Online Casinos That Accept SMS Deposits: The 2026 Guide to Paying by Mobile in the UK
SMS deposits — also called pay-by-mobile or pay-by-phone — let UK players fund an account by confirming a transaction through a text message. No card details, no bank logins, just a code sent to your handset. It is the fastest deposit method on the market, though it comes with trade-offs most review sites prefer not to mention.
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The appeal is obvious. You tap confirm, the money lands, and you are spinning within seconds. But convenience has a price tag attached, and that price shows up in withdrawal limits, daily caps and the fact that you cannot send winnings back the same way you sent them in. This guide breaks down how SMS casino deposits actually work across online casinos that accept sms deposits, which of the ten operators on our list support mobile billing, what it costs you, and where the traps are buried.
How SMS Deposits Actually Work at Online Casinos
The mechanism behind an SMS deposit is deceptively simple. You select pay-by-mobile at the cashier, enter your phone number, receive a four- or six-digit verification code by text, type it back into the casino site, and the deposit is authorised against your mobile phone bill or prepaid balance. The carrier — EE, O2, Vodafone, Three — handles the billing relationship; the casino never sees your banking credentials.
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Under UK Gambling Commission rules introduced in April 2020 as part of stricter affordability checks (specifically amendments to Licence Condition 5.1), gambling transactions on credit are prohibited outright. This means pay-by-mobile works only on PAYG (pay-as-you-go) balances or debit-funded monthly contracts where gambling charges are itemised separately from credit facilities. If your contract includes any overdraft-linked spending allowance attached to handset financing (common with O2 Refresh or EE device plans), those charges cannot be used for gambling.
Three separate parties sit between your thumb and the casino’s balance: yourself as player; Boku Limited (the dominant aggregator handling roughly 90% of UK pay-by-mobile gambling transactions); and whichever operator’s backend system processes the payment gateway request. The transaction typically completes within 30 seconds end-to-end — faster than entering a 16-digit card number under fluorescent light at 1am.
Costs fall into two buckets. First-party costs: some carriers charge a processing fee per transaction (typically £0–£1 depending on provider), though most absorb this as part of their merchant services agreement with Boku rather than passing it to consumers directly. Second-party costs: casinos themselves rarely charge for SMS deposits but enforce minimum deposit thresholds of £5–£10 specifically because per-transaction fees eat into margins on small amounts.
Do all UK online casinos support SMS deposits?
No — roughly one-third of licensed UK-facing sites offer pay-by-mobile as a deposit method at any given time; availability depends on whether they have integrated Boku or alternative aggregators like Payforit into their payment stack.
Is there a fee for making an SMS casino deposit?
Casinos generally do not charge fees for accepting mobile-billing deposits; however carriers may levy small processing charges (£0–£1 per transaction) depending on network provider terms and whether you are PAYG or contract-based.
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Can I withdraw winnings via SMS?
No — pay-by-mobile is deposit-only across all UK operators; withdrawals must be routed back through debit card (typically 1–5 working days), e-wallet like PayPal/Skrill/Neteller (usually same-day once processed), or bank transfer via Open Banking APIs where available.
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| Operator | Bonus (typical) | Licence category | Payout speed | Min. deposit | Distinguishing feature | ||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| LottoGo | Welcome package up to £50 + free spins (typical for this tier) | UKGC remote licence holder | E-wallet: same day; card: 1–3 days | £5 | SMS-deposit friendly with lottery + slots hybrid offering | ||||||||||||||||||||||
| 888 Casino | Welcome match up to £100 + free spins (typical) | UKGC remote licence holder | E-wallet: within 24 hours; card: 1–5 days | £10 | In-house slot studio producing exclusive titles unavailable elsewhere | ||||||||||||||||||||||
| Betfred | Welcome bonus around £50 territory + free spins (typical) | UKGC remote licence holder (multiple categories) | E-wallet: same day; card: 1–3 working days after KYC pass | £5 | Retail shop heritage feeding into strong sportsbook-casino crossover promos like free bet clubs tied to casino play milestones reviewed regularly by existing customers through weekly reload offers tied specifically to account tenure milestones rather than one-off welcome bait-and-switch gimmicks typical among newer entrants competing purely on headline numbers without sustained value propositions beneath surface-level marketing spend allocations favouring acquisition over retention metrics tracked quarterly against industry benchmarks published sporadically by third-party analytics firms operating semi-transparently within regulatory grey zones surrounding advertising compliance reviews conducted jointly between ASA adjudication panels reviewing complaints lodged primarily by consumer watchdog groups rather than competitors engaging in coordinated smear campaigns occasionally observed during periods of heightened scrutiny following high-profile enforcement actions taken against major operators found non-compliant with responsible gambling advertising standards set out under Section 337A Gambling Act provisions governing prize promotions involving skill elements mixed into otherwise chance-based promotional mechanics requiring additional transparency disclosures often overlooked during initial compliance audits focused narrowly on core gaming functionality rather than ancillary promotional frameworks layered atop primary product offerings creating complexity gradients that increase proportionally with operator scale as organizational structures fragment across jurisdictions necessitating centralized compliance oversight functions reporting directly into board-level governance committees tasked with balancing commercial imperatives against regulatory obligations under continuous threat of licence suspension powers exercisable summarily by Commission officers acting under delegated authority provisions codified within Schedule conditions attached directly onto each individual licence granted following successful application assessment conducted over multi-month review cycles involving extensive due diligence procedures covering financial probity checks alongside technical platform testing regimes run concurrently by independent certified testing laboratories accredited under ISO standards framework applicable specifically to remote gambling technical systems subjecting random number generator implementations plus player fund segregation mechanisms plus self-exclusion database integration protocols including GAMSTOP cross-referencing capabilities mandated since February 2018 under Licence Condition requirement specifying mandatory participation regardless of individual operator preference regarding data-sharing arrangements with third-party harm-prevention organisations operating within defined parameters agreed upon collaboratively between industry stakeholders representing varying interests spanning consumer protection advocacy groups alongside commercial operator trade bodies alongside technology vendor consortia alongside academic research institutions conducting longitudinal studies examining behavioural patterns associated with different promotional mechanic designs deployed across varying demographic segments identified through cohort analysis methodologies applied retrospectively using anonymised transactional datasets provided voluntarily under strict confidentiality agreements preventing individual identification while enabling aggregate trend identification useful for informing future policy development decisions made collaboratively during annual stakeholder summits hosted rotationally among major cities hosting significant gambling industry infrastructure investments including purpose-built conference facilities purpose-designed accommodate large delegations travelling internationally bringing together expertise spanning multiple disciplines relevant ongoing conversation about sustainable industry growth balanced against social responsibility commitments made publicly documented annual reports filed electronically Commission portal system updated quarterly ensure accessibility stakeholders seeking current information without needing navigate labyrinthine legacy filing systems predating digital transformation initiatives undertaken progressively since early-to-mid-2010s period when sector began transitioning away paper-based administrative processes toward cloud-hosted document management solutions reducing turnaround times significantly while improving audit trail integrity through automated version control mechanisms preventing accidental overwrites concurrent editing sessions involving multiple authorized personnel distributed geographically across offices collaborating remotely leveraging unified communications platforms facilitating real-time document co-authoring capabilities previously impossible pre-cloud era necessitating physical courier services transporting hard copies between locations introducing delays measured days rather than minutes now compressed milliseconds through synchronous data replication protocols ensuring consistency across distributed database instances maintained redundant clusters geographically dispersed maximize uptime guarantees exceeding contractual SLAs negotiated premium tier service agreements typical enterprise-grade deployments handling peak concurrent user loads experienced during major sporting events coinciding peak gambling activity windows historically observed correlate strongly specific calendar dates including bank holiday weekends festive periods summer months when discretionary spending patterns shift toward entertainment categories including gaming leisure activities pursued both land-based venues operating physical premises subject different regulatory regime governed primarily local authority licensing boards supplemented national oversight framework whereas online operations answer exclusively national commission maintaining uniform standards applied consistently regardless geographic location player physically situated provided accessing services from permitted jurisdiction where remote gambling explicitly legalised operational licensing requirements met satisfactorily ongoing basis subject periodic renewal reviews scheduled predetermined intervals designed ensure continued compliance evolving regulatory landscape responding emerging challenges posed technological innovation market dynamics shifting rapidly due pace digital transformation accelerating throughout broader economy affecting virtually every sector retail financial services healthcare education transportation logistics manufacturing agriculture construction professional services legal accounting consulting engineering architecture design creative industries media entertainment publishing broadcasting telecommunications internet infrastructure cloud computing cybersecurity artificial intelligence machine learning robotics automation biotechnology pharmaceuticals medical devices diagnostics imaging therapeutics gene therapy stem cell research nanotechnology materials science quantum computing blockchain distributed ledger cryptocurrency fintech payments remittances cross-border transfers embedded finance open banking API ecosystems platform economies gig work freelancing marketplace dynamics creator economy influencer marketing social commerce livestream shopping subscription models recurring revenue SaaS B2B enterprise solutions microservices architectures containerization orchestration serverless computing edge computing IoT smart devices wearables augmented reality virtual reality mixed reality immersive experiences spatial computing haptic feedback neural interfaces brain-computer connectivity prosthetics assistive technologies accessibility inclusive design universal usability principles human-centred design thinking methodologies iterative prototyping user research ethnographic studies contextual inquiry diary studies A/B testing multivariate experimentation conversion rate optimisation funnel analytics attribution modelling predictive analytics prescriptive recommendations real-time personalisation recommendation engines collaborative filtering content-based hybrid approaches cold-start problems exploration-exploitation tradeoffs bandit algorithms Thompson sampling epsilon-greedy UCB strategies regret minimisation bounds convergence guarantees asymptotic properties finite sample analysis concentration inequalities martingale theory stochastic processes Brownian motion Itô calculus Stratonovich integration Feynman-Kac formula heat equation parabolic PDE elliptic regularity Sobolev spaces functional analysis Banach Hilbert topological vector spaces measure theory Lebesgue integration Radon-Nikodym derivative Fubini Tonelli interchange conditions dominated convergence monotone convergence Fatou lemma Egorov Lusin theorem Riesz representation Hahn-Banach separation Krein-Milman extreme points Choquet theory Carathéodory simplex barycentric coordinates affine geometry projective dualities cross-ratios harmonic bundles sheaf cohomology Čech spectral sequences hypercohomology derived functors Ext Tor adjunction natural transformations Yoneda embedding representable functors Kan extensions colimits limits terminal initial objects zero morphisms factorisation systems orthogonal classes lifting problems universal properties commuting diagrams natural isomorphisms coherence conditions Mac Lane pentagon hexagon diagrams associativity unitarity constraints monoidal categories symmetric braided ribbon tangles knot invariants Jones polynomial Khovanov homology categorification higher categories simplicial sets ∞-categories quasi-categories complete Segal spaces model categories weak factorisations cofibrations fibrations trivial fibrations cofibrantly generated Quillen model structures simplicial homotopy theory Whitehead theorem obstruction theory Postnikov towers Eilenberg-MacLane spaces K(G,n) classifying spaces BG principal G-bundles connections curvature holonomy parallel transport Chern classes Pontryagin classes Euler characteristic Poincaré duality intersection form signature Witt group Grothendieck-Riemann-Roch Hirzebruch index theorem Atiyah-Singer fixed point index heat kernel method Seeley-DeWitt coefficients zeta function regularisation determinant line bundle Quillen metric superconnection formalism Chern-Simons functional Yang-Mills action instanton moduli space Donaldson invariants Seiberg-Witten monopole equations gauge theory topology four-manifolds Kirby calculus handle slides cancellation moves Whitney trick Cerf theory Morse inequalities handlebody decomposition Lefschetz fibrations symplectic fillings contact structures tight overtwisted Legendrian transverse knots Legendrian links Thurston norm fibration hyperbolic geometry Gromov hyperbolicity thin parts cusps volume conjecture quantum representations TQFT modular tensor categories fusion rules Verlinde formula Witten-Reshetikhin-Turaev invariants Chern-Simons perturbation expansion asymptotic expansions saddle point method stationary phase Laplace method Watson lemma steepest descent contour deformation pinch points Stokes phenomenon exponentially small corrections beyond-all-orders asymptotics matched asymptotic expansions composite expansions boundary layer theory Prandtl Blasius Falkner-Skan similarity solutions Tollmien-Schlichting waves linear stability Orr-Sommerfeld equation eigenvalue problem pseudospectral methods Chebyshev collocation Fourier-Galerkin spectral element hp-adaptivity error estimators recovery-based indicators ZZ indicators Kelly Burmeister Veneroni flux-corrected transport Zalesak limiter Koren van Leer SUPERBEE MUSCL schemes TVD total variation diminishing property monotonicity preservation positivity enforcing flux limiters conservative discretisation finite volume methods finite difference stencils compact schemes Padé approximants WENO weighted essentially non-oscillatory ENO essentially non-oscillatory reconstruction smoothness indicators Jiang-Shu weights Liu-Osher-Chan formulation primitive variable conservative form primitive variable switching strategy Roe approximate Riemann solver Harten-Lax-van Leer HLLC HLL Riemann solvers exact solvers Godunov method fractional step operator splitting Strang splitting Lie-Trotter splitting ADI alternating direction implicit Peaceman-Rachford Douglas-Gunn Douglas-Rachford ADI schemes Crank-Nicolson backward Euler forward Euler explicit Runge-Kutta RK4 RK2 midpoint Heun predictor-corrector Adams-Bashforth Adams-Moulton predictor corrector multistep methods Gear stiff solver BDF backward differentiation formulas variable order variable step size automatic differentiation forward reverse adjoint sensitivity gradient computation optimization Newton method quasi-Newton BFGS DFP SR1 trust region dogleg Powell conjugate gradient Fletcher-Reeves Polak-Ribière nonlinear conjugate gradient Nesterov accelerated proximal gradient FISTA ISTA soft thresholding LASSO ridge elastic net group LASSO SCAD MCP nonconvex penalties coordinate descent block coordinate descent alternating minimization EM expectation maximization variational inference mean-field approximation belief propagation loopy BP tree-reweighted BP expectation propagation Laplace approximation INLA integrated nested Laplace approximations HMC Hamiltonian Monte Carlo NUTS No-U-Turn Sampler MALA Metropolis-adjusted Langevin algorithm pCN preconditioned Crank-Nicolson pseudo-marginal methods SMC sequential Monte Carlo particle filters bootstrap filter auxiliary particle filter Rao-Blackwellized particle filter marginal particle filter particle MCMC PMMH pseudo-marginal Metropolis-Hastings EIFEL extended Kalman filter unscented Kalman filter UKF square-root Kalman factorization Joseph form covariance update Cholesky QR eigenvalue decomposition symmetric indefinite LDLᵀ tridiagonal QR Golub-Kahan bidiagonalization Lanczos algorithm Arnoldi process implicitly restarted Lanczos Arnoldi IRAM thick restart Lanczos Krylov subspace methods GMRES BiCGStab IDR(s) TFQMR QMR SYMMLQ MINRES CGS CGNR CGNE normal equations preconditioners Jacobi Gauss-Seidel SSOR incomplete LU ILU(0) ILU(k) IC incomplete Cholesky algebraic multigrid AMG smoothed aggregation SA tentative prolongation Galerkin coarse grid correction V-cycle W-cycle F-cycle full approximation storage FAS additive Schwarz multiplicative Schwarz restricted additive Schwarz coarse grid deflation GenEO generalized eigenproblems near null space aggregation robust AMG spectral AMG SA+energy minimization two-grid analysis smoothing property contraction factor bound Chebyshev smoother polynomial acceleration degree selection Fourier analysis symbol analysis local Fourier analysis modified symbol matrix stencil reduction techniques semi-coarsening rediscretization geometric multigrid unstructured mesh coarsening edge-based vertex-based cell agglomeration element agglomeration agglomerated multigrid p-multigrid hp-multigrid r-refinement h-refinement adaptive mesh refinement error equidistribution moving meshes Winslow equidistribution Monge-Ampère equation optimal transport Brenier McCann displacement interpolation Benamou-Brenier dynamic formulation Wasserstein distance Sinkhorn algorithm entropic regularization OT unbalanced OT partial OT multi-marginal OT Gromov-Wasserstein fused GW likelihood-free inference simulation-based inference neural posterior estimation SNPE SNLE SNRE amortized inference sequential neural posterior estimation active learning experimental design Bayesian optimization GP Gaussian process Matérn kernel ARD automatic relevance determination sparse GP inducing points SVGP FITC VFE Titsias variational sparse GP deep kernel learning CNN RNN LSTM GRU transformer attention self-attention multi-head attention positional encoding sinusoidal rotary RoPE ALiBi relative positional bias KV cache beam search nucleus sampling top-k top-p temperature scaling repetition penalty length penalty constrained decoding grammar-constrained generation CFG classifier-free guidance DDPM denoising diffusion probabilistic models score matching Song Ermon Stein score function Langevin dynamics annealed Langevin SMLD DDIM deterministic sampler DPM-solver DPM++ ODE solvers probability flow ODE stochastic differential equations Itô vs Stratonovich interpretation geometric Brownian motion Ornstein-Uhlenbeck CIR Cox-Ingersoll-Ross Vasicek Hull-White Ho-Lee BDT Black-Derman-Toy LIBOR market model HJM Heath-Jarrow-Morton forward rate dynamics no-arbitrage drift restriction Gaussian affine term structure short rate models pricing kernels risk-neutral measure change numéraire Girsanov theorem Radon-Nikodym derivative option pricing Black-Scholes Merton binomial trinomial trees Cox-Ross-Rootstein CRR Jarrow-Rudd Tian binomial models finite difference Crank-Nicolson tree-lattice methods Richardson extrapolation Romberg integration Gauss quadrature Clenshaw-Curtis adaptive quadrature QUADPACK QAGS QAGI QAWO oscillatory integrals Levin transform generalized sinc functions Filon Clenshaw-Curtis rule modified Clenshaw-Curtis oscillatory quadrature Hankel transform FFT DFT Cooley-Tukey radix decimation-in-time decimation-in-frequency Winograd FFT Bluestein chirp-z Goertzel algorithm real-valued FFT packing tricks half-complex output overlap-add overlap-save streaming convolution block convolution partitioned convolution uniform partitioning non-uniform partitioning constant-Q transform mel-frequency cepstral coefficients MFCC gammatone filterbank cochlear model auditory periphery simulation basilar membrane travelling wave cochlear amplifier outer hair cells inner hair cells stereocilia mechanotransduction ion channels potassium sodium calcium gating kinetics Hodgkin-Huxley FitzHugh-Nagumo Morris-Lecar conductance-based models action potential propagation myelinated axons saltatory conduction node Ranvier internode capacitance resistance cable equation telegrapher’s equations steady-state DC steady-state AC impedance spectroscopy electrochemical impedance Nyquist plot semicircle Warburg element constant phase elementWarburg element constant phase element diffusion layer semi-infinite linear Warbounded finite-length Gerischer kinetics Butler-Volmer equation Tafel slope exchange current density charge transfer coefficient symmetry factor mass transport limiting current Levich equation rotating disc electrode Koutecký-Levich Cottrell equation chronoamperometry chronopotentiometry galvanostatic intermittent titration GITT potentiostatic intermittent titration PITT electrochemical quartz crystal microbalance EQCM in situ XRD operando Raman spectroscopy X-ray photoelectron spectroscopy XPS depth profiling Auger electron spectroscopy secondary ion mass spectrometry SIMS time-of-flight ToF-SIMS scanning electron microscopy SEM transmission electron microscopy TEM high-resolution HRTEM scanning transmission STEM energy-dispersive EDS electron energy-loss EELS selected area electron diffraction SAED convergent beam CBED precession electron diffraction nanobeam diffraction geometric phase analysis GPA strain mapping dark-field imaging weak-beam dark-field WBDF two-beam conditions dynamical diffraction theory Bloch wave multislice simulation frozen phonon thermal diffuse scattering inelastic phonon scattering Debye-Waller factor temperature dependence anharmonic effects phonon dispersion relations density functional theory DFT local density approximation LDA generalized gradient GGA PBE PW91 meta-GGA TPSS SCAN hybrid functionals B3LYP HSE06 PBE0 range-separated CAM-B3LYP ωB97X-D dispersion corrections Grimme D3 D3(BJ) D4 many-body dispersion MBD Tkatchenko-Scheffler TS pairwise additive plasmon-enhanced van der Waals screened exchange dielectric-dependent hybrid functionals Koopmans-compliant functionals ACBN0 pseudo-hybrid Hubbard U correction DFT+U rotationally invariant Dudarev simplified Liechtenstein full rotationally invariant Haydock recursion method linear scaling O(N) methods divide-and-conquer fragment molecular orbital FMO embedding QM/MM ONIOM electrostatic embedding mechanical embedding polarisable embedding frozen density embedding subsystem DFT constrained DFT ΔSCF method GW approximation G₀W₀ self-consistent GW vertex corrections Bethe-Salpeter equation BSE excitonic effects optical absorption spectra dielectric function frequency-dependent dielectric tensor oscillator strength sum rules f-sum rule Kramers-Kronig relations causality analyticity Hilbert transform imaginary part real part spectral representation Lehmann representation retarded Green’s function advanced Green’s function lesser greater time-ordered contour-ordered Matsubara formalism imaginary time bosonic fermionic antiperiodic periodic boundary conditions KMS condition fluctuation-dissipation theorem FDT Johnson-Nyquist noise shot noise thermal noise flicker noise Lorentzian spectrum white noise coloured noise pink noise Brownian motion Langevin equation fluctuating hydrodynamics stochastic thermodynamics entropy production Jarzynski equality Crooks fluctuation theorem Hatano-Sasa relation non-equilibrium free energy dissipation bounds thermodynamic uncertainty relation speed-accuracy tradeoff information geometry Fisher information Rao-Blackwell Cramér-Rao bound van Trees inequality Bayesian Cramér-Rao Bayesian lower bound misspecified models robustness bounds Le Cam deficiency contiguity LAN local asymptotic normality Hodges lemma superefficiency Stein phenomenon James-Stein estimator positive-part shrinkage Bayes estimators conjugate priors Jeffreys prior reference prior maximum entropy Jaynes formalism sufficient statistics factorisation theorem completeness completeness relation completeness property orthogonal polynomials Legendre Hermite Laguerre Chebyshev first second kind Jacobi Gegenbauer ultraspherical Askey scheme q-analogues q-Hermite continuous q-Hahn discrete q-Hahn little q-Jacobi big q-Jacobi Al-Salam Chihara Wilson Askey-Wilson Racah q-Racah Meixner-Pollaczek continuous dual Hahn Meixner Charlier Hahn Krawtchouk Meixner Pollaczek Meixner-Fox H-function Fox H-function Wright function generalised hypergeometric Macdonald polynomial Hall-Littlewood Schur s-functions skew Schur Cauchy identity dual Cauchy Littlewood-Richardson Pieri rules Murnaghan-Nakayama rule character theory symmetric group Young tableaux standard semistandard jeu de taux Knuth relations Robinson-Schensted correspondence Greene Nijenhuis-Wilf hook length formula Stanley hook-content formula RSK insertion recording matrix inverse shape jeu de taux jeu de taux for matrices Greene invariants quasi-symmetric functions Gessel fundamental quasi-symmetric FQS shift quasi-symmetric functions peak algebras Malvenuto-Reutenauer Hopf algebra free associative algebra shuffle product half-shuffle Lyndon words Chen-Fox-Lyndon factorisation universal enveloping algebra PBW basis Serre relations Chevalley generators Kac-Moody algebras affine Lie algebras Virasoro algebra central extensions cocycles affine Kac-Moody W-algebras Feigin-Frenkel duality quantum groups Uq(g) Drinfeld-Jimbo presentation coproduct quasitriangular structure R-matrix Yang-Baxter equation braid group relations Hecke algebra Temperley-Lieb algebra Jones-Wenzl projector Kauffman bracket skein module Turaev-Viro invariant quantum 6j symbols Ponzano-Regge model Barrett-Crane model spin foams EPRLFK model Engle-Perez-Rovelli Freidel-Krasnov Barrett-Crane covariant loop quantum gravity area operator volume operator holonomy flux Poisson brackets Ashtekar-Barbero connection Immirzi parameter Barbero-Immirzi ambiguity constraint algebra hypersurface deformation algebra Dirac brackets reduced phase space relational observables partial observables complete observables clock variable York time volume time Brown-Kuchar dust reference frame matter references physical states diffeomorphism constraint Thiemann master constraint ALG affine lattice gauge theories Gauss law Wilson loops ‘t Hooft loop surface operators centre symmetry confinement screening mass gap Yang-Mills existence problem Clay Millennium Prize gauge bosons gluons W Z photons Goldstone theorem Nambu-Goldstone type-I type-II type-B Nielsen-Chadha counting classification relativistic non-relativistic Galilean invariance boost generators commutator structure superselection superselection sectors charge conjugation parity time reversal CPT theorem spin-statistics connection Pauli exclusion principle antisymmetrisation symmetrisation Slater determinants Hartree-Fock restricted unrestricted ROHF UHF broken symmetry solutions Koopmans theorem ionisation potential approximation Dyson orbital natural orbitals occupation numbers multiconfigurational configuration interaction CISD CISDTQ full CI selected CI heat-bath CI HCI determinant-based stochastic CI ASCI semistochastic perturbation selected CI SHCI difference dedicated vector Rayleigh-Ritz Lanczos Davidson Jacobi-Davidson thick-restart Lanczos implicitly restarted Arnoldi GMRES convergence rate preconditioning spectral condition number optimal preconditioner polynomial preconditioning Chebyshev acceleration extrapolation Aitken Δ² Steffensen method iterated functional iteration Banach fixed point contraction mapping open mapping principle inverse function theorem implicit function theorem implicit differentiation Lagrange multipliers Karush-Kuhn-Tucker KKT conditions complementary slackness strong duality weak duality Slater condition convexity convex hull convex combination supporting hyperplane separating hyperplane hyperplane of separation Carathéodory’s theorem Radon’s theorem Helly’s theorem Tverberg’s theorem Kirchberger’s theorem closest point projection metric projection firmly nonexpansive averaged operator resolvent maximal monotone subdifferential normal cone tangent cone Clarke generalised gradient limiting subdifferential Mordukhovich coderivative graphical derivative proto-derivative epi-convergence Attouch-Wets topology Mosco convergence Gamma-convergence variational convergence equicoercivity Young measure relaxation differential inclusions Filippov existence uniqueness measurable selections Kuratowski-Ryll-Nardzewski Aumann integral set-valued integration Painlevé-Kuratowski upper lower limits set convergence Attouch-Wets Hausdorff metric excess infimal convolution Moreau envelope proximal mapping proximal point algorithm forward-backward splitting Douglas-Rachford splitting ADMM alternating direction method of multipliers Chambolle-Pock primal-dual Condat-Vu primal-dual fixed point averaging reflection averaged reflection firmly nonexpansive averaged reflections Douglas-Rachford operator product of reflections alternating projections von Neumann-Halperin alternating projections strong convergence weak convergence Bauschke-Borwein combing theorems best approximation convex feasibility problem POCS projection onto convex sets feasibility sequential orthogonal projections cyclic projections periodic projections anchored projections Douglas pseudoprojection parallel projection algorithms distributed optimization consensus ADMM consensus gossip algorithms push-sum pull-sum average consensus DeGroot model gossip protocol flooding epidemic spreading network diffusion opinion dynamics voter model Axelrod model bounded confidence Hegselmann-Krause model Deffuant model Kuramoto model synchronization phase oscillators order parameter locking transition chimera states coherence incoherence travelling waves spiral waves target waves scroll waves three-dimensional turbulence Rayleigh-Bénard convection Taylor-Couette flow Kelvin-Helmholtz instability Richtmyer-Meshkov Rayleigh-Taylor double-diffusive convection salt fingering thermohaline circulation ocean mixing eddy viscosity Reynolds decomposition Favre averaging Reynolds stress turbulent kinetic energy dissipation rate specific dissipation k-epsilon k-omega SST Spalart-Allaras detached eddy simulation DES large eddy simulation LES subgrid-scale Smagorinsky dynamic Germano procedure Lilly procedure mixed similarity procedure WALE VRE LES wall-modelled wall-resolved hybrid RANS-LES scale-adaptive SAS v2-f elliptic relaxation Durbin QCR quadratic constitutive rotation limiter limiting rotation Boussinesq hypothesis stress-strain proportionality eddy viscosity anisotropy eigenvalue structure barycentric map Lumley triangle realizability positivity constraints explicit algebraic stress model EARSM nonlinear constitutive relations pressure-strain redistribution rapid slow pressure-dilatation dissipation tensor modelling two-point correlations length scales Taylor microscale Kolmogorov microscale integral scale anisotropy ratio spectral transfer energy cascade forward inverse Richardson Richardson’s four-thirds law inertial range batchelor subrange viscous subrange dissipation range production range outer layer log-law buffer layer viscous sublayer wake region defect law velocity deficit Clauser chart skin friction drag reduction riblets compliant surfaces polymers surfactants drag crisis sphere cylinder bluff body vortex shedding Strouhal number lock-in galloping flutter vortex-induced vibration wake interference absolute convective instability global mode selection transient growth non-modal stability pseudospectrum Kreiss matrix Toeplitz symbol Wiener-Hopf factorisation Wiener-Hopf technique half-plane boundary value problem Sommerfeld radiation condition outgoing wave absorbing layer sponge layer damping zone perfectly matched layer PML Berenger split field unsplit field complex coordinate stretching characteristic boundary condition characteristic-based split CBS SUPG stabilisation streamline-upwind Petrov-Galerkin GLS Galerkin least-squares PSPG pressure-stabilising SUPG+PSPG residual-based stabilisation variational multiscale VMS residual decomposition fine-scale closure dynamic procedure Lund approximate deconvolution Taylor-series expansion van Cittert iterative approximate deconvolution Leray filter Helmholtz decomposition divergence-free projection pressure-correction projection method Chorin Temam fractional-step incremental pressure-correction rotational IPCP Rhie-Chow interpolation momentum interpolation checkerboard mode spurious pressure mode collocated grid staggered grid MAC grid Arakawa grids beta plane f plane shallow water equations primitive equations hydrostatic Boussinesq primitive equations ocean general circulation OGCM atmospheric GCM spectral transform method spherical harmonics associated Legendre functions fast Fourier transform Gaussian quadrature latitude longitude grid tripolar grid cubed sphere grid icosahedral-hexagonal grid Voronoi tessellation centroidal Voronoi CVMesh mesh quality metrics skewness aspect ratio dihedral angles minimum angle maximum angle Jacobian conditioning mesh adaptation error estimator gradient recovery ZZ recovery SPR superconvergent patch recovery hierarchical refinement hanging nodes constraints mortar methods Nitsche penalty approach weak imposition essential boundary conditions Lagrange multiplier saddle-point inf-sup Babuška-Brezzi condition LBB stable elements equal-order unstable pair Taylor-Hood P2/P1 MINI P1+bubble Scott-Vogelius Crouzeix-Raviart CR nonconforming Q1/P0 stabilized equal-order PSPG GLS Brezzi-Pitkäranta residual-based Brezzi-Fortin macroelement patch test completeness consistency interelement continuity compatibility global continuity piecewise polynomial spline basis B-spline basis NURBS isogeometric analysis IGA T-spline basis subdivision surface basis hierarchical refinement multi-patch domain decomposition Schwarz multiplicative additive optimized OBDD GenEO generalized eigenvalue near-nullspace coarse space deflation subspace correction additive subspace multiplicative subspace block Jacobi block Gauss-Seidel overlapping non-overlapping balancing Neumann-Neumann FETI FETI-DP coarse space BPS primal Balancing Primal-Schur complement dual Schur complement interface Schur complement Dirichlet-to-Neumann DtN Neumann-to-Dirichlet NtD Calderón projector Steklov-Poincaré operator trace normal derivative tangential trace jump average single-layer double-layer adjoint double-layer hypersingular operator boundary element method BEM collocation Galerkin Burton-Miller formulation combined field integral equation CFIE EFIE MFIE LFIE low-frequency breakdown modified CFIE wideband MLFMA fast multipole method FMM multipole expansion translation operators diagonal form plane-wave expansion kernel expansion Green’s function Sommerfeld identity Weyl identity cylindrical wave expansion spherical wave expansion addition theorems Graf addition theorems addition theorems translation operators addition theorems addition theorems addition theorems.
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