Tools for special functions and special numbers Graduate Student - - PowerPoint PPT Presentation

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Tools for special functions and special numbers Graduate Student - - PowerPoint PPT Presentation

Tools for special functions and special numbers Graduate Student Colloquium Tulane University, New Orleans Armin Straub October 15, 2013 University of Illinois & Max-Planck-Institut at UrbanaChampaign f ur Mathematik, Bonn ISC


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SLIDE 1

Tools for special functions and special numbers

Graduate Student Colloquium Tulane University, New Orleans Armin Straub October 15, 2013 University of Illinois

at Urbana–Champaign

& Max-Planck-Institut

f¨ ur Mathematik, Bonn

ISC — PSLQ — OEIS — CAD — WZ

Tools for special functions and special numbers Armin Straub 1 / 26
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SLIDE 2

THE TOOLS TODAY

ISC

Inverse Symbolic Calculator

PSLQ

Lattice Reduction Algorithm

OEIS

On-Line Encyclopedia of Integer Sequences

CAD

Cylindrical Algebraic Decomposition

WZ

Wilf–Zeilberger Theory

Tools for special functions and special numbers Armin Straub 2 / 26
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SLIDE 3

Random walks

  • We study random walks in the plane

consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Tools for special functions and special numbers Armin Straub 3 / 26
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SLIDE 4

Random walks

  • We study random walks in the plane

consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Tools for special functions and special numbers Armin Straub 3 / 26
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SLIDE 5

Random walks

  • We study random walks in the plane

consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Tools for special functions and special numbers Armin Straub 3 / 26
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SLIDE 6

Random walks

  • We study random walks in the plane

consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Tools for special functions and special numbers Armin Straub 3 / 26
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SLIDE 7

Random walks

  • We study random walks in the plane

consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Tools for special functions and special numbers Armin Straub 3 / 26
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SLIDE 8

Random walks

  • We study random walks in the plane

consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Tools for special functions and special numbers Armin Straub 3 / 26
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SLIDE 9

Random walks

d

  • We study random walks in the plane

consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Tools for special functions and special numbers Armin Straub 3 / 26
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SLIDE 10

Random walks

d

  • We study random walks in the plane

consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

  • We are interested in the distance

traveled in n steps. For instance, how large is this dis- tance on average?

Q

  • Probability density: pn(x)
Tools for special functions and special numbers Armin Straub 3 / 26
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SLIDE 11

Random walks are only about 100 years old

  • Karl Pearson asked for

pn(x) in Nature in 1905.

This famous question coined the term random walk.

Applications include:

  • dispersion of mosquitoes
  • random migration of

micro-organisms

  • phenomenon of laser speckle
Tools for special functions and special numbers Armin Straub 4 / 26
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SLIDE 12

Long random walks

pn(x) ≈ 2x n e−x2/n for large n

THM

Rayleigh, 1905

10 20 30 40 50 0.01 0.02 0.03 0.04 0.05 0.06

EG

p200

The lesson of Lord Rayleigh’s solution is that in open country the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point!

Karl Pearson, 1905

Tools for special functions and special numbers Armin Straub 5 / 26
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SLIDE 13

Densities of short walks

p2

0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8

p3

0.5 1.0 1.5 2.0 2.5 3.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

p4

1 2 3 4 0.1 0.2 0.3 0.4 0.5

p5

1 2 3 4 5 0.05 0.10 0.15 0.20 0.25 0.30 0.35

p6

1 2 3 4 5 6 0.05 0.10 0.15 0.20 0.25 0.30 0.35

p7

1 2 3 4 5 6 7 0.05 0.10 0.15 0.20 0.25 0.30 Tools for special functions and special numbers Armin Straub 6 / 26
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SLIDE 14

Densities of short walks

p2

0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8

p3

0.5 1.0 1.5 2.0 2.5 3.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

p4

1 2 3 4 0.1 0.2 0.3 0.4 0.5

p5

1 2 3 4 5 0.05 0.10 0.15 0.20 0.25 0.30 0.35

p6

1 2 3 4 5 6 0.05 0.10 0.15 0.20 0.25 0.30 0.35

p7

1 2 3 4 5 6 7 0.05 0.10 0.15 0.20 0.25 0.30 Tools for special functions and special numbers Armin Straub 6 / 26
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SLIDE 15

Moments

  • The moments of a RV X are E(X), E(X2), E(X3), . . .
  • If X has probability density f(x) then

E(Xs) = ∞

−∞

xsf(x) dx The moments E(Xs) are analytic in s.

(if, e.g., f(x) is compactly supported)

FACT

Tools for special functions and special numbers Armin Straub 7 / 26
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SLIDE 16

Moments

  • The moments of a RV X are E(X), E(X2), E(X3), . . .
  • If X has probability density f(x) then

E(Xs) = ∞

−∞

xsf(x) dx The moments E(Xs) are analytic in s.

(if, e.g., f(x) is compactly supported)

FACT

  • Represent the kth step by the complex number e2πixk.
  • The sth moment of the distance after n steps is:

Wn(s) :=

  • [0,1]n
  • n
  • k=1

e2πxki

  • s

dx In particular, Wn(1) is the average distance after n steps.

Tools for special functions and special numbers Armin Straub 7 / 26
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SLIDE 17

Average distance traveled in two steps

  • Numerically: W2(1) ≈ 1.2732395447351626862
Tools for special functions and special numbers Armin Straub 8 / 26
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SLIDE 18

Average distance traveled in two steps

  • Numerically: W2(1) ≈ 1.2732395447351626862
Tools for special functions and special numbers Armin Straub 8 / 26
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SLIDE 19

Average distance traveled in two steps

  • Numerically: W2(1) ≈ 1.2732395447351626862
Tools for special functions and special numbers Armin Straub 8 / 26
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SLIDE 20

Average distance traveled in two steps

  • Numerically: W2(1) ≈ 1.2732395447351626862
Tools for special functions and special numbers Armin Straub 8 / 26
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SLIDE 21

The simple two-step case confirmed

  • The average distance in two steps:

W2(1) = 1 1

  • e2πix + e2πiy

dxdy = ?

Tools for special functions and special numbers Armin Straub 9 / 26
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SLIDE 22

The simple two-step case confirmed

  • The average distance in two steps:

W2(1) = 1 1

  • e2πix + e2πiy

dxdy = ? = 1

  • 1 + e2πiy

dy

Tools for special functions and special numbers Armin Straub 9 / 26
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SLIDE 23

The simple two-step case confirmed

  • The average distance in two steps:

W2(1) = 1 1

  • e2πix + e2πiy

dxdy = ? = 1

  • 1 + e2πiy

dy = 1 2 cos(πy)dy

  • 1 + e2πiy
  • =
  • 1 + (cos πy + i sin πy)2
  • = 2 cos(πy)
Tools for special functions and special numbers Armin Straub 9 / 26
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SLIDE 24

The simple two-step case confirmed

  • The average distance in two steps:

W2(1) = 1 1

  • e2πix + e2πiy

dxdy = ? = 1

  • 1 + e2πiy

dy = 1 2 cos(πy)dy = 4 π ≈ 1.27324

  • 1 + e2πiy
  • =
  • 1 + (cos πy + i sin πy)2
  • = 2 cos(πy)
Tools for special functions and special numbers Armin Straub 9 / 26
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SLIDE 25

The simple two-step case confirmed

  • The average distance in two steps:

W2(1) = 1 1

  • e2πix + e2πiy

dxdy = ? = 1

  • 1 + e2πiy

dy = 1 2 cos(πy)dy = 4 π ≈ 1.27324

  • Mathematica 7 and Maple 14 think the double integral is 0.

Better: Mathematica 8 and 9 just don’t evaluate the double integral.

  • 1 + e2πiy
  • =
  • 1 + (cos πy + i sin πy)2
  • = 2 cos(πy)
Tools for special functions and special numbers Armin Straub 9 / 26
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SLIDE 26

The simple two-step case confirmed

  • The average distance in two steps:

W2(1) = 1 1

  • e2πix + e2πiy

dxdy = ? = 1

  • 1 + e2πiy

dy = 1 2 cos(πy)dy = 4 π ≈ 1.27324

  • Mathematica 7 and Maple 14 think the double integral is 0.

Better: Mathematica 8 and 9 just don’t evaluate the double integral.

  • This is the average length of a random arc on a

unit circle.

  • 1 + e2πiy
  • =
  • 1 + (cos πy + i sin πy)2
  • = 2 cos(πy)
Tools for special functions and special numbers Armin Straub 9 / 26
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SLIDE 27

Moments of random walks

The sth moment Wn(s) of the density pn: Wn(s) := ∞ xspn(x) dx =

  • [0,1]n
  • e2πix1 + . . . + e2πixn

s dx

DEF

Tools for special functions and special numbers Armin Straub 10 / 26
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SLIDE 28

Moments of random walks

The sth moment Wn(s) of the density pn: Wn(s) := ∞ xspn(x) dx =

  • [0,1]n
  • e2πix1 + . . . + e2πixn

s dx

DEF

  • On a desktop:

W3(1) ≈ 1.57459723755189365749 W4(1) ≈ 1.79909248 W5(1) ≈ 2.00816

Tools for special functions and special numbers Armin Straub 10 / 26
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SLIDE 29

Moments of random walks

The sth moment Wn(s) of the density pn: Wn(s) := ∞ xspn(x) dx =

  • [0,1]n
  • e2πix1 + . . . + e2πixn

s dx

DEF

  • On a desktop:

W3(1) ≈ 1.57459723755189365749 W4(1) ≈ 1.79909248 W5(1) ≈ 2.00816

  • On a supercomputer:

Lawrence Berkeley National Laboratory, 256 cores

W5(1) ≈ 2.0081618

Tools for special functions and special numbers Armin Straub 10 / 26
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SLIDE 30

Moments of random walks

The sth moment Wn(s) of the density pn: Wn(s) := ∞ xspn(x) dx =

  • [0,1]n
  • e2πix1 + . . . + e2πixn

s dx

DEF

  • On a desktop:

W3(1) ≈ 1.57459723755189365749 W4(1) ≈ 1.79909248 W5(1) ≈ 2.00816

  • On a supercomputer:

Lawrence Berkeley National Laboratory, 256 cores

W5(1) ≈ 2.0081618

  • Hard to evaluate numerically to high precision.

Monte-Carlo integration gives approximations with an asymptotic error of O(1/ √ N) where N is the number of sample points.

Tools for special functions and special numbers Armin Straub 10 / 26
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SLIDE 31

Moments of random walks

The sth moment Wn(s) of the density pn: Wn(s) := ∞ xspn(x) dx =

  • [0,1]n
  • e2πix1 + . . . + e2πixn

s dx

DEF

n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1.273 2.000 3.395 6.000 10.87 20.00 37.25 3 1.575 3.000 6.452 15.00 36.71 93.00 241.5 4 1.799 4.000 10.12 28.00 82.65 256.0 822.3 5 2.008 5.000 14.29 45.00 152.3 545.0 2037. 6 2.194 6.000 18.91 66.00 248.8 996.0 4186.

Tools for special functions and special numbers Armin Straub 10 / 26
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SLIDE 32

Moments of random walks

The sth moment Wn(s) of the density pn: Wn(s) := ∞ xspn(x) dx =

  • [0,1]n
  • e2πix1 + . . . + e2πixn

s dx

DEF

n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1.273 2.000 3.395 6.000 10.87 20.00 37.25 3 1.575 3.000 6.452 15.00 36.71 93.00 241.5 4 1.799 4.000 10.12 28.00 82.65 256.0 822.3 5 2.008 5.000 14.29 45.00 152.3 545.0 2037. 6 2.194 6.000 18.91 66.00 248.8 996.0 4186. W2(1) = 4

π

Tools for special functions and special numbers Armin Straub 10 / 26
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SLIDE 33

Moments of random walks

The sth moment Wn(s) of the density pn: Wn(s) := ∞ xspn(x) dx =

  • [0,1]n
  • e2πix1 + . . . + e2πixn

s dx

DEF

n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1.273 2.000 3.395 6.000 10.87 20.00 37.25 3 1.575 3.000 6.452 15.00 36.71 93.00 241.5 4 1.799 4.000 10.12 28.00 82.65 256.0 822.3 5 2.008 5.000 14.29 45.00 152.3 545.0 2037. 6 2.194 6.000 18.91 66.00 248.8 996.0 4186. W2(1) = 4

π

W3(1) = 1.57459723755189 . . . = ?

Tools for special functions and special numbers Armin Straub 10 / 26
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SLIDE 34

Moments of random walks

The sth moment Wn(s) of the density pn: Wn(s) := ∞ xspn(x) dx =

  • [0,1]n
  • e2πix1 + . . . + e2πixn

s dx

DEF

n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1.273 2.000 3.395 6.000 10.87 20.00 37.25 3 1.575 3.000 6.452 15.00 36.71 93.00 241.5 4 1.799 4.000 10.12 28.00 82.65 256.0 822.3 5 2.008 5.000 14.29 45.00 152.3 545.0 2037. 6 2.194 6.000 18.91 66.00 248.8 996.0 4186. W2(1) = 4

π

W3(1) = 1.57459723755189 . . . = ? For instance, the sequence W3(2k) is 1, 3, 15, 93, 639, 4653, . . .

Tools for special functions and special numbers Armin Straub 10 / 26
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SLIDE 35

The integer sequence database

Tools for special functions and special numbers Armin Straub 11 / 26
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SLIDE 36

The integer sequence database

Tools for special functions and special numbers Armin Straub 11 / 26
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SLIDE 37

Advertisement

  • Based on the observation that

W3(2k) =

k

  • j=0

k j 22j j

  • ,

knowledge of modular forms allows us to deduce: W3(1) = 3 16 21/3 π4 Γ6 1 3

  • + 27

4 22/3 π4 Γ6 2 3

  • = 1.57459723755189 . . .

THM

Borwein- Nuyens- S-Wan, 2010

Tools for special functions and special numbers Armin Straub 12 / 26
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SLIDE 38

Modular forms

Modular forms are functions on the complex plane that are in-

  • rdinately symmetric. They satisfy so many internal symmetries

that their mere existence seem like accidents. But they do exist.

Barry Mazur (BBC Interview, “The Proof”, 1997)

Actions of γ = a b

c d

  • ∈ SL2(Z):
  • on τ ∈ H by

γ · τ = aτ + b cτ + d,

  • on f : H → C by

(f|kγ)(τ) = (cτ + d)−kf(γ · τ).

DEF

SL2(Z) is generated by T = ( 1 1

0 1 ) and S =

0 −1

1 0

  • .

T · τ = τ + 1, S · τ = −1 τ

EG

Tools for special functions and special numbers Armin Straub 13 / 26
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SLIDE 39

Modular forms

There’s a saying attributed to Eichler that there are five funda- mental operations of arithmetic: addition, subtraction, multipli- cation, division, and modular forms.

Andrew Wiles (BBC Interview, “The Proof”, 1997)

A function f : H → C is a modular form of weight k if

  • f|kγ = f for all γ ∈ SL2(Z),
  • f is holomorphic (including at the cusp i∞).

DEF

f(τ + 1) = f(τ), τ −kf(−1/τ) = f(τ).

EG

  • Similarly, MFs w.r.t. finite-index Γ SL2(Z)
  • Spaces of MFs finite dimensional, Hecke operators, . . .
Tools for special functions and special numbers Armin Straub 14 / 26
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SLIDE 40

Modular forms: a prototypical example

  • The Dedekind eta function

(q = e2πiτ)

η(τ) = q1/24

n1

(1 − qn) transforms as η(τ + 1) = eπi/12η(τ), η(−1/τ) = √ −iτη(τ). ∆(τ) = (2π)12η(τ)24 is a modular form of weight 12.

EG

Tools for special functions and special numbers Armin Straub 15 / 26
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SLIDE 41

Modularity of the three-step moments

  • The even moments

1, 3, 15, 93, 639, . . .

W3(2k) =

k

  • j=0

k j 22j j

  • have the modular parametrization

η6(2τ)η(3τ) η3(τ)η2(6τ)

modular form

=

  • k0

W3(2k) η(τ)η2(6τ) η2(2τ)η(3τ) 4k

modular function

.

Tools for special functions and special numbers Armin Straub 16 / 26
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SLIDE 42

Modularity of the three-step moments

  • The even moments

1, 3, 15, 93, 639, . . .

W3(2k) =

k

  • j=0

k j 22j j

  • have the modular parametrization

η6(2τ)η(3τ) η3(τ)η2(6τ)

modular form

=

  • k0

W3(2k) η(τ)η2(6τ) η2(2τ)η(3τ) 4k

modular function

. The values of modular functions at quadratic irrationalities τ ∈ Q( √ −d) are algebraic!

PSLQ predicts that for the above modular function x(τ), the value x(i/3) ≈ 0.52754 has minimal polynomial 1 − 6x4 − 24x6 − 3x8.

EG

Tools for special functions and special numbers Armin Straub 16 / 26
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SLIDE 43

Integer relation algorithms

  • How does the ISC recognize numbers?
Tools for special functions and special numbers Armin Straub 17 / 26
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SLIDE 44

Integer relation algorithms

  • How does the ISC recognize numbers?
  • PSLQ takes numbers x = (x1, x2, . . . , xn) and tries to find integers

m = (m1, m2, . . . , mn), not all zero, such that x · m = m1x1 + . . . + mnxn = 0. The vector m is called an integer relation for x.

In case that no relation is found, PSLQ provides a lower bound for the norm of any potential integer relation.

Tools for special functions and special numbers Armin Straub 17 / 26
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SLIDE 45

Integer relation algorithms

  • How does the ISC recognize numbers?
  • PSLQ takes numbers x = (x1, x2, . . . , xn) and tries to find integers

m = (m1, m2, . . . , mn), not all zero, such that x · m = m1x1 + . . . + mnxn = 0. The vector m is called an integer relation for x.

In case that no relation is found, PSLQ provides a lower bound for the norm of any potential integer relation.

Is x = 0.31783724519578224473 . . . algebraic?

In[1]:= PSLQ[{1, x, x2, x3, x4}] Out[1]= {1, 0, −10, 0, 1}

That is, x likely has minimal polynomial 1 − 10x2 + x4.

Therefore, x = √ 3 − √ 2. EG

Tools for special functions and special numbers Armin Straub 17 / 26
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SLIDE 46

Using PSLQ to find functional relations

  • A well-known fact: sin((2n − 1)x) is a linear combination of

sin(x), sin3(x), . . . , sin2n−1(x)

Tools for special functions and special numbers Armin Straub 18 / 26
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SLIDE 47

Using PSLQ to find functional relations

  • A well-known fact: sin((2n − 1)x) is a linear combination of

sin(x), sin3(x), . . . , sin2n−1(x)

In[1]:= With[{x = 1}, PSLQ[

N[{Sin[5x], Sin[x], Sin[x]3, Sin[x]5}, 20]]]

Out[1]= {−1, 5, −20, 16}

In other words, sin(5x) = 5 sin(x) − 20 sin3(x) + 16 sin5(x).

EG

Tools for special functions and special numbers Armin Straub 18 / 26
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SLIDE 48

Cylindrical Algebraic Decomposition

Arithmetic mean geometric mean

In[1]:= CylindricalDecomposition[(a+b)/2 Sqrt[ab], {a, b}] Out[1]= a 0 ∧ b 0

EG

Tools for special functions and special numbers Armin Straub 19 / 26
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SLIDE 49

Cylindrical Algebraic Decomposition

Arithmetic mean geometric mean

In[1]:= CylindricalDecomposition[(a+b)/2 Sqrt[ab], {a, b}] Out[1]= a 0 ∧ b 0

EG

If the sum of four positive numbers is 4c and the sum of their squares is 8c2, then none of the numbers can exceed (1 + √ 3)c.

EG

Tools for special functions and special numbers Armin Straub 19 / 26
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SLIDE 50

Cylindrical Algebraic Decomposition

Arithmetic mean geometric mean

In[1]:= CylindricalDecomposition[(a+b)/2 Sqrt[ab], {a, b}] Out[1]= a 0 ∧ b 0

EG

If the sum of four positive numbers is 4c and the sum of their squares is 8c2, then none of the numbers can exceed (1 + √ 3)c.

In[2]:= CylindricalDecomposition[Exists[{a2, a3, a4},

a1 a2 a3 a4 > 0 ∧ a1 + a2 + a3 + a4 == 4c ∧ a2

1 + a2 2 + a2 3 + a2 4 == 8c2], {c, a1}]

Out[2]= c > 0 ∧ 2c < a1 (1 +

√ 3)c

EG

Tools for special functions and special numbers Armin Straub 19 / 26
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SLIDE 51

Positivity of rational functions

  • A rational function

F(x1, . . . , xd) =

  • n1,...,nd0

an1,...,ndxn1

1 · · · xnd d

is positive if an1,...,nd > 0 for all indices.

Tools for special functions and special numbers Armin Straub 20 / 26
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SLIDE 52

Positivity of rational functions

  • A rational function

F(x1, . . . , xd) =

  • n1,...,nd0

an1,...,ndxn1

1 · · · xnd d

is positive if an1,...,nd > 0 for all indices. An obviously positive rational function: 1 1 − x − y + xy = 1 (1 − x)(1 − y)

EG

Tools for special functions and special numbers Armin Straub 20 / 26
slide-53
SLIDE 53

Positivity of rational functions

  • A rational function

F(x1, . . . , xd) =

  • n1,...,nd0

an1,...,ndxn1

1 · · · xnd d

is positive if an1,...,nd > 0 for all indices. An obviously positive rational function: 1 1 − x − y + xy = 1 (1 − x)(1 − y)

EG

1 1 − x − y + λxy is positive if and only if λ 1.

THM

Tools for special functions and special numbers Armin Straub 20 / 26
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SLIDE 54

Positivity of rational functions

  • A rational function

F(x1, . . . , xd) =

  • n1,...,nd0

an1,...,ndxn1

1 · · · xnd d

is positive if an1,...,nd > 0 for all indices. An obviously positive rational function: 1 1 − x − y + xy = 1 (1 − x)(1 − y)

EG

The following rational function is positive: 1 1 − (x + y + z + w) + 2

3(xy + xz + xw + yz + yw + zw)

This is a rescaled version of 1/e2(1 − x, 1 − y, 1 − z, 1 − w). CONJ

Askey– Gasper 1972

Tools for special functions and special numbers Armin Straub 20 / 26
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SLIDE 55

Positivity of rational functions

The Askey–Gasper rational function A(x, y, z) and the Szeg˝

  • rational function S(x, y, z) are positive.

A(x, y, z) = 1 1 − (x + y + z) + 4xyz S(x, y, z) = 1 1 − (x + y + z) + 3

4(xy + yz + zx)

EG

There is a positivity-preserving operator T such that T ·A = S.

THM

S 2007

Tools for special functions and special numbers Armin Straub 21 / 26
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SLIDE 56

Positivity of rational functions

The Askey–Gasper rational function A(x, y, z) and the Szeg˝

  • rational function S(x, y, z) are positive.

A(x, y, z) = 1 1 − (x + y + z) + 4xyz S(x, y, z) = 1 1 − (x + y + z) + 3

4(xy + yz + zx)

EG

There is a positivity-preserving operator T such that T ·A = S.

THM

S 2007

The diagonal Taylor terms of A are given by [xnynzn]A(x, y, z) =

n

  • k=0

n k 3 .

By WZ, both sides satisfy the recurrence (n + 1)2an+1 = (7n2 + 7n + 2)an + 8n2an−1. EG

Tools for special functions and special numbers Armin Straub 21 / 26
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SLIDE 57

Positivity of rational functions

The diagonal Taylor terms of S(2x, 2y, 2z), namely 1, 12, 198, 3720, 75690, 1626912, . . . , satisfy the recurrence

2(n + 1)2sn+1 = 3

  • 27n2 + 27n + 8
  • sn − 81(3n − 1)(3n + 1)sn−1.

EG

Tools for special functions and special numbers Armin Straub 22 / 26
slide-58
SLIDE 58

Positivity of rational functions

The diagonal Taylor terms of S(2x, 2y, 2z), namely 1, 12, 198, 3720, 75690, 1626912, . . . , satisfy the recurrence

2(n + 1)2sn+1 = 3

  • 27n2 + 27n + 8
  • sn − 81(3n − 1)(3n + 1)sn−1.

EG

To prove positivity from the recurrence, apply CAD to the formula (∀n, A, B) n 1, A 0, B λA = ⇒ C λB where 2(n + 1)2C = 3(27n2 + 27n + 8)B − 81(3n − 1)(3n + 1)A.

Tools for special functions and special numbers Armin Straub 22 / 26
slide-59
SLIDE 59

Positivity of rational functions

The diagonal Taylor terms of S(2x, 2y, 2z), namely 1, 12, 198, 3720, 75690, 1626912, . . . , satisfy the recurrence

2(n + 1)2sn+1 = 3

  • 27n2 + 27n + 8
  • sn − 81(3n − 1)(3n + 1)sn−1.

EG

To prove positivity from the recurrence, apply CAD to the formula (∀n, A, B) n 1, A 0, B λA = ⇒ C λB where 2(n + 1)2C = 3(27n2 + 27n + 8)B − 81(3n − 1)(3n + 1)A.

In[1]:= With[{C = . . .},

CylindricalDecomposition[ForAll[{n, A, B}, n 1 ∧ B λA ∧ A 0, C λB], {λ}]]

Out[1]= 27/2 λ 3/8(31 +

√ 385)

Tools for special functions and special numbers Armin Straub 22 / 26
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SLIDE 60

Positivity of rational functions

  • The Kauers–Zeilberger rational function

1 1 − (x + y + z + w) + 2(yzw + xzw + xyw + xyz) + 4xyzw is conjectured to be positive.

  • Its positivity implies the positivity of the Askey–Gasper function

1 1 − (x + y + z + w) + 2

3(xy + xz + xw + yz + yw + zw).

The Kauers–Zeilberger function has diagonal coefficients dn =

n

  • k=0

n k 22k n 2 .

PROP

S-Zudilin 2013

Tools for special functions and special numbers Armin Straub 23 / 26
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SLIDE 61

Positivity of rational functions

Under what condition(s) is the positivity of a rational function h(x1, . . . , xd) = 1 d

k=0 ckek(x1, . . . , xd)

implied by the positivity of its diagonal?

Q

  • Is the positivity of h(x1, . . . , xd−1, 0) a sufficient condition?

1 1+x+y has positive diagonal coefficients but is not positive.

EG

Tools for special functions and special numbers Armin Straub 24 / 26
slide-62
SLIDE 62

Positivity of rational functions

Under what condition(s) is the positivity of a rational function h(x1, . . . , xd) = 1 d

k=0 ckek(x1, . . . , xd)

implied by the positivity of its diagonal?

Q

  • Is the positivity of h(x1, . . . , xd−1, 0) a sufficient condition?

1 1+x+y has positive diagonal coefficients but is not positive.

EG

h(x, y) = 1 1 + c1(x + y) + c2xy is positive iff h(x, 0) and the diagonal of h(x, y) are positive.

THM

S-Zudilin 2013

Tools for special functions and special numbers Armin Straub 24 / 26
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SLIDE 63

Drunken birds

Tools for special functions and special numbers Armin Straub 25 / 26
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SLIDE 64

Drunken birds

A drunk man will find his way home, but a drunk bird may get lost forever.

Shizuo Kakutani, 1911–2004 ”

Tools for special functions and special numbers Armin Straub 25 / 26
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SLIDE 65

THANK YOU!

Slides for this talk will be available from my website: http://arminstraub.com/talks

  • A. Straub, W. Zudilin

Positivity of rational functions and their diagonals Preprint, 2013

  • J. Borwein, A. Straub, J. Wan, W. Zudilin (appendix by D. Zagier)

Densities of short uniform random walks Canadian Journal of Mathematics, Vol. 64, Nr. 5, 2012, p. 961-990

  • J. Borwein, D. Nuyens, A. Straub, J. Wan

Some arithmetic properties of short random walk integrals The Ramanujan Journal, Vol. 26, Nr. 1, 2011, p. 109-132

  • A. Straub

Positivity of Szeg¨

  • ’s rational function

Advances in Applied Mathematics, Vol. 41, Issue 2, Aug 2008, p. 255-264

Tools for special functions and special numbers Armin Straub 26 / 26