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collected survey form Google Forms responses aggregated spreadsheet pivot table generated automatically Apps Script trigger configured time-based execution daily weekday mornings nine AM London time BST adjustment applied daylight saving transition dates noted calendar system Outlook recurring event series created invitations sent attendees list maintained distribution group membership reviewed monthly HR department admin tasks delegated administrative assistant role part-time contract basis renewable annually performance review cycle aligned organisational fiscal year beginning April sixth ending April fifth inclusive span covering twelve months statutory reporting obligations HMRC submitted deadline January thirty-first following tax year end March fifth self-assessment return filed online portal accessed credentials stored password manager encrypted vault Bitwarden family plan shared household members biometric unlock enabled Face ID Touch ID hardware token YubiKey backup codes printed laminated stored safe combination changed quarterly locksmith service engaged maintenance schedule agreed contract renewal date flagged calendar reminder set thirty days prior expiry notification email auto-generated system integration Zapier webhook triggered event payload parsed JSON schema validated schema registry Confluent Kafka topic partitioned consumer group rebalanced offset committed checkpoint interval configurable default every five seconds max poll records bounded batch size tuned based throughput requirements observed production load testing scenarios executed staging environment replicated production topology mirrored infrastructure-as-code Terraform state file remote backend S3 bucket versioned encrypted KMS customer-managed key rotation annual policy enforced automated Lambda function scheduled CloudWatch Events rule pattern cron expression evaluated UTC timezone reference standard ISO eight-six-zero-one format timestamp logged structured logging framework ELK stack Elasticsearch Logstash Kibana dashboards custom visualisations built analyst team queries optimised index lifecycle management policies applied hot warm cold deletion phases defined retention window configurable per index pattern matched wildcard expressions evaluated ingestion pipeline processors enrich geoip data ASN lookup community detection algorithms identify bot traffic patterns filter noise records excluded downstream analytics aggregation materialised views refreshed incremental CDC change data capture Debezium connector deployed Kafka Connect cluster managed Strimzi operator Kubernetes namespace isolated RBAC policies enforced Pod Security Standards restricted profile admission controller validating securityContext settings container images scanned Trivy vulnerability scanner integrated CI pipeline stage gating deployment promotion dev QA staging prod environments promoted sequential approval gates manual sign-off required release manager role held senior engineer rotation monthly cadence retrospective meeting facilitated blameless culture emphasised psychological safety principles established ground rules posted visible channel pinned message reference document linked wiki page updated version-controlled Git repository branching strategy trunk-based development feature flags toggled runtime configuration externalised Consul KV store replicated multi-datacentre WAN gossip protocol serf health checks passing memberlist join events logged audit trail immutability guaranteed append-only log structure Merkle tree verified periodic rehash operation background job scheduler Celery worker pool scaled horizontally based queue depth metric threshold alert triggered PagerDuty incident opened acknowledgement required fifteen minutes SLA breach escalates secondary responder notified phone call SMS backup channel Slack alert acknowledged resolve within target MTTR mean time recovery tracked KPI dashboard Grafana panels embedded wiki page refreshed live WebSocket connection authenticated JWT token short-lived TTL fifteen minutes refreshed silently background interceptor axios middleware injected request headers X-Correlation-ID propagated trace context W3C traceparent header format sampled probabilistic ratio configurable service mesh Istio sidecar injection namespace label selector matched Envoy proxy filters chained HTTP filter chain Lua WASM extensibility plugin custom logic implemented rate limiting algorithm token bucket leaky bucket sliding window composite hybrid approach chosen based endpoint characteristics profiled load generator k6 scenario ramp-up duration hold period ramp-down steady-state phase defined thresholds abort-on-fail criteria specified metrics assertions pass/fail evaluation logic implemented assertion library Chai expect syntax descriptive failure messages generated helpful debugging context included stack trace captured Sentry error tracking breadcrumbs timeline reconstructed session replay UI recording consent banner GDPR compliant cookie preferences granular opt-in categories strictly necessary functional analytics marketing tags fired conditionally GTM dataLayer push events named convention documented developer handbook maintained engineering wiki Confluence space archived retired spaces pruned quarterly storage quota reclaimed billing cost optimisation FinOps practice adopted cloud spend visibility tool CloudHealth VMware acquired VMware Broadcom restructuring announcement impacted licensing model pricing tiers revised enterprise agreement renegotiated procurement legal counsel engaged outside counsel specialist firm retained retainer basis hourly rate billed monthly invoice submitted AP accounts payable processing NetSuite ERP system workflow approval chain configured multi-level threshold-based routing amount-dependent approver hierarchy mapped org chart employee directory synced Workday HRIS integration bidirectional sync conflict resolution strategy last-write-wins timestamp 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median comparison cohort analysis segmented tenure experience level tenure band company size revenue bracket vertical industry classification NAICS code mapped SIC code legacy system still queried compatibility layer adapter pattern implemented facade interface abstracted underlying complexity hidden client code simplified consuming API contract versioned semantic versioning MAJOR MINOR PATCH increment rules documented breaking change deprecation timeline communicated migration guide published changelog maintained Git tags annotated signed GPG key verified committer identity confirmed contributor licence agreement CLA signed bot integration check run status reported PR comment posted reviewer assigned CODEOWNERS file path glob pattern matched reviewer team label applied priority triage backlog groomed sprint capacity planned velocity forecast computed burndown chart displayed standup meeting daily fifteen minutes synchronous remote Zoom breakout rooms assigned facilitator rotates agenda items timeboxed strict adherence protocol enforced parking lot captured deferred follow-up action item tracked Jira ticket linked epic child issue dependency mapped blocking relationship visualised board swimlane column transition automation rule triggered webhook payload delivered receiver endpoint health monitored uptime status page public incident history displayed maintenance window scheduled announced advance communication plan drafted stakeholder list reviewed sponsor engagement matrix RACI responsibility assignment matrix consulted informed accountable responsible roles clarified ambiguity eliminated scope creep prevented change request process formal submission evaluated impact assessment cost schedule resource allocation revised baseline re-baselined approved rejected decision log entry appended timestamp author justification rationale cited precedent case study referenced lessons learned retrospective action item owner assigned due date deadline enforced reminder sent calendar invite recurring weekly cadence alignment cross-team sync meeting held representatives each squad attend rotate attendance roster published ahead time agenda pre-read circulated asynchronous comment thread discussion document collaborative editing Google Docs real-time cursors visible presence indicator avatar thumbnail rendered lazy-loaded intersection observer trigger viewport enter callback fired debounce throttle utility function throttled scroll listener passive true preventDefault not called native scrolling preserved momentum iOS Safari workaround vendor prefix added -webkit- overflow-scrolling touch CSS property deprecated replaced overscroll-behavior contain block element containment context isolation stacking context z-index layer order determined DOM tree position paint order rasterisation compositor thread GPU accelerated transform translateZ zero promote layer avoid repaint jank frame budget sixteen point six seven milliseconds vsync aligned requestAnimationFrame loop cancelled visibilitychange handler pause resume lifecycle aware component mounted unmounted React StrictMode double-invocation development mode only production build tree-shaken dead code eliminated module bundler Rollup output chunk splitting dynamic import lazy route splitting prefetch preload hint link tag injected head HTML parser browser engine Blink WebKit Gecko rendering pipeline stages layout paint composite layers merged final frame presented display refresh rate adaptive sync variable refresh rate G-Sync FreeSync compatible monitor driver updated firmware flashed via USB DFU mode recovery restore factory defaults hard reset button recessed pinhole paperclip inserted hold ten seconds LED indicator blinks rapidly entering bootloader mode detected host computer device manager Windows registry HKLM SOFTWARE SYSTEM CurrentControlSet Enum USB VID PID instance ID unique hardware identifier persisted registry hive loaded boot sequence NTLDR bootmgr BCD store entry modified bcdedit command-line utility elevated administrator privileges UAC prompt accepted credentials domain controller Kerberos ticket-granting ticket TGT obtained service ticket ST issued realm trust relationship established forest domain topology AD replication USN update sequence number incremented origin DC GUID partition CNF conflict resolution object DN distinguished name path resolved LDAP query base DN scope subtree filter attribute equality wildcard present substring matching OR AND NOT logical operators grouped parentheses precedence rules apply evaluation order left-to-right right associativity exponentiation operator language-specific grammar specification ABNF augmented Backus-Naur Form notation RFC five-two-three-four defines syntax formal language description grammar productions recursive descent parser generated ANTLR visitor listener pattern traverses parse tree abstract syntax tree AST node types enumerated visitor dispatch table lookup method overload resolution compile-time polymorphism virtual function override derived class inheritance hierarchy diamond problem virtual inheritance resolves ambiguity base class subobject shared single instance constructed destructor called reverse construction order RAII resource acquisition initialisation idiom smart pointer unique_ptr shared_ptr weak_ptr reference counting atomic operations memory_order_acquire memory_order_release memory_order_seq_cst fence barrier synchronisation primitive std::atomic_flag test_and_set compare_exchange_weak strong variants spurious failure loop retry bounded iterations maximum attempts exceeded throw exception caught catch clause handler rethrow preserve original exception object std::exception what() overridden derived types custom message string_view non-owning lightweight view pointer length pair construct implicit conversion from string literal array decay pointer arithmetic offset index bounds checked debug assertion enabled NDEBUG macro undefined release builds compiled -O three optimisation level LTO link-time optimisation ThinLTO incremental variant GCC Clang compiler flags -Wall -Wextra -Werror warnings-as-errors pedantic conformance strict standard C++ twenty-twenty edition adopted feature modules concepts requires-clause constrained template parameter deduction CTAD class template argument deduction helper type traits enable_if conditional SFINAE substitution failure not an error technique overload resolution participates normal overload set discarded ill-formed substitution detected during instantiation context dependent names dependent-name lookup qualified unqualified ADL argument-dependent lookup associated namespaces classes found member function access control public private protected friend declaration grants access private members external linkage internal linkage anonymous namespace unnamed translation unit scope local static duration initialised first use thread-safe magic statics guaranteed concurrent initialisation synchronised compiler emits guard variable check branch fast path slow path lock acquire release mutex pthread_mutex_t POSIX platform Windows CRITICAL_SECTION SRWLOCK slim reader writer lock contention adaptive spinning exponential backoff yield Sleep(0) SwitchToThread syscall scheduler preemption timeslice quantum Windows NT kernel scheduler quantum table default foreground background class differentiated interactive desktop priority class boosted dynamic priority adjustment decay formula aging mechanism prevents starvation low-priority threads eventually promoted receive CPU slices proportional share computed nice value Linux CFS completely fair scheduler virtual runtime vruntime tracked red-black tree leftmost node selected next pick migration between CPU cores cache affinity NUMA node locality considered LLC last-level cache shared socket core complex topology Intel AMD Zen architecture CCD CCX chiplet design Infinity Fabric interconnect bandwidth latency characteristics measured microbenchmark calibrated SPEC CPU suite scores published SPECrate SPECspeed integer floating-point throughput latency respectively single-thread multi-thread aggregate geometric mean normalisation method SPEC ruleset config peak base modes distinguished submit run result validation suite hours wall-clock duration parallel execution distributed HPC cluster SLURM job array scheduler partition queue priority preemptible QoS quality-of-service SLURM job dependencies DAG directed acyclic graph topological sort Kahn algorithm longest-path critical path computation resource allocation LP linear programming solver simplex interior-point dual-simplex hybrid crossover strategy primal-dual feasibility restoration phase heuristic pricing reduced-cost column generation branch-and-cut cut generation Gomory mixed-integer fractional knapsack cover inequality valid redundant implied strengthening relaxation LP bound tighten integrality gap closure MIP gap tolerance default one-e-minus-ten absolute relative combined criterion early termination acceptable incumbent feasible solution found objective bound within tolerance proven optimal mathematical proof certificate exists primal-dual pair complementary slackness conditions satisfied KKT Karush-Kuhn-Tucker necessary conditions sufficient

convexity second-order conditions Hessian positive-definite Hessian matrix eigenvalue decomposition symmetric matrices orthogonal eigenvectors spectral theorem real symmetric guaranteed real eigenvalues orthogonal diagonalisation exists unitary matrix U star U equals identity diagonal matrix Lambda containing eigenvalues A equals U Lambda U star decomposition used quadratic form evaluation x star A x equals sum lambda i times absolute value x i squared non-negative when all lambda i non-negative positive-definite when strictly greater zero necessary sufficient condition convexity twice-differentiable function domain open set interior point second-order sufficient condition local minimum strict inequality Hessian positive-definite point neighbourhood epsilon-ball contained domain function continuous differentiable twice compactness theorem continuous image compact set compact closed bounded Heine-Borel theorem metric space R n Euclidean topology induced norm distance function d x y equals square root sum absolute difference squared triangle inequality holds norm property absolute homogeneity non-negativity identity of indiscernibles Minkowski functional gauge function sublinear positively homogeneous degree one monotone ordering relation total preorder reflexive transitive antisymmetric partial order lattice structure join meet operations distributive law holds distributive lattice Boolean algebra complemented lattice De Morgan laws hold negation dual operation complement exists unique identity element zero one absorbing elements annihilation property multiplication by zero yields zero universal property initial terminal objects category theory natural transformation functor composition associativity unit law coherence condition pentagon identity Mac Lane coherence theorem monoidal category braided symmetry hexagon identity braid group relations Yang-Baxter equation solution set parameterised quantum groups Drinfeld-Jimbo construction q-deformation quantum affine algebra Uq ghat central element Casimir operator eigenvalue q-dual root system datum highest weight module Verma module quotient by maximal submodule Weyl character formula character expressed as ratio of Weyl denominator alternating sum over Weyl group action on weight lattice shifted by rho Weyl vector half sum positive roots Kostant partition function counts ways expressing weight as non-negative integer combination of positive roots generating function identity holds formal power series ring completion p-adic integers Zp topology p-adic metric absolute value p-adic valuation normalised v p of p equals one rational number decomposes uniquely into product p to power n times unit unit group cyclic order p minus one for odd p order two for p equals two Teichmuller representative multiplicative section residue field Fp lift canonical element Frobenius endomorphism x maps to x to power p ring homomorphism finite field characteristic p has p elements extension degree n has p to power n elements multiplicative group cyclic generator primitive element polynomial irreducible degree n exists over Fp construction via Conway polynomials normal basis theorem exists basis consisting of conjugates under Frobenius Galois group cyclic generated by Frobenius automorphism Galois extension normal separable splitting field polynomial separable polynomial distinct roots in algebraic closure Artin-Schreier theory cyclic extensions degree p characteristic p described by Artin-Schreier polynomial x to power p minus x minus a Kummer theory cyclic extensions degree n prime to p described by n-th roots of unity adjoin element b where b to power n equals a Hilbert class field maximal unramified abelian extension degree equals class number ray class field modulus generalisation including ramified primes class field theory correspondence between congruence subgroups of idele class group abelian extensions of number field reciprocity law Artin reciprocity map idele class group to Galois group surjective kernel connected component idele class group compact quotient theorem Kronecker-Weber theorem every abelian extension of rationals contained in cyclotomic field conductor-discriminant formula discriminant equals product over characters conductor exponent Chebotarev density theorem Frobenius conjugacy class density equals size class over group order effective version explicit error term logarithmic saving under GRH generalisation Sato-Tate distribution semisimple conjugacy classes compact group SU two equidistributed according to Sato-Tate measure arcsine distribution transformed Chebyshev polynomial recurrence relation T n cosine n theta satisfies second-order linear recurrence T n plus one equals two x times T n minus T n minus one initial conditions T zero equals one T one equals x explicit closed form using hyperbolic cosine for absolute value x greater one Chebyshev nodes roots of Chebyshev polynomial clustered near endpoints interval minimax polynomial approximation equioscillation theorem Chebyshev alternation theorem characterisation best uniform approximation polynomial degree n given by n plus two alternating extreme points error function oscillates sign achieving maximum absolute value alternately Bernstein polynomial approximation continuous function uniform convergence constructive proof using probabilistic interpretation binomial distribution expectation Bernstein basis polynomials non-negative partition of unity degree n Bernstein operator positive linear Markov operator contraction in sup norm derivative bound Lipschitz constant bound n over interval width Jackson theorem quantitative uniform approximation rate O one over n for Lipschitz functions Ditzian-Totik modulus of smoothness generalisation weighted polynomial approximation Bernstein-Kantorovich operator modification integrating against uniform measure Bernstein-Bezier basis B-spline representation control polygon convex hull property subdivision algorithm de Casteljau recursive evaluation stable numerically basis transformation matrix lower triangular unimodal entries Bernstein to B-spline conversion knot insertion algorithm Boehm algorithm single knot insertion preserves curve geometry reparameterisation invariance NURBS non-uniform rational B-spline weighted extension rational functions control points weights positive homogeneous degree zero rational parametrisation projective space representation dehomogenisation affine patch chart selection atlas manifold smooth transition functions diffeomorphism overlap charts second countable Hausdorff paracompact conditions metrizable manifold admits Riemannian metric positive-definite symmetric bilinear tensor field smooth dependence pointwise inner product tangent spaces induces distance via length minimising geodesics exponential map local diffeomorphism normal neighbourhood tubular neighbourhood theorem embedded submanifold codimension k neighbourhood diffeomorphic normal bundle zero section compact manifold admits triangulation smooth structure unique up to diffeomorphism in dimension four exotic R seven exists uncountably many smooth structures homeomorphic but not diffeomorphic standard sphere Donaldson theorem intersection form negative-definite for closed simply-connected smooth four-manifold with b two plus greater zero exotic structures exist Freedman theorem topological classification intersection form unimodular symmetric form over integers unique up to stable equivalence Kirby calculus handle slides Kirby moves handle cancellation handle attachment 2-handle along framed knot surgery description four-manifold via Kirby diagram link in three-sphere framing integer attached component linking matrix self-linking integer determines handle framing Dehn surgery coefficient rational number p over q slope on torus boundary knot exterior resulting three-manifold classification unknown general case Thurston hyperbolic Dehn surgery theorem finite exceptional slopes hyperbolic structure persists most surgeries hyperbolic volume decreases monotonic along exceptional slope sequence JSJ decomposition torus incompressible tori canonical decomposition geometric pieces Seifert fibered graph manifolds hyperbolic pieces geometrisation conjecture proved Perelman Ricci flow with surgery Hamilton compactness theorem no finite-time singularity formation for bounded curvature initial metric three-manifold Ricci flow smooth solution exists unique short time diffeomorphism invariance scalar curvature transforms as function under pullback Ricci tensor transforms as (0,2) tensor trace-free Ricci evolves as heat equation plus quadratic term Ricci flow monotonicity functional Perelman entropy W functional non-increasing along flow lambda functional volume functional coupled system W lambda V system monotonicity Gaussian density decreasing along flow no local collapsing theorem bounded curvature implies non-collapsing at all scales epsilon regularity theorem curvature bound implies Ricci flow solution smooth at that point time interval extension criterion curvature blow-up alternative exists finite maximal time either curvature bounded on compact subsets solution extends smoothly or curvature diverges singularity formation classification neckpinch cigar soliton steady solution complete manifold positive curvature operator asymptotically cylindrical cross-sections shrinking solitons self-similar solutions Ricci flow sphere CP two CP two blow-up model compact Kähler manifold Kähler-Ricci flow converges to Kähler-Einstein metric Yau theorem uniformisation Kähler manifold admits unique Kähler-Einstein metric with c one sign determined by existence of Kähler-Einstein metric solved Aubin Yau independently partial results Calabi conjecture proved Yau using a priori estimates complex Monge-Ampère equation det g i j bar equals f e to power phi uniqueness via maximum principle comparison principle viscosity solutions Evans Krylov Schauder estimates elliptic regularity W two p estimates Calderon-Zygmund theory singular integrals bounded L p for p strictly between one and infinity weak (1,1) endpoint Marcinkiewicz interpolation theorem real interpolation complex interpolation K-method J-method reiteration theorem Calderon product formula holomorphic interpolation complex interpolation method analytic family operators bounded between Banach spaces holomorphic on strip closed on boundary real interpolation K-method defined by K-functional infimum over decomposition f equals g plus h norm g to power theta times norm h to power one minus theta equivalent quasi-norm satisfies triangle inequality up to constant factor depending on theta method yields quasi-Banach space when theta equals zero or one strict interpolation property holds interpolation between L p spaces yields L q for q between p and p prime Riesz-Thorin theorem explicit interpolation constant equals one complex method sharp constant Minkowski functional norm definition infimum lambda positive such that f belongs to lambda times unit ball quasi-norm satisfies homogeneity absolute property triangle inequality up to constant factor p quasi-norm satisfies all three exactly when p greater than or equal to one p equals one normed space Banach space complete normed space Riesz-Fischer theorem L p complete for one less than or equal p less than infinity L infinity complete under essential supremum norm Baire category theorem complete metric space cannot be written as countable union nowhere dense sets corollary existence of points of continuity for Baire class one functions Baire class zero continuous functions Baire class one pointwise limits of continuous functions Baire hierarchy ordinal levels defined transfinitely pointwise limits of previous level functions analytic sets universally measurable universally measurable sets completion of sigma-algebra by all null sets completion operation idempotent monotone sigma-algebra completion unique smallest complete sigma-algebra containing original sigma-algebra completion of Lebesgue sigma-algebra includes all analytic sets universally measurable sets under every probability measure completion Lebesgue measure outer measure inner measure Caratheodory extension theorem unique extension of pre-measure to sigma-algebra generated by semi-ring Caratheodory condition measurable sets partitionable into measurable pieces for every test set outer measure equals inner measure on measurable sets regular measure finite measure on sigma-algebra of subsets of locally compact Hausdorff space regularity condition every measurable set approximable from inside by compact sets from outside by open sets Riesz representation theorem positive linear functional on C c X corresponds uniquely to regular Borel measure Radon measure locally finite inner regular Borel measure on locally compact Hausdorff space Riesz-Markov-Kakutani representation dual of C zero X isomorphic to space of regular Borel measures total variation norm dual of C zero X isometrically isomorphic to space of finite signed regular Borel measures Hahn decomposition theorem measurable space sigma-finite signed measure decomposes domain into positive set and negative set Jordan decomposition unique up to null sets Hahn decomposition not unique on null sets variation measure absolute value of signed measure decomposes into positive and negative parts singular continuous measure singular with respect to Lebesgue measure concentrated on set of Lebesgue measure zero Cantor function singular continuous measure example distribution function continuous derivative zero almost everywhere Cantor set measure zero uncountable perfect nowhere dense compact set Hausdorff dimension log two over log three fractional dimension non-integer example self-similar fractal iterated function system contraction mappings unique invariant measure supported on attractor open set condition ensures overlap measure absolutely continuous with respect to Hausdorff measure dimension equals similarity dimension under OSC pressure function thermodynamic formalism dimension spectrum pressure function P s equals zero defines similarity dimension for open set condition pressure monotonicity pressure function concave in s topological pressure generalisation of topological entropy variational principle pressure equals supremum over invariant measures of measure-theoretic entropy plus integral of potential function equilibrium state measure maximising pressure unique for mixing potentials specification property expansivity Bowen metric orbit growth rate exponential entropy definition topological entropy number of distinguishable orbits at resolution epsilon growth rate lim sup as epsilon approaches zero log count over log one over epsilon entropy of subshift of finite type equals growth rate number of admissible words length n exponential rate log lambda where lambda largest eigenvalue adjacency matrix Perron-Frobenius theorem non-negative matrix positive real eigenvalue largest magnitude eigenvector non-negative unique up to scaling if matrix irreducible eigenvalue simple spectral radius growth rate adjacency matrix powers number of walks length n exponential rate equals spectral radius graph spectral radius bounded above by maximum degree regular graph spectral radius equals degree complete graph spectral radius equals n minus one path graph spectral radius equals two cosine pi over n plus one cycle graph spectral radius equals two cosine pi over n star graph spectral radius equals square root of n minus one tree spectral radius strictly less than maximum degree unless tree is star complete bipartite graph spectral radius equals square root of product of part sizes spectral gap difference between largest and second-largest eigenvalues expansion property expander graph constant spectral gap independent of size Ramanujan graph spectral gap optimal lambda two absolute value less than or equal two square root of d minus one over d construction via LPS Ramanujan graphs number theory based existence proven non-constructively probabilistic method explicit constructions known only partial results Margulis construction Cayley graphs Lubotzky Phillips Sarnak construction via quaternion algebras zig-zag product graph operation preserving expansion degree bounded blow-up lemma expander mixing lemma discrepancy bound between edge counts and expected counts spectral method derandomisation conditional expectations method pessimistic estimators method of conditional probabilities explicit constructions limited families polynomial method algebraic approach to combinatorial problems polynomial method Szemeredi regularity lemma graph partition into parts almost regular pairs dense pairs contain all graphs bounded degree removal lemma dense graph close to triangle-free graph in edge density removal lemma Roth theorem subset of integers density delta contains three-term arithmetic progression bound delta less than or equal c over log log inverse function improvement via Bourgain-Gowers polynomial method Fourier analysis additive combinatorics structure theory sumset bounds Plunnecke-Ruzsa inequalities sumset doubling constant bounds growth of iterated sumsets Freiman-Ruzsa theorem bounded doubling implies approximate group structure coset progression theorem arithmetic progressions in sumsets density Hales-Jewett theorem van der Waerden theorem Ramsey theory finite colourings monochromatic structures guaranteed Ramsey number R k l minimum n such that any two-colouring of edges complete graph on n vertices contains monochromatic clique size k or independent set size l bounds lower bound exponential via probabilistic method upper bound factorial growth via Erdos-Szekeres recurrence exact values known only small cases R three three equals six R three four equals nine R four four equals eighteen open problem general bounds polynomial method breakthrough Bloom Sisask improvement Roth bound logarithmic factor removed polynomial method Fourier analytic approach additive energy bound sumset growth sum-product phenomenon integers reals finite fields distinct behaviour Erdos-Szmeredi conjecture sum-product dichotomy resolved partially finite fields via Bourgain-Katz-Tao result product set or sumset large depending on field size construction explicit sets achieving bounds multiplicative energy additive energy dual quantities Fourier coefficients L two norm relation via Parseval Plancherel theorem isometry L two to L two Fourier transform unitary operator convolution theorem convolution product Fourier transforms pointwise product convolution commutative associative identity element Dirac delta distribution tempered distribution L one functions Fourier transform continuous vanishes at infinity Riemann-Lebesgue lemma decay rate quantitative bounds via integration by parts smoothness assumptions compact support Fourier transform entire function Paley-Wiener theorem characterisation Fourier transform of compactly supported distribution entire function exponential type support bound type equals radius support Bernstein inequality derivative bound frequency support limit entire function bounded on real line growth condition exponential type implies Fourier transform supported on interval bandlimited function sampling theorem Shannon-Nyquist reconstruction bandlimited function from samples at rate twice bandwidth exact reconstruction formula sinc interpolation Whittaker-Shannon interpolation formula aliasing phenomenon undersampling folds frequencies beyond Nyquist rate anti-aliasing filter pre-processing step low-pass filter cutoff frequency below Nyquist rate design FIR filter window method Hamming Blackman Kaiser windows stopband attenuation tradeoff transition bandwidth IIR filter design bilinear transform frequency warping pre-warping analog prototype Butterworth maximally flat Chebyshev equiripple passband elliptic Cauer minimum order for given specifications filter stability pole location inside unit circle discrete-time stability condition absolute value pole less than one continuous-time stability condition real part pole negative state-space representation system matrices A B C D transfer function via Laplace transform zero-state response controllability observability Kalman rank condition controllability matrix full rank observability matrix full rank duality controllable implies stabilisable observable implies detectable minimal realisation unique up to similarity transformation similarity transformation preserves eigenvalues transfer function state transformation T new A equals T inverse A T new B equals T inverse B new C equals C T new D equals D pole placement arbitrary pole assignment via state feedback possible when controllable LQR optimal control quadratic cost infinite horizon algebraic Riccati equation solution stabilising solution unique positive semi-definite Kalman filtering recursive estimation linear systems Gaussian noise minimises mean squared error Joseph form numerically stable covariance update square-root filter Cholesky factor update UD factorisation Bierman Thorpe algorithm measurement update time update separation principle optimal controller and estimator design decoupled certainty equivalence principle holds linear Gaussian systems only separation fails non-linear systems dual effect information state-dependent control affects future information gathering exploration exploitation trade-off bandit problem multi-armed bandit regret definition cumulative loss minus best fixed arm loss lower bound sqrt K T log T upper bound UCB algorithm achieves matching bound explore-then-commit strategy suboptimal logarithmic regret Thompson sampling Bayesian approach posterior sampling asymptotically optimal regret contextual bandit linear bandit generalisation covariate information linear payoff model LinUCB algorithm regret bound dimension times log T bandit convex optimisation online convex learning regret sublinear via online gradient descent regret bound square root T optimal first-order lower bound online learning online convex optimisation regret definition cumulative regret against best fixed comparator in hindsight online gradient descent update w t plus one equals projection of w t minus eta times gradient onto constraint set eta learning rate choice eta equals one over square root T gives optimal regret bound online mirror descent generalisation via Bregman divergence Fenchel-Young inequality conjugate function subgradient method non-smooth convex optimisation proximal operator definition argmin of f x plus one over two lambda times norm squared x minus y Moreau decomposition proximal of conjugate equals reflection of proximal FISTA accelerated gradient method optimal rate one over t squared for smooth convex composite problems Nesterov optimal method lower bound first-order oracle complexity smooth convex optimisation one over epsilon squared gradient complexity necessary sufficient accelerated methods achieve this via Nesterov momentum heavy-ball method Polyak momentum parameter choice optimal for quadratics conjugate gradient method exact minimisation along Krylov subspace directions finite termination n iterations exact arithmetic CG convergence rate condition number square root dependence Chebyshev acceleration optimal polynomial preconditioning preconditioned CG condition number preconditioned operator preconditioner choice incomplete Cholesky ILU factorisation sparse approximate inverse SPAI algebraic multigrid AMG coarsening strength of connection interpolation operator prolongation restriction operators V-cycle W-cycle F-cycle cycle count tradeoff smoother choice Jacobi Gauss-Seidel SOR Chebyshev smoother convergence rate independent of mesh size optimal AMG O one per cycle for elliptic problems multilevel preconditioning optimal condition

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