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ABC conjectureabstractabstract lang completenessactionadditive combinatoricsadele ringadic valuationaffine groupaffine spacesahiv22ahiv22supportalgebraic geometryalgorithmallow denyallows termalmost orthogonalityalmost periodicityanalysisanalytic number theoryanalytic setangular momentumangular velocityantichain operatorantichain tile countappendapprox hom pfrapproximationarcaritharithmetic geometryarithmetic progressionsas tensorasymptoticsasync runtimeauthorizationauthorizeauthorizerautomata theoryautomorphismauxiliaryball extensionball scalingbase changebase permbasicbasicsbasisbasis linearbatched evaluatorbehavioural theoryberlekamp welchbig opbijectivitybilinear formbindingbirthdaybisimulationbivariateblock rel distancebmobn254borel caratheodorybounding mutualbraketbridgebrownian motionbrun titchmarshbuchi congrbuchi congruencebuchi interbundledbundled semanticscaching logging oraclecaching oraclecadlagcadlag modificationcalculationscampanatocan reach 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Theoremfgfibringfidelityfieldfield strengthfinitary carlesonfinitefinite adele ringfinite coverfinite dimensionalfinite fieldfinitenessfinset termsfirstfirst estimatefive quantaflattenflex approxfoldingfolding polynomialforest complementforest unionforestsforkformal languagesformal logicfoundationsfourierFourier analysisfree semigroupfree statefrey packagefrobeniusfrom mathlibfrom unaryfull beta confluencefull beta eta confluencefull etafunctiongadgetsgauge transformationgen heapgen setsgeneralgeneralizedgenerate seedgeneric liftghost mapgradgradinggrid structuregromovgroupgroup theory stuffguruswamiguruswami sudanhaltinghamiltonianhandler from unaryhandler specsharmonic analysisharmonic waveharnackhas tauhas var adjointhash commitmentheat kernelheat solutionheat solution propertyhelperherbrand quotienthilbert kernelhilbert strong typehilbert94histholder normholder van der corputhom pfrhomotopyhopshprinciplehundred percenthypothesis testingideal gasidentical until badimmersionimplimplicationsimproved pfrindicatorinduction scheme delta1inductive constructionsinequalitiesinf modelinf occinfinite graph ramseyinfinite wireinflationinputinsert and contractinsert and contract natinsert noneinsert someinsertion sortinstanceinstancesintegralintegral closureinteractive protocolsinterleaved codeintervalinverse iterationis dist boundedis extremais plane waveis viableisqbridgeiteriterated laplacianiterationiwaniec kowalski ch1kemdemkey encap mechkinetic energykinetic termknowledge soundnesskoszul signkullbackkummers lemmal2estimatelagrangianlanguagelaplace inversionlaplace runge lenz vectorlarge separationleakagelemmaslevelslex'orderlie traceline in cubiclinear codelinear mv extensionlinearizedlinftylipschitz normlistlocallocal fieldslocal john nirenberglocalisation datalocallyloglog gradientlogging oraclelooplp function toolkitls corelyubashevsky seilermac from prfmachinemagnetic fieldmainmain resultsmain theoremmain thmmapmap unionmarcinkiewicz 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varrestricted provabilityrestricted weak typerestrictionresultantresultsrho functionalrough inputround consistencyrowsrqrudinsafetysampleable typesamplingscaled ball estimatesschur triangulationschwartz submoduleschwartz zippel countingsec expsecond estimatesection2section4securityseeded forkseeded oraclesegmentself adjointsemanticssend challengesend claimsend witnessseparation logicseq composeserre approximationserre derivative slashsetset theoryshamirsigmasignature algsimple processsimulate qsimulationsinc kernel error boundssingle roundslash action auxilslicesmooth approximationsmooth embeddingsmooth existencesmooth testssobolevsobolev poincaresobolev spacessolid spheresolutionsoundnessspectral projection boundsphere packingsplit foldsplit multiplicative reductionsplitting modulesquare integrablessastage onestampacchia truncationstandard formstandard reductionstate projectionstate separatingstatementstatic contractstatic wick termstd do bridgestepsstochastic intervalstochastic processesstr actionstraight linestrong normstrong pntstructuressub specsubdomainsubtypesucc succ abovesumsumcheck phasesuper commutesupportsurroundingsymbolic compilationsymbolizersymm enc algsymmetrictake droptanhtate curvetate curve base changetate curve constructiontate parametertau functionaltau in queriesten quantatensor producttensorialtest functionsthermo quantitiesthree apthree dimensiontile correlationtile existencetile structuretime contracttime contractiontime liketime ordertime slicetiseto complexto fft domainto nato singleto single acceptto solstopologytorsion endgametotaltotal derivative equivalencetpetracetrace inner producttrace vanishingtrajectorytraversaltrivial cohomologytriviality criteriontruncationttransformtvdisttwotwo statetypecheckertypecheckingtypingudfultraproductunaryunbalancingunboundedunbundleduncertaintyuncontracted listuniform integrableuniquenessunitunit tensorunitaryuniversal algebraunpredictabilityunramifiedup downup singletupdatesutransformvalid refsvalidationvan der corputvandermondevariancevcdimensionvector bundlesverificationverifierversion spaceversion space latticeweak calderon zygmundweak derivativeweak derivativesweak harnackweak pfrweightedweighted estimateswell typedwell typed okwell typed verificationwfwhole space sobolevwick contractionswick termwicks theorem normalwienerwired runwith abortwitnesseswriter costyukawazero consensuszero count crudezeta boundszf

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Φ₀ S transform

φ₀_S_transform

Plain-language statement

The S-transformation formula for φ₀: φ₀(-1/z) = φ₀(z) - (12i/π)(1/z)φ₋₂(z) - (36/π²)(1/z²)φ₋₄(z) This is Blueprint Lemma 7.2.

sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

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