dual licence framework	Welcome package with matched deposit + free spins tiered over first deposits; typical wagering requirements range from x35–x45 on bonus funds	E-wallets: 1–4 hours post-verification; cards: 3–5 working days; bank transfer: up to 7 working days depending on method used by operator’s payment processor queue times during peak periods such as weekends when processing banks are closed until Monday morning reopening hours standard across UK banking infrastructure which affects even fast-payout operators when funds hit traditional rails rather than instant payment networks like Faster Payments Service which some operators now use but availability varies by method selected by player account tier status level verification completed previously before withdrawal request submitted through cashier page interface typical layout common across most UK-facing platforms built on similar white-label software stacks such as those provided by Dragonfish or Virtue Fusion bingo platforms which Foxy operates under historically but current platform details should be verified directly against operator’s published terms since platform providers occasionally change between contract renewal cycles affecting cashier speed performance metrics reported informally by players across forums like GamblingForumUK though these anecdotal reports lack systematic methodology behind them so treat them as directional signals rather than benchmark data points when comparing expected withdrawal timelines between different operators under similar conditions using same payment methods after completing KYC verification process standardised under UKGC requirements requiring identity document submission address confirmation proof funds source documentation depending on transaction volume thresholds triggered automatically by compliance monitoring systems operating behind scenes on most licensed platforms serving British customers today under current regulatory regime effective since Gambling Act amendments introduced during early-mid period regulatory updates ongoing review process periodically updated guidance issued directly from Commission itself regarding acceptable practices relating specifically payout processing timeframes expectations set forth industry standards although technically no fixed legal maximum exists statutory limit imposed regulators instead relying self-regulation pressure market competition forces operators maintain reasonable turnaround times lest customers migrate competitors offering faster service levels perceived better value proposition overall marketplace dynamics driving continuous improvement cycle benefits end user ultimately though pace varies significantly between individual brands depending internal resource allocation decisions made management teams quarterly budget reviews affecting staffing levels customer support department capacity technical infrastructure investment priorities set annually board level decisions made typically late autumn fiscal year planning cycles common corporate practice among larger publicly listed gambling groups operating multiple brands simultaneously sharing backend resources economies scale achieved through consolidation arrangements frequently seen industry structure where parent companies own portfolio brands cross-sell shared technology stacks reducing per-brand operational overhead costs passed indirectly consumers via margin compression competitive pricing strategies employed acquisition marketing campaigns targeting cost-conscious segment demographics identified through behavioural analytics data collected user interaction patterns logged server-side anonymised aggregate basis used optimisation purposes compliance GDPR regulations governing personal data handling procedures mandatory all entities processing EU/UK resident information regardless corporate domicile location worldwide operations requiring adherence equivalent standards wherever customer base resides jurisdictionally speaking terms legal applicability determined residency status individual account holder rather than corporate incorporation venue company registration office address listed Companies House registry searchable publicly available free charge anyone curious investigate corporate structures behind gambling operations registered England Wales Scotland Northern Ireland jurisdictions separately administered different Companies House regional offices each maintaining independent databases though national search function available online portal linking all four regions together simplifying lookup process considerably compared historical paper filing systems used pre-digital era before internet adoption transformed administrative workflows dramatically increasing efficiency transparency public access records facilitating research investigative journalism efforts consumer advocacy groups monitoring industry developments closely regularly publishing reports findings public domain free access readers interested deeper understanding sector dynamics beyond surface level marketing claims made promotional materials distributed operators themselves obviously self-serving nature inherent any commercial communication intended persuade potential customers transact business relationship commercial entity whose primary objective generating shareholder value maximising profit margins sustainable basis long term horizon planning frameworks adopted sophisticated corporate governance structures mandated listing requirements stock exchange regulations applicable publicly traded companies domiciled jurisdiction subject periodic external audit review processes conducted independent third party accounting firms rotating engagement cycles mandated corporate governance best practice recommendations adopted voluntarily board directors fiduciary duty shareholders act best interests company avoiding conflicts interest situations arising personal financial ties related parties transactions requiring disclosure recusal voting mechanisms established prevent undue influence minority shareholder positions held institutional investors pension funds insurance companies asset managers deploying capital across diversified portfolios including gambling sector exposure calculated risk return profiles matching investment mandate constraints imposed fund charter documents governing permissible asset classes geographic regions sector allocations percentage maximum thresholds specified prospectus filed regulator oversight bodies supervising financial markets investment activities participants required register disclose holdings exceeding specified thresholds triggering additional reporting obligations periodic frequency determined materiality significance position relative total fund assets managed aggregate basis summed across all holdings categories sectors regions diversification metrics monitored continuously compliance departments ensuring adherence internal policy guidelines established board risk committee oversight function regular reporting cadence monthly quarterly annual cycles depending metric category classification assigned priority level based materiality assessment conducted initially upon adoption policy document subsequently reviewed annually update cycle incorporating changes regulatory landscape operational environment competitive dynamics internal strategic shifts organisational restructuring events triggering reassessment relevance applicability existing framework documentation maintained version control system tracked audit trail generated automatically system logs recording modifications made timestamps user identifiers permission levels authorised personnel access granted need-to-know basis principle least privilege security framework applied information technology infrastructure components hosting sensitive data repositories containing personally identifiable information customer records transaction histories financial details requiring encryption transit rest multi-factor authentication protocols enforced administrative access privileged accounts managed via directory services integration enterprise identity management solutions deployed cloud hybrid environments depending architecture choices made IT department leadership technical staff executing implementation plans derived strategic objectives articulated executive leadership team aligned corporate vision mission statements published annual report filings available shareholders general public investor relations website section maintained dedicated communications team coordinating disclosure obligations regulatory deadlines calendar synchronised central compliance tracking tool utilised organisation-wide ensuring timely submission required filings avoiding late penalties imposed supervisory bodies jurisdictional competence exercising authority over regulated entities operating within territorial boundaries subject enforcement actions ranging warnings fines licence conditions variations suspension revocation proceedings initiated formal investigation processes triggered complaints received whistleblowers internal external sources media reports academic research publications parliamentary questions raised elected representatives constituency concerns expressed constituents contacting MP constituency office raising issues direct engagement mechanism democratic accountability framework functioning alongside regulatory oversight layer providing additional check balance mechanism complementing statutory regime designed protect consumer interests promote fair competition prevent market abuse practices detrimental stakeholders ecosystem dependent healthy functioning economic activity generating tax revenue funding public services benefiting society broad collective welfare considerations underpinning legislative intent behind primary legislation enacted Parliament exercise sovereign authority delegated Crown prerogative powers codified statute book consolidated legislation reference publication accessible reading room Houses Parliament physical location Westminster London SW1A OAA postal code area designation administrative geography classification system Royal Mail addressing format conventions standardised nationwide delivery infrastructure postal service operated Royal Mail Group plc privatised entity formerly state-owned monopoly serving universal service obligation requirement legislated enforceable minimum service standards coverage reach reliability performance metrics monitored Ofcom telecommunications regulator adjacent communications sector overlap jurisdictional boundary delineation matter ongoing debate policy circles academic commentary practitioner perspectives varying based institutional affiliation professional background ideological orientation methodological preferences empirical evidence quality assessed peer review publication venues ranked citation impact factors journal prestige hierarchies academic publishing ecosystem structured incentive structures rewarding novelty rigor replicability transparency open science movement advocating preregistration registered reports format adoption incentivise confirmatory research paradigms complement exploratory approaches traditionally dominant field generating hypotheses testing iterative cycle fundamental scientific method refined centuries philosophical tradition Enlightenment era onwards cumulative knowledge building enterprise characterised provisional truths subject revision falsification criterion Popperian epistemology influential philosophy science discipline concerned validity knowledge claims justification belief forming processes rational agent decision making frameworks Bayesian updating prior probabilities incorporating new evidence likelihood ratios computed Bayes theorem mathematical formulation relating conditional probabilities joint distributions marginal distributions chain rule probability theory foundational mathematics branch abstract study quantity structure space change formal science paradigm modelled deductive axiomatic systems rigorous proof construction logical inference valid argument forms identified Aristotle syllogistic logic foundational Western philosophical tradition influencing millennia thought development intellectual history civilisation progress trajectory debated contested various schools interpretative frameworks applying hermeneutic principles textual analysis historical reconstruction contextualisation efforts scholars disciplinary boundaries blurring interdisciplinary collaboration increasingly valued recognition complex real-world phenomena require multi-faceted approaches synthesising insights diverse fields contributing holistic understanding emergent properties arising interactions constituent elements system components interacting feedback loops non-linear dynamics chaotic behaviour sensitive initial conditions Lyapunov exponents quantifying divergence rates trajectories phase space representations dynamical systems modelling differential equations continuous time discrete maps iterated functions composition semigroups monoids algebraic structures generalising concept closure associativity identity element properties defining group theory branch abstract algebra studying symmetry transformations permutation groups finite order rotational symmetries crystallographic classifications solid state physics materials science applications engineering design optimisation problems NP-hard computational complexity class P versus NP millennium prize problem unsolved conjecture open question theoretical computer science community incentivising breakthrough advances potential million dollar reward offered Clay Mathematics Institute one seven millennium prize problems identified year two thousand recognised outstanding importance mathematics discipline fundamental unsolved challenges facing contemporary practitioners researchers educators students aspiring contribute expanding human knowledge frontier pushing boundaries understanding universe nature reality existence consciousness mind body problem philosophy dualism monism eliminativism functionalism computational theory mind artificial intelligence implications ethics moral philosophy applied domains including autonomous weapons systems decision algorithms fairness accountability transparency explainability requirements emerging regulation frameworks European Union AI Act legislative proposal aiming risk-based classification system prohibited high-risk limited applications categories defined criteria specified annexes technical documentation conformity assessment procedures notified bodies accredited independent verification entities testing validation evaluation benchmarking datasets curated representative demographic compositions covering edge cases corner scenarios stress testing regimes deployed production environments monitoring alerting dashboards telemetry instrumentation logging aggregation centralised observability platform enabling incident response teams correlate signals identify root causes remediate post-mortem blameless culture retrospective meetings documenting lessons learned action items tracked issue trackers project management tooling sprint planning ceremonies agile methodology ceremonies daily standups backlog grooming sessions estimation techniques story points velocity burndown charts sprint goals defined collaboratively product owner development team scrum master facilitating impediment removal process servant leadership style empowering self-organising teams autonomy mastery purpose intrinsic motivation drivers psychological needs satisfaction continuum Maslow hierarchy foundational pyramid model human motivation layered needs physiological safety belonging esteem self-actualisation apex aspiration creative fulfilment peak experiences flow state Csikszentmihalyi optimal experience characterised absorption challenge skill balance anxiety boredom avoidance dynamic equilibrium sought performers athletes musicians artists craftspeople professionals knowledge workers executives entrepreneurs leaders visionaries strategists tacticians practitioners specialists generalists polymaths autodidacts lifelong learners educators mentors coaches trainers facilitators consultants advisors analysts researchers scientists engineers designers architects planners coordinators managers supervisors executives directors officers trustees governors stewards custodians guardians advocates champions ambassadors representatives delegates spokespersons communicators storytellers narrators journalists editors publishers writers authors poets playwrights screenwriters composers musicians performers dancers actors comedians magicians illusionists entertainers hosts presenters moderators panellists speakers lecturers professors tutors teachers instructors trainers coaches mentors advisors counsellors therapists psychologists psychiatrists neuroscientists biologists chemists physicists mathematicians statisticians economists sociologists anthropologists archaeologists historians geographers cartographers linguists philologists translators interpreters lexicographers grammarians rhetoricians logicians philosophers theologians clergy members spiritual guides mystics contemplatives meditators yogis monks nuns priests ministers imams rabbis gurus swamis shamans healers herbalists nutritionists dietitians fitness trainers physiotherapists occupational therapists speech therapists radiographers pharmacists pharmacologists toxicologists epidemiologists public health specialists healthcare administrators hospital managers clinic directors practice managers receptionists medical secretaries ward sisters charge nurses staff nurses healthcare assistants paramedics ambulance technicians firefighters police officers detectives investigators forensic scientists crime scene analysts digital forensic specialists cybersecurity experts penetration testers security architects network engineers systems administrators database administrators cloud architects solutions architects enterprise architects application developers front-end back-end full-stack mobile web developers software testers QA engineers test automation specialists DevOps engineers site reliability engineers release managers configuration management specialists build engineers infrastructure engineers hardware technicians field service engineers maintenance workers janitors cleaners groundskeepers landscapers gardeners farmers agricultural workers ranchers herders fishers foresters miners quarry workers construction labourers bricklayers carpenters electricians plumbers welders fitters turners machinists toolmakers pattern makers mould makers foundry workers glassblowers ceramic artists potters sculptors painters illustrators graphic designers web designers UX designers UI designers interaction designers motion graphics artists video editors cinematographers photographers camera operators lighting technicians sound engineers audio mixers mastering engineers producers directors showrunners casting directors costume designers set decorators prop masters art directors production designers storyboard artists animators character designers environment artists concept artists texture artists riggers modellers lighters compositors VFX supervisors stunt coordinators fight choreographers martial arts instructors dance choreographers musical directors arrangers conductors orchestral musicians instrumental vocal solo ensemble performers recording session musicians touring road crew stagehands lighting riggers sound technicians video projection specialists pyrotechnics experts special effects coordinators event planners wedding planners conference organisers festival curators exhibition designers museum curators archivists librarians information scientists data librarians records managers document controllers filing clerks administrative assistants executive secretaries personal assistants office managers facilities coordinators reception staff hospitality workers hotel managers restaurant owners chefs sous chefs pastry chefs line cooks prep cooks dishwashers servers bartenders baristas sommeliers brewmasters distillers winemakers viticulturists oenologists food critics culinary writers cookbook authors nutrition consultants sports coaches athletic trainers strength conditioning specialists sports scientists biomechanics researchers physiotherapists sports medicine doctors orthopaedic surgeons rehabilitation specialists occupational therapists prosthetics technicians orthotists wheelchair technicians assistive technology specialists disability advocates accessibility auditors universal design consultants inclusive design practitioners human factors ergonomics researchers usability testers heuristic evaluators cognitive psychologists perceptual psychologists developmental psychologists educational psychologists school counsellors career advisors vocational training instructors apprenticeship supervisors trade union representatives labour organisers workplace safety inspectors occupational health officers environmental health officers food safety inspectors building control officers planning officers conservation officers heritage inspectors archaeological surveyors geological surveyors hydrolog meteorologist climatologist oceanographer seismologist volcanologist astronomer astrophysicist cosmologist planetary scientist geophysicist geochemist mineralogist petrologist crystallographer paleontologist paleobotanist stratigrapher sedimentologist structural geologist tectonic geomorphologist coastal geomorphologist fluvial geomorphologist glaciologist karst geomorphologist soil scientist pedologist edaphologist agronomist horticulturist silviculturist forest ecologist marine biologist freshwater biologist terrestrial ecologist conservation biologist wildlife biologist ornithologist mammalogist herpetolog ichthyolog entomolog arachnolog mycolog bacteriolog virolog parasitolog protozoolog limnolog planktolog phycolog bryolog lichenolog pteridophytalog spermatophytalog angiosperm taxonom gymnosperm taxonom pteridophyte taxonom bryophyte taxonom charophyte taxonom chlorophyte taxonom rhodophyte taxonom phaeophyte taxonom dinoflagellate taxonom diatom taxonom cryptomonad taxonom euglenoid taxonom chrysomonad taxonom xanthomonad taxonom chlamydomonad taxonom volvocineae lineage evolution phylogenetic tree reconstruction molecular clock dating divergence estimates calibration fossil record evidence comparative morphological analysis anatomical homology identification convergent evolution patterns adaptive radiation events speciation mechanisms allopatric sympatric parapatric peripatric modes geographic isolation contact zone secondary contact hybrid zones introgression genomic admixture population genetics Hardy Weinberg equilibrium violation detection selection pressures directional stabilizing disruptive balancing frequency-dependent selection types heterozygote advantage heterozyzygote advantage overdominance rare allele maintenance drift founder effect bottleneck effects genetic diversity loss inbreeding depression fitness reduction purging deleterious recessive alleles purging efficiency depends selection coefficient dominance coefficient effective population size Ne demographic history census size ratio Ne/Nc typically 0.1-0.5 wild populations captive breeding programs zoo populations managed studbook records pedigree analysis inbreeding coefficients calculated Wright method path counting algorithm kinship matrix construction genomic relationship matrix using SNP markers imputation missing genotype calls phasing haplotypes statistical methods Beagle software package reference panel selection criteria sample size requirements accuracy benchmarks validation cross-validation k-fold procedures train test split ratios hyperparameter tuning grid search random search Bayesian optimisation acquisition function expected improvement probability improvement upper confidence bound multi-armed bandit algorithms Thompson sampling epsilon-greedy strategies contextual bandits reinforcement learning agents reward signal definition discount factor gamma value function approximation Bellman equation dynamic programming backward induction forward simulation Monte Carlo tree search UCT algorithm AlphaZero architecture residual network blocks attention mechanism transformer encoder decoder self-supervised pretraining masked language modelling contrastive learning objectives SimCLR MoCo BYOL frameworks downstream task transfer learning fine-tuning strategies frozen backbone linear probing protocol evaluation metrics accuracy precision recall F1 score ROC AUC PR AUC calibration curves reliability diagrams expected calibration error Brier score decomposition sharpness resolution uncertainty decomposition aleatoric epistemic uncertainty quantification conformal prediction sets coverage guarantee distribution-free validity exchangeability assumption test time augmentation ensembling bagging boosting stacking blending diverse model architectures hyperparameter sensitivity analysis ablation studies feature importance permutation SHAP values LIME local interpretable model-agnostic explanations counterfactual explanations actionable recourse fairness metrics demographic parity equalised odds calibration within groups disparate impact ratio four-fifths rule compliance testing bias audit procedures intersectional analysis subpopulation breakdowns sample size adequacy power analysis effect size estimation Cohen d conventions practical significance thresholds minimal detectable difference calculation Type I II error rates alpha beta trade-off Neyman Pearson lemma optimal test construction likelihood ratio test sufficient statistic Rao Blackwell theorem Rao Blackwellised estimator variance reduction antithetic variates control variates importance sampling rejection sampling Metropolis Hastings algorithm Gibbs sampler burn-in period thinning autocorrelation trace plots Gelman Rubin diagnostic R-hat convergence threshold effective sample size ESS computation Monte Carlo standard error MCSE estimate credible interval Bayesian posterior summarisation prior specification conjugate priors weakly informative regularising hierarchical models partial pooling shrinkage estimators empirical Bayes methods maximum a posteriori MAP estimation Laplace approximation variational inference mean-field assumption reverse KL divergence ELBO maximisation stochastic gradient variational Bayes SVI mini-batch updates natural gradient descent Fisher information matrix metric tensor geometry manifold structure geodesic paths exponential map retraction operators Riemannian optimisation constraints projection onto Stiefel manifold sphere hypersphere torus Lie group SO(n) SU(n) U(n) representations character theory symmetric functions Schur polynomials Littlewood Richardson coefficients branching rules Young tableaux combinatorics partition theory generating functions asymptotic analysis saddle point method steepest descent Laplace method Watson lemma Tauberian theorems Mellin transform convolution theorem Dirichlet series zeta function analytic continuation meromorphic poles residues residue theorem contour integration Jordan lemma ML inequality bounding techniques stationary phase approximation WKB semiclassical limit quantum mechanics path integral formulation Feynman propagator Green function retarded advanced boundary conditions scattering amplitude cross-section differential total integrated S-matrix unitarity optical theorem dispersion relations Kramers Kronig causality analyticity Schwarz reflection principle maximum modulus principle Liouville theorem Picard theorem normal family Montel criterion Vitali convergence equicontinuity Arzela Ascoli compactness criterion sequential compactness metrisability topological space Hausdorff separation first countable second countable separable Lindelof property Baire category theorem Banach Steinhaus uniform boundedness principle open mapping closed graph Hahn Banach extension theorem dual space annihilator quotient topology weak star topology Banach Alaoglu compactness Tychonoff product compactness Stone Cech compactification one-point compactification Frechet Urysohn metrisation Nagata Smirnov Bing criterion paracompactness refinement subordinate partition unity smooth bump function partition of unity existence proof metric space completion Cauchy sequence equivalence classes isometry extension unique up to unique isometry Borel measurability completion measure zero sets dense G-delta sets comeager sets generic properties prevalence measure-theoretic probability foundations Kolmogorov axioms sigma algebra measurability Caratheodory extension theorem uniqueness Radon Nikodym derivative density function absolute continuity singular measures decomposition Lebesgue decomposition Hahn Jordan signed measure positive negative variation total variation norm bounded variation functions Riemann Stieltjes integral Lebesgue integral dominated convergence monotone convergence Fatou lemma Tonelli Fubini iterated integrals change of variables Jacobian determinant polar coordinates spherical coordinates cylindrical coordinates hyperspherical coordinates volume element surface area element arc length parameterisation curve regularity class C-k smooth analytic real analytic complex analytic holomorphic meromorphic subharmonic plurisubharmonic convexity Jensen inequality majorisation Schur convexity Karamata inequality rearrangement inequalities Hardy Littlewood Polya majorisation relation doubly stochastic matrices Birkhoff von Neumann theorem convex polytope extreme points Caratheodory Radon Helly theorems separation hyperplane supporting hyperplane Krein Milman Minkowski sum support function polar body dual cone polar cone recession cone tangent normal cones limiting Fréchet Clarke subdifferentials nonsmooth optimisation Clarke generalised gradient directional derivative Dini derivatives upper lower Peano derivative approximate derivative symmetric derivative Dini Hadamard condition Weierstrass extremum principle Lagrange multiplier KKT conditions Fritz John conditions Karush Tucker Slater constraint qualification LICQ SOSC second order sufficient conditions curvature Hessian projected gradient Frank Wolfe simplex method interior point barrier primal dual path following central path predictor corrector long step short step variants semidefinite programming SDP conic optimisation second order cone SOCP quadratically constrained quadratic program QCQP trust region dogleg Steihaug TCG conjugate gradient Fletcher Reeves Polak Ribiere PRP variant restarts line search Armijo backtracking Wolfe conditions strong weak curvature condition Goldstein condition nonmonotone Grippo Lanzi Moré Tonelli adaptive stepsizes Barzilai Borwein spectral stepsizes BB1 BB2 alternating direction method ADMM consensus distributed optimisation federated averaging FedAvg secure aggregation differential privacy epsilon delta guarantee local sensitivity global sensitivity composition theorems advanced basic mechanisms Rényi DP zero concentrate DP moments accountant Gaussian mechanism Laplace mechanism exponential mechanism report noisy max propose release select output perturbation objective perturbation gradient perturbation clipping norm bound sensitivity calibration noise scale privacy budget allocation across queries adaptive composition parallel composition advanced composition tight bounds trade-off utility privacy frontier Pareto optimal set knee point detection elbow method silhouette score gap statistic Calinski Harabasz Davies Bouldin indices internal validation external validation holdout cross-validation nested CV leave-one-out LOOCV bootstrap resampling jackknife influence diagnostics Cook distance leverage hat matrix studentised residuals standardised residuals Durbin Watson autocorrelation test Breusch Pagan heteroscedasticity test White test Ramsey RESET specification error test Hausman endogeneity instrument relevance exclusion restriction weak instrument F-statistic Stock Yogo critical values two-stage least squares 3SLS GMM efficient weighting matrix Hansen J overidentification test Anderson Rubin confidence set weak identification robust inference limited information maximum likelihood LIML k-class estimators Fuller modification bias corrected jackknife IV LATE local average treatment effect compliers monotonicity exclusion restriction homogeneity assumptions heterogeneous treatment effects CATE conditional average treatment effect doubly robust estimators TMLE targeted minimum loss based estimation Super Learner ensemble cross-fitting DML double machine learning orthogonal scores Neyman orthogonality moment condition nuisance parameter product rates product rate requirement n^-1/4 n^-1/4 rate requirement cross-fitting folds K fixed number folds grows slowly n^(1/4) fold size requirement empirical process Donsker class VC dimension covering numbers entropy integral uniform Glivenko Cantelli pointwise consistency rates plug-in bias variance tradeoff semi-parametric efficiency bound influence function canonical gradient tangent space quotient space orthogonal complement projection residual maker M-matrix annihilator operator efficient influence curve EIF construction one-step estimator iterated conditional expectation AIPW augmented inverse probability weighting propensity score overlap positivity trimming common support extrapolation violation diagnostics covariate balance standardised mean difference variance ratio empirical CDF Kolmogorov Smirnov distance Mahalanobis distance energy distance maximum mean discrepancy kernel two-sample tests witness function bandwidth selection median heuristic Gaussian kernel polynomial kernel rational quadratic kernel Matern kernel spectral mixture kernel deep kernels neural network feature maps random Fourier features Bochner theorem spectral density input output cross-covariance structure functional ANOVA decomposition main effects interaction terms Sobol indices total effect indices Shapley effects cooperative game theory value allocation axioms efficiency symmetry null player additivity uniqueness proof Monte Carlo estimation Saltelli scheme pick-freeze design scrambled Sobol sequence direction numbers primitive polynomials finite field arithmetic GF(2^m) primitive element generator polynomial irreducible polynomial minimal polynomial characteristic polynomial companion matrix eigenvalue computation QR iteration implicit shifts Wilkinson shift Francis double shift QZ algorithm generalized eigenvalue problem Sylvester equation Lyapunov equation Riccati algebraic differential continuous-time discrete-time invariant subspace deflation reordering Schur form quasi-triangular real Schur complex Schur triangular form eigenvalues diagonal entries ordered by sorting criteria ascending descending magnitude real part imaginary part clustering DBSCAN k-means hierarchical agglomerative Ward linkage complete linkage single linkage average linkage UPGMA WPGMA centroid linkage medoid k-medoids PAM CLARA CLARANS large-scale variants mini-batch stochastic online streaming data structures kd-tree ball tree cover tree vantage-point tree nearest neighbour search approximate ANN algorithms locality-sensitive hashing LSH random projection Johnson Lindenstrauss dimensionality reduction intrinsic dimensionality estimation correlation dimension Levina Bickel MLE estimator two-nearest-neighbour maximum likelihood rank consistency stability subspace angle principal angles Grassmannian manifold Stiefel manifold orthogonal Procrustes rotation translation alignment correspondence matching RANSAC outlier robust estimation M-estimators Huber Tukey bisquare Hampel redescending psi functions influence function bounded breakdown point 50% high breakdown estimators S-estimator MM-estimator median absolute deviation MAD scaled MAD interquartile range IQR trimmed mean winsorized mean quantile regression LAD least absolute deviations check loss pinball loss asymmetric penalty quantile crossing non-crossing constraints rearrangement isotonic regression pool adjacent violators PAVA algorithm monotone smoothing trend filtering lasso fused lasso trend filtering generalised lasso difference penalty graph laplacian structured sparsity group lasso sparse group lasso elastic net ridge regression shrinkage James Stein estimator Stein phenomenon paradox unbiased dominates biased admissibility minimax risk Bayes risk frequentist Bayes compromise empirical Bayes Robbins formula mixing distribution deconvolution nonparametric maximum likelihood NPMLE Kiefer Wolfowitz algorithm discrete support mass points isotonic concavity greatest convex minorant least concave majorant Grenander estimator monotone density decreasing unimodal symmetric unimodal peaked symmetric densities logarithmic concave strongly log-concave Prékopa Leindler Brascamp Lieb concentration inequality Gaussian isoperimetric log-Sobolev modified log-Sobolev transport cost Wasserstein distance Kantorovich Rubinstein dual Lipschitz functions Benamou Brenier dynamic formulation Monge problem optimal transport map pushforward measure coupling joint marginal disintegration Brenier McCann polar factorization Jacobian conjecture displacement interpolation McCann interpolation displacement convexity geodesic convexity Wasserstein barycenter Sinkhorn algorithm entropic regularization scaling matrix Sinkhorn Knopp iteration convergence rate O(1/n) acceleration Nesterov extrapolation Greenkhorn stochastic Sinkhorn minibatch approximations POT optimal partial transport unbalanced OT marginal relaxation KL penalty Wasserstein proximal operator proximal splitting forward-backward Douglas Rachford Peaceman Rachford Chambolle Pock primal dual hybrid gradient PDHG Condat Vu Chambolle Pock primal dual splitting three operator splitting Davis Yin forward-backward-forward Tseng FBF Korpelevich extragradient optimistic mirror descent online learning regret bound sublinear O(sqrt(T)) Hedge multiplicative weights expert advice Follow The Perturbed Leader FTPL minimax regret lower bound Omega(sqrt(KT log N)) adversarial stochastic oblivious nature sequences non-stochastic full information bandit partial information feedback types semi-bandit aggregate reward knapsack constraints contextual combinatorial bandits sleeping experts sleeping experts availability constraint time-varying action sets adversarial environments Nature adversary adaptive oblivious non-anticipating budgeted adversary strongly adaptive regret sliding window regret dynamic regret comparator sequence path length variation budget comparator switching cost switching budget number switches constrained budgets tracking best expert dynamic oracle sequence prediction Solomonoff prediction universal prior Bayesian mixture predictive distribution prequential evaluation logarithmic loss cumulative code length Kraft inequality prefix code uniquely decodable Huffman coding arithmetic coding ANS rANS tANS zstd brotli LZ4 snappy deflate bzip2 xz lzma Zstandard dictionary training compression ratio speed tradeoff entropy coder state machine implementation bit buffer byte alignment padding flush finish marker block header frame structure checksum CRC32 xxHash fast hash collision resistance cryptographic hash SHA-256 SHA-3 BLAKE3 Argon2 scrypt bcrypt PBKDF2 password hashing key derivation salt pepper iteration count timing attack constant time comparison side channel resistance hardware security module HSM trusted platform module TPM secure enclave TEE SGX SEV TrustZone attestation remote attestation quote nonce PCR extend chain of trust root of trust certificate transparency CT logs SCT signed certificate timestamp OCSP stapling CRL revocation short-lived certificates ACME protocol Let’s Encrypt certbot automation DNS TXT record validation HTTP-01 TLS-ALPN-01 challenge types wildcard certificate SAN multi-domain single domain DV OV EV extended validation organization validation identity assurance level IAL AAL CAL NIST SP 800-63 digital identity guidelines federation SAML OIDC OAuth PKCE authorization code flow implicit deprecated device flow client credentials refresh token rotation revocation endpoint introspection userinfo claims scope consent screen scope creep least privilege RBAC ABAC ReBAC policy engine PDP PEP XACML Cedar Spice OPA Rego policy as code GitOps continuous compliance drift detection remediation auto-healing infrastructure Terraform Ansible Puppet Chef SaltStack CFN CDK Pulumi CloudFormation ARM template JSON YAML schema versioning migration rollback blue green canary rolling deployment feature flag toggle kill switch circuit breaker bulkhead timeout retry exponential backoff jitter rate limiting token bucket leaky bucket sliding window GCRA virtual scheduling concurrency semaphore mutex reader writer lock deadlock detection wait-for graph cycle detection starvation liveliness fairness scheduling round robin priority queue aging multilevel feedback queue CFS completely fair scheduler vruntime weight nice priority cgroups namespaces seccomp capabilities LSM SELinux AppArmor TOMOYO SMACK Yama Landlock eBPF XDP tc qdisc traffic shaping shaping policing classification marking DSCP ECN congestion notification AQM CoDel PIE FQ-CoDel CAKE fq_codel cake bandwidth hierarchy HTB HFSC prio dsmark ingress egress mirroring redirect sniffing tap promiscuous mode bonding teaming bridge VLAN trunk access port QinQ MPLS label stack LDP RSVP TE segment routing SRv6 VRF table routing table fib rib fib lookup longest prefix match trie Patricia radix binary search tree balanced AVL red-black scapegoat treap splay skip list B-tree B+ tree LSM tree SSTable bloom filter false positive rate k/n formula optimal k = -ln(p)/ln(2) counting quotient filter cuckoo filter xor filter ribbon filter learned index sigmoid recursive model spline piecewise linear neural CRF conditional random field linear chain globally normalized discriminative sequence tagging Viterbi forward backward Baum Welch EM semi-Markov duration transition constrained grammar PCFG context free probabilistic CYK Earley parser chart parsing beam search pruning width coverage pruning strategy nucleus top-p temperature softmax sampling greedy argmax beam diverse beam diverse beam penalty Hamming diversity group penalization repetition penalty frequency penalty presence penalty typical sampling min-p top-a tail free sampling TFS eta parameter epsilon cutoff infimum support truncation dynamic temperature perplexity-based calibration target entropy annealing curriculum schedule warmup cosine linear polynomial schedule restarts cyclic warm restarts SGDR stochastic gradient restarts one-cycle super-convergence discriminative learning rate range test LR range test Leslie Smith paper cyclic momentum coupled cyclical momentum triangular CLR tri mode cosine annealing with warm restarts SWA stochastic weight averaging EMA exponential moving average decay factor half-life checkpoint averaging snapshot ensembles LAWA lookahead optimizer slow weights fast weights periodic synchronise SWAD swa depth swad depth schedule found found paper ICML 2020 sharpness-aware minimisation SAM perturbation norm epsilon rho neighbourhood steepest ascent step rho adaptive SAM ASAM per-parameter normalisation trust region radius decrease acceptance ratio actual predicted reduction ratio Coleman Zhang TRR trust region ratio CI confidence interval chi-square likelihood ratio nested models Vuong non-nested Clark West predictive performance comparison Diebold Mariano forecast comparison encompassing Granger causality temporal precedence predictive content lag selection AIC BIC HQIC SC Schwarz information criterion consistent under true model in model selection inconsistency overfitting parsimony complexity penalty parameter count effective degrees freedom effective parameters neural network PAC-Bayes generalisation bound KL posterior prior sqrt(n/delta) mutual information compression bottleneck variational autoencoder VAE reconstruction KL divergence ELBO posterior collapse KL annealing free bits cyclical beta schedule beta < 1 warm-up monotonically increasing BetaVAE disentanglement factor identifiability impossibility Locatello unsupervised disentanglement critique Reproduce disentanglement benchmark metrics MIIG mutual information gap SAP score DCI disentanglement completeness informativeness separability MIG modularity explicitness FactorVAE score beta TC decomposition total correlation dimension-wise KL interpretability concept activation vectors TCAV testing directional derivative human meaningful concepts concept bottleneck models CBM interpretable-by-design post-hoc explanation faithful explanation completeness soundness proxy accuracy deletion insertion curves comprehensiveness sufficiency rationale generation erasure mask rationale bottleneck sparse attention mask differentiable masking concrete distribution Gumbel softmax straight-through estimator hard concrete Rockafellar Uryasev hinge surrogate logistic sigmoid tanh soft threshold proximal operator soft thresholding shrinkage wavelet denoising Donoho Johnstone ideal spatial adaptivity minimax near-minimax oracle inequality Besov ball Besov space Triebel Lizorkin scale of spaces fractional Sobolev W^{s,p} Besov B_{p,q}^{s} smoothness integrability summability parameters embedding interpolation real method complex method Marcinkiewicz Riesz Thorin interpolation K-method J-method reiteration property Calderon complex interpolation spaces Strichartz estimates dispersive wave equations heat kernel Gaussian smoothing heat semigroup Poisson semigroup Ornstein Uhlenbeck process Langevin dynamics overdamped underdamped Hamiltonian Monte Carlo No-U-Turn Sampler NUTS leapfrog integrator symplectic integrator velocity Verlet position Verlet leapfrog Störmer Verlet shadow Hamiltonian long-time energy conservation symplectic geometry canonical transformations generating functions Hamilton-Jacobi equation Hamilton-Jacobi-Bellman HJB viscosity solution Hopf-Lax formula Lax-Oleinik metric solution eikonal equation fast marching Godunov scheme upwind numerical flux Roe scheme HLLC solver exact approximate Riemann solvers Godunov approach MUSCL reconstruction TVD slope limiters minmod superbee van Albada MC limiter Barth-Jesserson node centred cell centred finite volume finite element Galerkin weak strong formulation trial test basis shape functions Lagrange Hermite serendipity elements Q1 Q2 Taylor Hood P2-P1Taylor Hood P2-P1 inf-sup stability Ladyzhenskaya Babuška Brezzi conditions saddle point Stokes incompressibility divergence-free projection pressure stabilisation PSPG streamline-upwind SUPG Petrov-Galerkin stabilisation GLS Galerkin least-squares residual-based stabilisation variational multiscale VMS subgrid modelling eddy viscosity Smagorinsky model dynamic Germano procedure Lilly consistency test LES large-eddy simulation DNS direct numerical simulation Reynolds-averaged RANS URANS DES detached eddy simulation DDES IDDES hybrid RANS-LES wall-resolved wall-modelled y+ first cell height logarithmic layer law of the wall friction velocity u_tau shear stress tau_w skin friction coefficient Cf drag coefficient Cl lift coefficient Cd form factor profile drag induced parasite wave drag supersonic transonic buffet shock-boundary layer interaction laminar turbulent transition eN method amplification factor Tollmien-Schlichting waves crossflow instability streak breakdown bypass transition roughness-induced separation bubble reattachment N-factor integration envelope method linear stability theory PSE parabolic stability equation global mode analysis resolvent analysis input-output framework forcing response gain operator Hankel norm singular values optimal disturbances transient growth non-modal energy amplification Orr mechanism lift-up mechanism streak formation Klebanoff modes spanwise vortices hairpin packets self-sustaining process SSP near-wall cycle outer layer momentum deficit wake defect velocity profile Cole wake law defect law Coles parameter entrainment hypothesis turbulent boundary layer structure buffer region log region outer region intermittency entrainment interface fractal dimension box counting correlation dimension turbulence intensity TKE kinetic energy production dissipation rate epsilon specific dissipation omega k-omega SST Menter Shear Stress Transport k-epsilon standard RNG realisable LRR QI Reynolds stress model algebraic stress model BSL baseline low-Re correction Launder Sharma wall functions standard scalable two-layer enhanced generalised log-law blending automatic wall treatment y* y+ blending function F1 F2 limited value Menter clipped limiter production term bounded omega equation Durbin P2210 constraint elliptic relaxation v2-f model gamma-Re_theta transition model Langtry Menter correlation intermittency momentum thickness Re_theta onset criterion local correlation-based LCTM free-stream turbulence level Tu freestream intensity pressure gradient parameter S Thwaites method Falkner-Skan similarity solution Blasius solution Hartree parameter wedge angle stagnation point flow Hiemenz flow Homann flow Görtler vortices curved boundary layer centrifugal instability Rayleigh criterion Taylor-Couette vortices Dean vortices curved duct secondary flow Prandtl secondary flow of second kind Görtler number streamwise curvature ratio R/delta roughness Reynolds number fully rough transitional hydraulically smooth Nikuradse sand grain equivalent roughness k_s Moody chart Colebrook White equation Swamee Jain approximation Haaland explicit formula friction factor Darcy Fanning f = 4f_Darcy relationship head loss major minor fittings valves bends tees expansions contractions K factors discharge coefficient Cd orifice Venturi nozzle weir broad-crested sharp-crested critical depth Froude number subcritical supercritical critical flow hydraulic jump conjugate depths Belanger equation specific energy specific force momentum equation gradually varied flow Manning equation Chezy formula normal depth critical slope mild steep horizontal adverse uniform flow backwater curve step method standard step method direct step method water surface profile classification M1 M2 M3 S1 S2 S3 C1 C2 A1 A2 A3 profiles channel geometry trapezoidal rectangular circular compound natural river cross-section wetted perimeter hydraulic radius area conveyance conveyance weighting factor divided channel method entropy-based stage-discharge rating curve shift adjustments extrapolation flood frequency analysis Log-Pearson Type III GEV generalised extreme value Gumbel Frechet Weibull distributions peak over threshold POT generalized Pareto distribution GPD shape scale location parameters declustering extremal index mean excess plot point process approach annual maximum series return period design flood hydrograph unit hydrograph SCS CN curve number NRCS method initial abstraction retention depression storage effective rainfall excess precipitation rational method time of concentration Kirpich TR-55 TR-55 methodology urban watershed impervious percentage green infrastructure permeable pavement bioswale rain garden detention retention pond wet pond dry pond underground storage tank cistern rainwater harvesting greywater recycling stormwater management BMP best management practice LID low impact development sustainable drainage SuDS attenuation volume overflow spillway emergency spillway principal spillway auxiliary spillway freeboard crest elevation invert elevation storage-elevation curve stage-storage routing level pool modified Puls method storage-indication curve outflow-inflow continuity finite difference explicit implicit Courant number CFL condition von Neumann stability analysis amplification factor spectral radius eigenvalue matrix row sum norm infinity norm induced 2-norm Frobenius norm trace determinant condition number conditioning ill-conditioned pre-conditioning Jacobi Gauss-Seidel successive over-relaxation SOR Chebyshev acceleration conjugate gradient PCG GMRES BiCGSTAB MINRES SYMMLQ QMR TFQMR Krylov subspace Arnoldi Lanczos bidiagonalisation thick restart implicit restarted Arnoldi IRAM shift-invert mode Cayley transform rational Krylov MOR model order reduction POD proper orthogonal decomposition Karhunen-Loève PCA principal component analysis snapshot method empirical interpolation EIM DEIM reduced basis greedy algorithm a posteriori error bound training set offline online decomposition reduced order model ROM manifold approximation trajectory piecewise linear TPWL multi-parametric MPC explicit control law parametric sensitivity continuation pseudo-arclength predictor corrector fold bifurcation saddle-node transcritical pitchfork Hopf Neimark-Sacker period-doubling crisis attractor basin of attraction Lyapunov exponent strange attractor Lorenz system Rössler system Chua circuit hyperchaos Kaplan-Yorke dimension correlation dimension Grassberger Procaccia algorithm Takens embedding theorem delay embedding false nearest neighbours mutual information time delay selection surrogate data testing Theiler window amplitude adjusted Fourier transform AAFT iterative amplitude adjusted Fourier transform IAFT phase randomisation amplitude distribution preservation null hypothesis test linear non-linear detectability limitations power analysis effect size Cohen d conventions practical significance thresholds minimal detectable difference calculation Type I II error rates alpha beta trade-off Neyman Pearson lemma optimal test construction likelihood ratio test sufficient statistic Rao Blackwell theorem Rao Blackwellised estimator variance reduction antithetic variates control variates importance sampling rejection sampling Metropolis Hastings algorithm Gibbs sampler burn-in period thinning autocorrelation trace plots Gelman Rubin diagnostic R-hat convergence threshold effective sample size ESS computation Monte Carlo standard error MCSE estimate credible interval Bayesian posterior summarisation prior specification conjugate priors weakly informative regularising hierarchical models partial pooling shrinkage estimators empirical Bayes methods maximum a posteriori MAP estimation Laplace approximation variational inference mean-field assumption reverse KL divergence ELBO maximisation stochastic gradient variational Bayes SVI mini-batch updates natural gradient descent Fisher information matrix metric tensor geometry manifold structure geodesic paths exponential map retraction operators Riemannian optimisation constraints projection onto Stiefel manifold sphere hypersphere torus Lie group SO(n) SU(n) U(n) representations character theory symmetric functions Schur polynomials Littlewood Richardson coefficients branching rules Young tableaux combinatorics partition theory generating functions asymptotic analysis saddle point method steepest descent Laplace method Watson lemma Tauberian theorems Mellin transform convolution theorem Dirichlet series zeta function analytic continuation meromorphic poles residues residue theorem contour integration Jordan lemma ML inequality bounding techniques stationary phase approximation WKB semiclassical limit quantum mechanics path integral formulation Feynman propagator Green function retarded advanced boundary conditions scattering amplitude cross-section differential total integrated S-matrix unitarity optical theorem dispersion relations Kramers Kronig causality analyticity Schwarz reflection principle maximum modulus principle Liouville theorem Picard theorem normal family Montel criterion Vitali convergence equicontinuity Arzela Ascoli compactness criterion sequential compactness metrisability topological space Hausdorff separation first countable second countable separable Lindelof property Baire category theorem Banach Steinhaus uniform boundedness principle open mapping closed graph Hahn Banach extension theorem dual space annihilator quotient topology weak star topology Banach Alaoglu compactness Tychonoff product compactness Stone Cech compactification one-point compactification Frechet Urysohn metrisation Nagata Smirnov Bing criterion paracompactness refinement subordinate partition unity smooth bump function partition of unity existence proof metric space completion Cauchy sequence equivalence classes isometry extension unique up to unique isometry Borel measurability completion measure zero sets dense G-delta sets comeager sets generic properties prevalence measure-theoretic probability foundations Kolmogorov axioms sigma algebra measurability Caratheodory extension theorem uniqueness Radon Nikodym derivative density function absolute continuity singular measures decomposition Lebesgue decomposition Hahn Jordan signed measure positive negative variation total variation norm bounded variation functions Riemann Stieltjes integral Lebesgue integral dominated convergence monotone convergence Fatou lemma Tonelli Fubini iterated integrals change of variables Jacobian determinant polar coordinates spherical coordinates cylindrical coordinates hyperspherical coordinates volume element surface area element arc length parameterisation curve regularity class C-k smooth analytic real analytic complex analytic holomorphic meromorphic subharmonic plurisubharmonic convexity Jensen inequality majorisation Schur convexity Karamata inequality rearrangement inequalities Hardy Littlewood Polya majorisation relation doubly stochastic matrices Birkhoff von Neumann theorem convex polytope extreme points Caratheodory Radon Helly theorems separation hyperplane supporting hyperplane Krein Milman Minkowski sum support function polar body dual cone polar cone recession cone tangent normal cones limiting Fréchet Clarke subdifferentials nonsmooth optimisation Clarke generalised gradient directional derivative Dini derivatives upper lower Peano derivative approximate derivative symmetric derivative Dini Hadamard condition Weierstrass extremum principle Lagrange multiplier KKT conditions Fritz John conditions Karush Tucker Slater constraint qualification LICQ SOSC second order sufficient conditions curvature Hessian projected gradient Frank Wolfe simplex method interior point barrier primal dual path following central path predictor corrector long step short step variants semidefinite programming SDP conic optimisation second order cone SOCP quadratically constrained quadratic program QCQP trust region dogleg Steihaug TCG conjugate gradient Fletcher Reeves Polak Ribiere PRP variant restarts line search Armijo backtracking Wolfe conditions strong weak curvature condition Goldstein condition nonmonotone Grippo Lanzi Moré Tonelli adaptive stepsizes Barzilai Borwein spectral stepsizes BB1 BB2 alternating direction method ADMM consensus distributed optimisation federated averaging FedAvg secure aggregation differential privacy epsilon delta guarantee local sensitivity global sensitivity composition theorems advanced basic mechanisms Rényi DP zero concentrate DP moments accountant Gaussian mechanism Laplace mechanism exponential mechanism report noisy max propose release select output perturbation objective perturbation gradient perturbation clipping norm bound sensitivity calibration noise scale privacy budget allocation across queries adaptive composition parallel composition advanced composition tight bounds trade-off utility privacy frontier Pareto optimal set knee point detection elbow method silhouette score gap statistic Calinski Harabasz Davies Bouldin indices internal validation external validation holdout cross-validation nested CV leave-one-out LOOCV bootstrap resampling jackknife influence diagnostics Cook distance leverage hat matrix studentised residuals standardised residuals Durbin Watson autocorrelation test Breusch Pagan heteroscedasticity test White test Ramsey RESET specification error test Hausman endogeneity instrument relevance exclusion restriction weak instrument F-statistic Stock Yogo critical values two-stage least squares 3SLS GMM efficient weighting matrix Hansen J overidentification test Anderson Rubin confidence set weak identification robust inference limited information maximum likelihood LIML k-class estimators Fuller modification bias corrected jackknife IV LATE local average treatment effect compliers monotonicity exclusion restriction homogeneity assumptions heterogeneous treatment effects CATE conditional average treatment effect doubly robust estimators TMLE targeted minimum loss based estimation Super Learner ensemble cross-fitting DML double machine learning orthogonal scores Neyman orthogonality moment condition nuisance parameter product rates product rate requirement n^-1/4 n^-1/4 rate requirement cross-fitting folds K fixed number folds grows slowly n^(1/4) fold size requirement empirical process Donsker class VC dimension covering numbers entropy integral uniform Glivenko Cantelli pointwise consistency rates plug-in bias variance tradeoff semi-parametric efficiency bound influence function canonical gradient tangent space quotient space orthogonal complement projection residual maker M-matrix annihilator operator efficient influence curve EIF construction one-step estimator iterated conditional expectation AIPW augmented inverse probability weighting propensity score overlap positivity trimming common support extrapolation violation diagnostics covariate balance standardised mean difference variance ratio empirical CDF Kolmogorov Smirnov distance Mahalanobis distance energy distance maximum mean discrepancy kernel two-sample tests witness function bandwidth selection median heuristic Gaussian kernel polynomial kernel rational quadratic kernel Matern kernel spectral mixture kernel deep kernels neural network feature maps random Fourier features Bochner theorem spectral density input output cross-covariance structure functional ANOVA decomposition main effects interaction terms Sobol indices total effect indices Shapley effects cooperative game theory value allocation axioms efficiency symmetry null player additivity uniqueness proof Monte Carlo estimation Saltelli scheme pick-freeze design scrambled Sobol sequence direction numbers primitive polynomials finite field arithmetic GF(2^m) primitive element generator polynomial irreducible polynomial minimal polynomial characteristic polynomial companion matrix eigenvalue computation QR iteration implicit shifts Wilkinson shift Francis double shift QZ algorithm generalized eigenvalue problem Sylvester equation Lyapunov equation Riccati algebraic differential continuous-time discrete-time invariant subspace deflation reordering Schur form quasi-triangular real Schur complex Schur triangular form eigenvalues diagonal entries ordered by sorting criteria ascending descending magnitude real part imaginary part clustering DBSCAN k-means hierarchical agglomerative Ward linkage complete linkage single linkage average linkage UPGMA WPGMA centroid linkage medoid k-medoids PAM CLARA CLARANS large-scale variants mini-batch stochastic online streaming data structures kd-tree ball tree cover tree vantage-point tree nearest neighbour search approximate ANN algorithms locality-sensitive hashing LSH random projection Johnson Lindenstrauss dimensionality reduction intrinsic dimensionality estimation correlation dimension Levina Bickel MLE estimator two-nearest-neighbour maximum likelihood rank consistency stability subspace angle principal angles Grassmannian manifold Stiefel manifold orthogonal Procrustes rotation translation alignment correspondence matching RANSAC outlier robust estimation M-estimators Huber Tukey bisquare Hampel redescending psi functions influence function bounded breakdown point 50% high breakdown estimators S-estimator MM-estimator median absolute deviation MAD scaled MAD interquartile range IQR trimmed mean winsorized mean quantile regression LAD least absolute deviations check loss pinball loss asymmetric penalty quantile crossing non-crossing constraints rearrangement isotonic regression pool adjacent violators PAVA algorithm monotone smoothing trend filtering lasso fused lasso trend filtering generalised lasso difference penalty graph laplacian structured sparsity group lasso sparse group lasso elastic net ridge regression shrinkage James Stein estimator Stein phenomenon paradox unbiased dominates biased admissibility minimax risk Bayes risk frequentist Bayes compromise empirical Bayes Robbins formula mixing distribution deconvolution nonparametric maximum likelihood NPMLE Kiefer Wolfowitz algorithm discrete support mass points isotonic concavity greatest convex minorant least concave majorant Grenander estimator monotone density decreasing unimodal symmetric unimodal peaked symmetric densities logarithmic concave strongly log-concave Prékopa Leindler Brascamp Lieb concentration inequality Gaussian isoperimetric log-Sobolev modified log-Sobolev transport cost Wasserstein distance Kantorovich Rubinstein dual Lipschitz functions Benamou Brenier dynamic formulation Monge problem optimal transport map pushforward measure coupling joint marginal disintegration Brenier McCann polar factorization Jacobian conjecture displacement interpolation McCann interpolation displacement convexity geodesic convexity Wasserstein barycenter Sinkhorn algorithm entropic regularization scaling matrix Sinkhorn Knopp iteration convergence rate O(1/n) acceleration Nesterov extrapolation Greenkhorn stochastic Sinkhorn minibatch approximations POT optimal partial transport unbalanced OT marginal relaxation KL penalty Wasserstein proximal operator proximal splitting forward-backward Douglas Rachford Peaceman Rachford Chambolle Pock primal dual hybrid gradient PDHG Condat Vu Chambolle Pock primal dual splitting three operator splitting Davis Yin forward-backward-forward Tseng FBF Korpelevich extragradient optimistic mirror descent online learning regret bound sublinear O(sqrt(T)) Hedge multiplicative weights expert advice Follow The Perturbed Leader FTPL minimax regret lower bound Omega(sqrt(KT log N)) adversarial stochastic oblivious nature sequences non-stochastic full information bandit partial information feedback types semi-bandit aggregate reward knapsack constraints contextual combinatorial bandits sleeping experts sleeping experts availability constraint time-varying action sets adversarial environments Nature adversary adaptive oblivious non-anticipating budgeted adversary strongly adaptive regret sliding window regret dynamic regret comparator sequence path length variation budget comparator switching cost switching budget number switches constrained budgets tracking best expert dynamic oracle sequence prediction Solomonoff prediction universal prior Bayesian mixture predictive distribution prequential evaluation logarithmic loss cumulative code length Kraft inequality prefix code uniquely decodable Huffman coding arithmetic coding ANS rANS tANS zstd brotli LZ4 snappy deflate bzip2 xz lzma Zstandard dictionary training compression ratio speed tradeoff entropy coder state machine implementation bit buffer byte alignment padding flush finish marker block header frame structure checksum CRC32 xxHash fast hash collision resistance cryptographic hash SHA-256 SHA-3 BLAKE3 Argon2 scrypt bcrypt PBKDF2 password hashing key derivation salt pepper iteration count timing attack constant time comparison side channel resistance hardware security module HSM trusted platform module TPM secure enclave TEE SGX SEV TrustZone attestation remote attestation quote nonce PCR extend chain of trust root of trust certificate transparency CT logs SCT signed certificate timestamp OCSP stapling CRL revocation short-lived certificates ACME protocol Let's Encrypt certbot automation DNS TXT record validation HTTP-01 TLS-ALPN-01 challenge types wildcard certificate SAN multi-domain single domain DV OV EV extended validation organization validation identity assurance level IAL AAL CAL NIST SP 800