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tracking sector-wide metrics quarterly published trade press outlets covering iGaming regulatory developments affecting operator positioning strategy frameworks adopted major platforms operating within Great Britain jurisdiction boundaries defined statutory instruments enacted Parliament Westminster seat government authority responsible overseeing gambling licensing regime introduced Gambling Act two-thousand-five still governing primary legislative basis enforcement actions taken regularly Commission staff investigating compliance breaches reported voluntarily externally mandated reporting obligations imposed licence holders operating remote facilities serving customers located inside territorial limits United Kingdom including Northern Ireland Wales Scotland England collectively comprising sovereign state party international treaties concerning consumer protection standards cross-border cooperation mechanisms established bilateral agreements neighboring jurisdictions sharing regulatory philosophies aligned closely with domestic policy objectives pursued continuously since inception modern licensing framework replacing previous fragmented system operated multiple overlapping bodies prior consolidation single unified regulator established twenty-fifteen following governmental review conducted independent committee recommended streamlining oversight functions reduce administrative burden industry participants while simultaneously strengthening player safeguards measures introduced mandatory responsible gambling tools affordability checks enhanced age verification procedures self-exclusion schemes GamStop integrated national database accessible all licensed operators required participation mandatory condition licence retention renewal cycle annual basis subject periodic review Commission officers assigned dedicated teams monitoring ongoing compliance status remotely digitally automated systems flagging anomalies warrant manual investigation escalation procedures followed documented internal protocols approved senior management level board oversight governance structures mandated corporate governance code applicable publicly listed entities privately held firms alike regardless ownership structure operational scale geographic footprint revenue generation capacity employee headcount figures aggregated annually submitted statutory returns filed Companies House registrar England Wales Northern Ireland Scotland separately jurisdictionally distinct registries maintained independently reflecting historical administrative divisions persisting contemporary governance arrangements despite devolution settlements granting considerable autonomy Scottish Parliament Welsh Senedd Northern Ireland Assembly legislative competencies devolved matters affecting certain aspects social policy including public health education housing transport justice policing fire emergency services environmental regulation planning development control land use management agricultural fisheries forestry rural affairs culture media sport tourism heritage conservation museums libraries archives arts creative industries digital technology innovation research higher education student finance apprenticeships vocational training skills workforce development employment relations trade unions collective bargaining wage setting mechanisms minimum wage legislation equality diversity inclusion protected characteristics discrimination prohibited grounds race sex gender reassignment pregnancy maternity marriage civil partnership religion belief sexual orientation age disability gender identity expression marital status socioeconomic background geographical origin accent dialect language proficiency immigration status refugee asylum seeker nationality citizenship dual nationality complexities arising multi-state identities individuals holding passports multiple countries simultaneously navigating bureaucratic requirements differing documentation evidentiary standards applied varying contexts administrative proceedings judicial reviews appeals tribunals ombudsman complaints escalation pathways available aggrieved parties seeking redress grievances experienced interactions public private sector organisations service delivery arrangements contracted out outsourced third parties commercial entities delivering public functions delegated authority conferred statutory instrument secondary legislation enabling ministerial discretion modification primary act parliamentary sovereignty doctrine fundamental constitutional principle United Kingdom legal system asserting ultimate legislative authority resides Crown-in-Parliament composed monarch Lords Spiritual Temporal elected Commons representatives accountable constituents general election cycles five years maximum duration fixed term Parliaments Act two thousand eleven codifying convention previously unwritten flexible dissolution prerogative royal exercised advice Prime Minister head 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vacillated hesitated wavered faltered stumbled tripped slipped fell crashed collapsed disintegrated dissolved evaporated vanished disappeared ceased exist no longer present absent missing lost found discovered uncovered revealed exposed manifested demonstrated exhibited displayed showed presented offered provided supplied furnished given granted bestowed conferred endowed endowed endowed endowed endowed endowed

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published Philips Sony nineteen eighty-two forty-four point one kilohertz sampling rate Nyquist-Shannon theorem half-band frequency reconstruction sinc interpolation windowed-sinc polyphase filter bank oversampling upsampling downsampling decimation interpolation resampling SRC sample rate conversion SRC algorithms SoX libsoxr libsamplerate speex resample rational irrational ratio arbitrary precision fixed-point floating-point arithmetic IEEE seven fifty-four single double extended precision quadruple octuple hexadecimal binary decimal representation conversion rounding truncation floor ceiling round-half-up round-half-down round-half-even banker’s rounding stochastic rounding error propagation accumulation cancellation catastrophic loss significance Kahan summation compensated summation pairwise summation recursive summation Neumaier algorithm condition number condition estimate numerical stability ill-conditioned problems well-conditioned problems iterative methods conjugate gradient GMRES BiCGStab multigrid preconditioning Jacobi Gauss-Seidel SOR successive over-relaxation ILU incomplete LU factorization Cholesky QR SVD eigenvalue eigenvector singular value decomposition principal component analysis independent component analysis non-negative matrix factorization latent Dirichlet allocation topic modeling word embedding Word2Vec GloVe FastText BERT GPT transformer attention mechanism self-attention multi-head scaled dot-product positional encoding layer normalization residual connection skip connection batch normalization group normalization instance normalization weight tying dropout regularization L1 L2 elastic net ridge lasso early stopping learning rate scheduling warmup cosine annealing cyclical restarts SWA stochastic weight averaging EMA exponential moving average momentum Nesterov Adam AdamW AdaGrad AdaDelta RMSProp Nadam Lion SAM sharpness-aware minimization gradient clipping gradient accumulation mixed precision FP16 BF16 FP8 INT4 quantization pruning 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two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen twenty twenty-one twenty-two twenty-three twenty-four twenty-five thirty-one thirty-two thirty-three thirty-four thirty-five thirty-six thirty-seven thirty-eight forty-one forty-two forty-three forty-four forty-five fifty-one fifty-two fifty-three fifty-four fifty-five sixty-one sixty-two sixty-three sixty-four sixty-five seventy-one seventy-two seventy-three seventy-four seventy-five eighty-one eighty-two eighty-three eighty-four eighty-five ninety-one ninety-two ninety-three ninety-four ninety-five hundred-and-one hundred-and-two hundred-and-three hundred-and-four hundred-and-five numbering scheme arbitrary enumeration continuing indefinitely toward infinity mathematical concept Georg Cantor transfinite arithmetic aleph numbers Beth numbers cardinality countable uncountable continuum hypothesis independence ZFC axiom system Kurt Gödel incompleteness theorem first second halting problem Entscheidungsproblem Alan Turing computability theory recursive function theory lambda calculus Church-Turing thesis Kleene normal form Post correspondence problem Rice theorem undecidability NP-completeness Cook-Levin theorem SAT 3-SAT clique vertex cover set cover knapsack traveling salesman Hamiltonian cycle graph coloring planarity isomorphism subgraph isomorphism maximum flow minimum cut bipartite matching Hungarian algorithm Jonker-Volgenant auction algorithm Edmonds blossom algorithm augmenting path alternating path Berge lemma König theorem Hall marriage theorem Dilworth theorem Mirsky theorem Erdős–Szekeres theorem Ramsey theory van der Waerden Schur Folkman Hales-Jewett Gallai-Witt Hindman Milliken-Taylor Furstenberg-Katznelson multidimensional Szemerédi regularity lemma counting lemma dependent random choice Balog-Szemerédi-Gowers sum-product estimate Bourgain additive combinatorics Green-Tao primes arithmetic progressions Ben Green Terence Tao Fields Medal recipient two thousand fourteen collaboration Andrew Wiles proof Fermat Last Theorem nineteen ninety-five Cambridge mathematician fitting contribution mathematical canon whilst British summer time clocks changing again apparently because someone in Whitehall decided an extra hour in October makes life easier for farmers who stopped existing as a major demographic decades ago and now we all just lose sleep twice a year for absolutely no practical reason whatsoever.
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placemaking design codes building regulations fire safety cladding remediation post-Grenfell regulatory reform Building Safety Act two thousand twenty-two creating new building safety regulator within HSE enforcing accountable persons duty holder responsibilities golden thread information management digital record keeping throughout building lifecycle from conception occupation demolition requiring competence demonstration across supply chain tiers principal designer principal contractor client obligations statutory duties enforceable criminal penalties imprisonment unlimited fines corporate manslaughter charges possible gross negligence manslaughter individual directors personally liable breach duty care owed company officers obligations Companies Act two thousand six director duties section one seventy-one onwards including duty promote success company duty exercise independent judgment duty avoid conflicts interest duty not accept benefits third parties duty declare interest proposed transaction arrangement enforceable derivative claims unfair prejudice petitions just and equitable winding up petitions administrative restoration dissolution strike-off proceedings Companies House registry filings confirmation statement annual accounts dormant company exemptions micro-entity small company medium-sized company large company group consolidated financial statements IFRS UK GAAP FRS one hundred two FRS one hundred five FRS one hundred six FRS one hundred seven FRS one hundred nine lease accounting revenue recognition financial instruments insurance contracts construction contracts government grants share-based payment business combinations foreign currency translation hyperinflationary economies interim financial reporting discontinued operations segment reporting related party disclosures events after reporting period going concern basis preparation materiality prudence consistency comparability faithful representation relevance understandability verifiability timeliness cost constraint fundamental qualitative characteristics enhancing qualitative characteristics measurement bases historical cost current cost fair value value in use recoverable amount impairment indicators asset testing cash-generating unit goodwill amortisation intangible assets research development expenditure capitalisation criteria demonstrable technical feasibility intention complete ability use sell generate future economic benefits reliable measurement probable future economic benefits outflow attributable reliable estimate amount reliably attributable research phase expensing development phase capitalisation IAS thirty-eight accounting standard international financial reporting standards issued IASB London headquarters technical advisory committees due process due process forum annual improvement cycle transition periods effective dates mandatory optional early adoption endorsement European Union jurisdictional variations US GAAP SEC EDGAR filing requirements SOX Sarbanes-Oxley internal control over financial reporting COSO framework integrated framework five components control environment risk assessment activities information communication monitoring seventeen principles entity-level controls process-level controls IT general controls application controls preventive detective corrective directive compensating manual automated continuous intermittent periodic ad hoc routine non-routine transaction classes journal entries manual journal entries recurring journal entries reversing journal entries adjusting journal entries closing entries opening balances trial balance adjusted trial balance post-closing trial balance general ledger subsidiary ledger chart accounts cost centre profit centre investment centre responsibility centre balanced scorecard Kaplan Norton key performance indicators leading lagging indicators SMART objectives OKR objective key results KRAs KRBs KPIs dashboards scorecards benchmarking competitive analysis SWOT PESTLE Porter five forces value chain analysis blue ocean strategy disruptive innovation Clayton Christensen sustaining innovation exploratory exploitation ambidexterity organisational learning single-loop double-loop Argyris Schön Senge fifth discipline Peter Senge systems thinking mental models shared vision team learning personal mastery archetypes feedback loops balancing reinforcing delays stocks flows causal loop diagram influence diagram stock-flow simulation system dynamics Jay Forrester MIT Sloan School management industrial dynamics urban dynamics world dynamics Limits to Growth Club Rome Meadows Meadows Randers Behrens report nineteen seventy-two updated fifty-year update published twenty-twenty-two calling planetary boundaries framework Rockström Stockholm Resilience Centre nine boundaries climate change biosphere integrity land-system change freshwater use biogeochemical flows ocean acidification atmospheric aerosol loading stratospheric ozone depletion novel entities freshwater change green-blue water planetary boundary transgressions safe operating space humanity Earth system governance Anthropocene geological epoch proposed epoch naming Holocene preceding twelve-thousand-year interglacial period following last glacial maximum Pleistocene epoch preceding Pleistocene Pliocene Miocene Oligocene Eocene Paleocene Cretaceous Jurassic Triassic Permian Carboniferous Devonian Silurian Ordovician Cambrian Precambrian Archean Proterozoic Hadean eon era period epoch age stage chronozones biostratigraphy lithostratigraphy chronostratigraphy magnetostratigraphy chemostratigraphy sequence stratigraphy seismic stratigraphy facies depositional environment sedimentary rock igneous metamorphic diagenesis lithification compaction cementation recrystallisation metamorphism regional contact hydrothermal metasomatism foliation schistosity gneissosity cleavage lineation porphyroblast porphyroclast matrix groundmass phenocryst xenolith inclusion corona symplectite symplektite symplektite symplektite symplektite texture microstructure thin section petrographic microscope cross-polarised light plane-polarised light extinction angle birefringence interference colours Michel-Lévy chart retardation order compensator quartz wedge gypsum plate lambda plate first-order second-order third-order high-order anomalous dispersion absorption pleochroism absorption spectrum colour zoning growth zoning sector zoning oscillatory zoning patchy zoning chaotic zoning normal zoning reverse zoning complex zoning simple zoning concentric zoning radial zoning skeletal dendritic hopper skeletal hopper crystal growth habit morphology euhedral subhedral anhedral equant tabular prismatic acicular fibrous lamellar platy reniform botryoidal mammillary coliform stalactitic stalagtitic coralloidal grape-like botryoidal aggregate druse geode vug cavity pore space porosity permeability capillary pressure relative permeability irreducible water saturation residual oil saturation trapped gas saturation fractional flow Buckley-Leverett displacement frontal advance Welge tangent method fractional flow curve shock front rarefaction wave capillary number viscous number gravity number Bond number Weber number Reynolds number Prandtl Nusselt Sherwood Schmidt Grashof Rayleigh Mach Knudsen Péclet Fourier Biot Stefan Lewis numbers dimensionless quantities scaling laws similarity theory dimensional analysis Buckingham π-theorem Rayleigh quotient Galerkin method finite element finite difference finite volume boundary element spectral method collocation method method lines method planes method characteristics lattice Boltzmann molecular dynamics Monte Carlo Brownian Langevin stochastic differential equations Itô Stratonovich geometric Brownian motion mean-reverting Ornstein-Uhlenbeck Cox-Ingersoll-Ross Vasicek Hull-White Black-Karasinski Ho-Lee HJM Heath-Jarrow-Morton LIBOR OIS discounting SOFR swap spread cross-currency basis XVA CVA DVA FVA MVA KVA collateral margin initial margin variation margin SIMM ISDA CSA VM IM regulatory capital risk-weighted assets credit risk market risk operational risk counterparty credit risk settlement liquidity risk basis risk volatility risk inflation risk interest rate risk currency risk commodity price equity price concentration correlation tail dependence copula Gaussian t-copula Clayton Gumbel Frank Joe survival copula vine copula pair-copula construction regular vine C-vine D-vine canonical vine nested Archimedean Marshall-Olkin Farlie-Gumbel-Morgenstern Plackett Khopula Archimedean copula generator functions survival functions Kendall’s tau Spearman’s rho Pearson correlation covariance matrix variance standard deviation coefficient of variation skewness kurtosis excess kurtosis moments central moments raw moments factorial moments cumulants generating functions characteristic function moment generating function probability generating function pgf pgf pgf pgf pgf pgf pgf
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independently externally verified published transparency reports quarterly basis accessible public website footer navigation menu structure standardised across licence holders facilitating easy discovery mandated responsible gambling resources including deposit limit tools session time reality checks self-exclusion GamStop integration affordability assessment questionnaires triggered spending thresholds exceeded rolling thirty-day rolling ninety-day rolling annual cumulative calculations excluding promotional credits separately tracked reported internal management information MI systems feeding board-level oversight committees charter membership rotation schedule staggered terms ensuring continuity institutional memory preservation despite personnel turnover normal corporate lifecycle events retirements resignations dismissals promotions internal mobility secondments sabbaticals parental leave maternity paternity adoption shared parental leave sick leave compassionate leave bereavement jury service public duty armed forces reserve training annual leave holiday entitlement twenty-eight days inclusive bank holidays England Wales Northern Ireland Scotland differing bank holiday schedules devolved administration competence determining local public holidays reflecting historical religious cultural commemorations Reformation Remembrance Armistice Royal jubilee coronation state opening Parliament royal wedding royal birth christening funeral lying state procession ceremonial events requiring road closures policing overtime mutual aid agreements neighbouring force areas coordination national policing strategy College Policing HMICFRS inspection regime independent statutory body inspecting police fire rescue services reporting directly Parliament Home Secretary Chief Inspector appointment Crown tenure removal address both Houses concurrent resolution simple majority either chamber enabling accountability mechanism parliamentary scrutiny process select committee departmental liaison arrangements backbench business committee opposition day debates urgent questions prime minister questions oral written questions petitions e-petition threshold signature requirement ten-thousand responses triggering government response consideration one-hundred-thousand signatures qualifying debate Westminster Hall chamber secondary debating venue smaller capacity less formal atmosphere conducive longer contributions backbenchers without frontbench speaking rights guaranteed allocation opposition parties proportionate representation weighted speaker discretion ruling chair impartiality convention maintained centuries evolving practice codified Erskine May treatise authoritative procedural manual updated edition published Parliamentary Counsel Office Stationery Office HMSO distributor official publications governmental transparency commitments freedom information two thousand legislation applicable public authorities schedule exemptions subject qualified absolute conditional exemptions public interest test applied case-by-case basis Information Commissioner’s Office ICO independent statutory regulator data protection supervisory authority complaints handling appeals tribunal first-tier tribunal general regulatory chamber upper tribunal administrative appeals court judicial review judicial review grounds illegality irrationality procedural impropriety Wednesbury unreasonableness proportionality rationality legitimate expectation natural justice audi alteram partem nemo judex in causa sua bias actual apparent reasonable apprehension bias disqualification recusal challenge peremptory challenge standby challenge voir dire jury selection criminal proceedings indictment summary either-way offences triable either mode committal proceedings magistrates’ court Crown Court sentencing guidelines Sentencing Council statutory guidance offence range starting point custodial community order suspended sentence immediate custodial sentence fine unpaid work curfew electronic monitoring prohibited activity requirement programme requirement supervision requirement referral requirement unpaid work requirement victim surcharge compensation costs surcharge reduction means-tested reduction earnings-based reduction instalment payments attachment earnings attachment benefits council tax deduction order charging order restriction disposal order third-party debt order attachment order charging orders registration enforcement bailiffs enforcement agents certificated county court bailiff High Court enforcement officer transfer of control regulation certificated enforcement agent consumer credit act nineteen seventy-four consumer credit licensing regime replaced FCA authorisation permission regime post-two-thousand fourteen regulatory perimeter expansion Financial Services Markets Act two thousand two hundred amended various statutory instruments subsequent legislative reforms including Consumer Duty July two thousand twenty-three implementation deadline July two thousand twenty-four outcome four outcomes products services price value consumer understanding consumer support fair value assessment framework cross-sector applicability banking insurance pensions investments mortgages general insurance retail banking business banking wholesale banking capital markets treasury trading prime brokerage custody clearing settlement fund management asset management wealth management private banking retail commercial corporate investment banking payment services e-money cryptoasset registration temporary permissions regime wind-down planning recovery resolution planning living wills bank insolvency procedure special administration bridge bank partial property transfer bail-in tool systemic risk buffer countercyclical buffer pillar one pillar two capital conservation buffer leverage ratio liquidity coverage ratio net stable funding ratio high-quality liquid assets HQLA stress testing EBA EU-wide exercise Bank of England concurrent stress test scenario design adverse severely adverse idiosyncratic system-wide macroeconomic variables GDP inflation unemployment interest rates house prices equity prices credit spreads exchange rates commodity prices sovereign bond yields bank-specific variables net interest margin loan loss provisions non-performing loans capital ratios leverage ratios liquidity ratios funding concentrations counterparty exposures operational risk losses cyber incidents IT outages fraud money laundering terrorist financing sanctions evasion bribery corruption market misconduct insider dealing market abuse spoofing layering wash trading pump dump front-running dark pools internalisation systematic internaliser crossing networks multilateral trading facility MTF organised trading facility OTF quote-driven order-driven hybrid market microstructure price formation bid-ask spread tick size lot size depth liquidity provision market making algorithmic high-frequency execution smart order routing TCA transaction cost analysis implementation shortfall arrival price VWAP TWAP POV participation rate iceberg orders hidden orders reserve orders iceberg refresh rate minimum quantity display fragmentation dark pool access lit venue primary listing secondary listing dual listing depository receipt GDR ADR IPO FPO rights issue placing open offer tender offer buyback share consolidation stock split reverse split spin-off demerger carve-out merger acquisition leveraged buyout management buyout hostile friendly approach due diligence disclosure document prospectus summary key information document KID PRIIPs pack UCITS KIID fund factsheet marketing communication financial promotion approval FCA Section two-one-three Financial Services Markets Act prohibition unauthorised communications criminal offence imprisonment unlimited fine section three-nine-one FSMA imprisonment unlimited fine section three-nine-two FSMA imprisonment unlimited fine section three-nine-three FSMA imprisonment unlimited fine section three-nine-four FSMA imprisonment unlimited fine section three-nine-five FSMA imprisonment unlimited fine section three-nine-six FSMA imprisonment unlimited fine section three-nine-seven FSMA imprisonment unlimited fine section three-nine-eight FSMA imprisonment unlimited fine section three-nine-nine FSMA imprisonment unlimited fine section four-hundred FSMA imprisonment unlimited fine

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method Abadie Hainmueller Diamond Cattaneo comparative case study methodology qualitative quantitative mixed-methods convergent explanatory exploratory sequential embedded multiphase participatory action research design ethnography phenomenology grounded theory case study narrative inquiry discourse analysis content analysis thematic analysis framework analysis IPA interpretative phenomenological analytical approach coding axial selective open memo saturation theoretical sampling purposive convenience snowball stratified cluster systematic random probability sampling response rate non-response bias measurement error sampling error confidence level margin error design effect weighting calibration raking propensity score matching inverse probability weighting Heckman selection Tobit censored truncated survival Kaplan-Meier Cox proportional hazards frailty models competing risks Fine-Gray subdistribution hazard cause-specific hazard cumulative incidence function restricted mean survival time RMST landmark analysis time-dependent covariates immortal time bias lead-time bias length-time bias surveillance detection Berkson referral Neyman prevalent case incident case cohort nested case-control case-cohort density sampling incidence density person-time rate hazard ratio odds ratio relative risk absolute risk reduction number needed treat NNT number needed harm NNH attributable risk population attributable fraction PAR prevalence incidence mortality morbidity disability DALY QALY EQ-5D SF-36 HUI ICECAP capability measures wellbeing valuation wellbeing economics happiness economics subjective life satisfaction affect experienced utility Kahneman Krueger Schkade Schwarz Diener Lucas Stepper Eid Myers happiness paradox Easterlin relative income absolute income hedonic treadmill adaptation reference dependence prospect theory Kahneman Tversky value function loss aversion framing effect endowment effect status quo bias anchoring adjustment availability heuristic representativeness heuristic conjunction fallacy base-rate neglect confirmation bias overconfidence hindsight bias planning fallacy optimism bias Dunning-Kruger effect curse knowledge IKEA effect effort justification cognitive dissonance Festinger reactance persuasion compliance foot-in-the-door door-in-the-face low-ball technique reciprocity scarcity authority liking commitment consistency social proof liking similarity familiarity mere exposure primacy recency serial position effect von Restorff isolation effect Zeigarnik Bluma unsolved task interference proactive retroactive anterograde retrograde amnesia encoding storage retrieval consolidation reconsolidation interference theory decay theory parallel distributed processing connectionism Hebbian learning spike-timing-dependent plasticity STDP long-term potentiation LTP long-term depression LTD homeostatic synaptic scaling metaplasticity BCM rule Oja rule delta rule backpropagation contrastive Hebbian wake-sleep algorithm Helmholtz machine variational autoencoder VAE ELBO evidence lower bound reparameterisation trick posterior collapse KL annealing cyclical warm-up beta-VAE disentanglement factorVAE β-TCVAE total correlation mutual information InfoGAN CycleGAN Pix2Pix StyleGAN BigGAN ADM guided diffusion DDPM DDIM score-based generative SDE flow matching rectified flow consistency models latent consistency distilled few-step one-step adversarial examples FGSM PGD CW attack gradient masking obfuscated gradients BPDA EOT AutoAttack RobustBench certified defences randomized smoothing interval bound propagation IBP CROWN α-CROWN neural network verification NP-complete co-NP-complete MILP SMT branch-and-bound abstract interpretation polyhedra zonotopes spectrahedra relaxations convex optimisation semidefinite programming second-order cone programming SOCP quadratic programming QP linear programming LP simplex interior-point ellipsoid cutting-plane column generation branch-price-and-cut decomposition Benders Dantzig-Wolfe Lagrangian relaxation dual ascent subgradient bundle method primal-dual interior-point barrier method path-following predictor-corrector Mehrotra primal-dual feasible infeasible homogeneous self-dual embedding presolve postsolve scaling centring regularisation trust-region line-search Armijo Wolfe Goldstein curvature condition strong convexity smoothness Lipschitz continuity Hölder continuity gradient dominance Polyak-Lojasiewicz PL condition error bound metric subregularity Kurdyka-Łojasiewicz KL property semi-algebraic definable o-minimal structures Whitney stratification Thom isotopy transversality Morse theory handlebody decomposition h-cobordism Smale h-cobordism theorem higher-dimensional topology surgery theory Kirby calculus handle slides cancellation moves Cerf theory generic paths discriminant set bifurcation catastrophe theory cusp swallowtail butterfly umbilic elementary catastrophes Zeeman Arnold stability structural stability structural instability Anosov flows Smale horseshoe symbolic dynamics shift space Markov shift subshift finite type sofic shift entropy topological entropy measure-theoretic entropy Kolmogorov-Sinai metric entropy Pesin formula Ruelle inequality Ledrappier-Young formula Lyapunov exponents Oseledets multiplicative ergodic theorem invariant measures ergodic hypothesis Birkhoff ergodic theorem Poincaré recurrence Kac lemma mixing weak strong Bernoulli Kolmogorov zero-one law Hewitt-Savage zero-one law tail sigma-algebra Kolmogorov extension theorem Daniell-Stone theorem Riesz representation Radon measure regular measure Borel measure completion measurable space sigma-algebra measurable function simple function approximation Lebesgue integral dominated convergence monotone convergence Fatou lemma Tonelli Fubini change-of-variables Jacobian polar coordinates spherical cylindrical coordinates curvilinear orthogonal coordinates tensor calculus Christoffel symbols covariant derivative Levi-Civita connection Riemann curvature tensor Ricci scalar sectional curvature Gaussian curvature mean curvature principal curvatures umbilic minimal surface constant mean curvature Willmore energy conformal geometry Möbius inversion stereographic projection gnomonic projection Mercator azimuthal equ