exp-pairs-0.2.1.0: Linear programming over exponent pairs

Math.ExpPairs.Pair

Contents

Description

Initial exponent pairs.

Provides a set of initial exponent pairs, consisting of two points (0, 1), (1/2, 1/2) and a triangle with vertices in (1/6, 2/3), (2/13, 35/52) and (13/84, 55/84). The triangle is represented as a list of nodes of a net, covering the triangle.

Below A and B stands for van der Corput's processes. See Math.ExpPairs.Process for explanations.

Synopsis

# Documentation

data Triangle Source #

Vertices of the triangle of initial exponent pairs.

Constructors

 Corput16 Usual van der Corput exponent pair (1/6, 2/3) = AB(0, 1). HuxW87b1 An exponent pair (2/13, 35/52) from Huxley M. N. Exponential sums and the Riemann zeta function' // Proceedings of the International Number Theory Conference held at Universite Laval in 1987, Walter de Gruyter, 1989, P. 417-423. Bourgain17 An exponent pair (13/84, 55/84) from Bourgain J. Decoupling, exponential sums and the Riemann zeta function // J. Amer. Math. Soc., 2017, 30, P. 205-224.
Instances
 Source # Instance detailsDefined in Math.ExpPairs.Pair Methods Source # Instance detailsDefined in Math.ExpPairs.Pair MethodsenumFrom :: Triangle -> [Triangle] #enumFromTo :: Triangle -> Triangle -> [Triangle] # Source # Instance detailsDefined in Math.ExpPairs.Pair Methods Source # Instance detailsDefined in Math.ExpPairs.Pair Methods(<) :: Triangle -> Triangle -> Bool #(>) :: Triangle -> Triangle -> Bool # Source # Instance detailsDefined in Math.ExpPairs.Pair MethodsshowList :: [Triangle] -> ShowS # Source # Instance detailsDefined in Math.ExpPairs.Pair Associated Typestype Rep Triangle :: Type -> Type # Methodsto :: Rep Triangle x -> Triangle # Source # Instance detailsDefined in Math.ExpPairs.Pair Methodspretty :: Triangle -> Doc ann #prettyList :: [Triangle] -> Doc ann # type Rep Triangle Source # Instance detailsDefined in Math.ExpPairs.Pair type Rep Triangle = D1 (MetaData "Triangle" "Math.ExpPairs.Pair" "exp-pairs-0.2.1.0-J4IGbuSTVwXCgBqjoU0P5n" False) (C1 (MetaCons "Corput16" PrefixI False) (U1 :: Type -> Type) :+: (C1 (MetaCons "HuxW87b1" PrefixI False) (U1 :: Type -> Type) :+: C1 (MetaCons "Bourgain17" PrefixI False) (U1 :: Type -> Type)))

data InitPair' t Source #

Type to hold an initial exponent pair.

Constructors

 Corput01 Usual van der Corput exponent pair (0, 1). Corput12 Usual van der Corput exponent pair (1/2, 1/2) = B(0, 1). Mix !t !t Point from the interior of Triangle. Exactly Mix a b = a * Corput16 + b * HuxW87b1 + (1-a-b) * Bourgain17
Instances
 Eq t => Eq (InitPair' t) Source # Instance detailsDefined in Math.ExpPairs.Pair Methods(==) :: InitPair' t -> InitPair' t -> Bool #(/=) :: InitPair' t -> InitPair' t -> Bool # Ord t => Ord (InitPair' t) Source # Instance detailsDefined in Math.ExpPairs.Pair Methodscompare :: InitPair' t -> InitPair' t -> Ordering #(<) :: InitPair' t -> InitPair' t -> Bool #(<=) :: InitPair' t -> InitPair' t -> Bool #(>) :: InitPair' t -> InitPair' t -> Bool #(>=) :: InitPair' t -> InitPair' t -> Bool #max :: InitPair' t -> InitPair' t -> InitPair' t #min :: InitPair' t -> InitPair' t -> InitPair' t # Show t => Show (InitPair' t) Source # Instance detailsDefined in Math.ExpPairs.Pair MethodsshowsPrec :: Int -> InitPair' t -> ShowS #show :: InitPair' t -> String #showList :: [InitPair' t] -> ShowS # Source # Instance detailsDefined in Math.ExpPairs.Pair Associated Typestype Rep (InitPair' t) :: Type -> Type # Methodsfrom :: InitPair' t -> Rep (InitPair' t) x #to :: Rep (InitPair' t) x -> InitPair' t # (Pretty t, Num t, Eq t) => Pretty (InitPair' t) Source # Instance detailsDefined in Math.ExpPairs.Pair Methodspretty :: InitPair' t -> Doc ann #prettyList :: [InitPair' t] -> Doc ann # type Rep (InitPair' t) Source # Instance detailsDefined in Math.ExpPairs.Pair type Rep (InitPair' t) = D1 (MetaData "InitPair'" "Math.ExpPairs.Pair" "exp-pairs-0.2.1.0-J4IGbuSTVwXCgBqjoU0P5n" False) (C1 (MetaCons "Corput01" PrefixI False) (U1 :: Type -> Type) :+: (C1 (MetaCons "Corput12" PrefixI False) (U1 :: Type -> Type) :+: C1 (MetaCons "Mix" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 t) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 t))))

Exponent pair built from rational fractions of Corput16, HuxW87b1 and Hux05

The set of initial exponent pairs. It consists of Corput01, Corput12 and 496 = sum [1..31] Mix-points, which forms a uniform net over Triangle.

Convert initial exponent pair from its symbolic representation as InitPair to pair of rationals.

Same as initPairToValue`, but immediately convert from Q^2 to PN^3.

# Orphan instances

 (Integral a, Show a) => Pretty (Ratio a) Source # Instance details Methodspretty :: Ratio a -> Doc ann #prettyList :: [Ratio a] -> Doc ann #