Use of Computer Technology for Insight and Proof

Use of Computer Technology for Insight and Proof

Strengths, Weaknesses and Practical Strategies(i) The role of CAS in analysis(ii) Four practical mechanisms(iii) Applications

Kent Pearce

Texas Tech University

Presentation: Fresno, California, 24 September 2010

Question

Consider

2

(

)

)

(

cosxg

f

x

x

e

x

Question

Consider

2

(

)

)

(

cosxg

f

x

x

e

x

Question

Consider

2 cos( )xh e x

Question

Consider

2 cos( )xh e x

Question

Given a function f on an interval [a, b], what does it take to show that f is non-negative on [a, b]?

Transcendental Functions

Consider

( ) cos( )g x x


Consider

( ) cos( )g x x

cos(0)1

cos(0.95)

0.5816830895

cos(0.95 + 2000000000*π)

0.5816830895

cos(0.95 + 2000000000.*π)

cos(0.95 + 2000000000.*π)


Blackbox Approximations

Transcendental / Special Functions

Polynomials/Rational Functions

CAS Calculations Integer Arithmetic

Rational Values vs Irrational Values

Floating Point Calculation

Question


(P)Lots of Dots

(P)Lots of Dots

(P)Lots of Dots

(P)Lots of Dots

1( )

2 1y f x

x

(P)Lots of Dots

1( )

2 1y f x

x

Question


Proof by Picture Maple, Mathematica, Matlab, Mathcad,

Excel, Graphing Calculators, Java Applets

Practical Methods

A. Sturm Sequence Arguments B. Linearity / Monotonicity Arguments C. Special Function Estimates D. Grid Estimates

Applications

"On a Coefficient Conjecture of Brannan," Complex Variables. Theory and Application. An International Journal 33 (1997) 51_61, with Roger W. Barnard and William Wheeler.

"A Sharp Bound on the Schwarzian Derivatives of Hyperbolically Convex Functions," Proceeding of the London Mathematical Society 93 (2006), 395_417, with Roger W. Barnard, Leah Cole and G. Brock Williams.

"The Verification of an Inequality," Proceedings of the International Conference on Geometric Function Theory, Special Functions and Applications (ICGFT) (accepted) with Roger W. Barnard.

"Iceberg-Type Problems in Two Dimensions," with Roger.W. Barnard and Alex.Yu. Solynin

Practical Methods


Iceberg-Type Problems


Dual Problem for Class Let and let

For let

and For 0 < h < 4, let

Find

0

( ) max area( )f hf

A h E H

{ | Re( ) }.hH z z h

0

0 1

1{ ( ) : is analytic,f z a a z f

z

univalent on }.D f \ ( )fE f D

0 { | 0 }.ff E

{ : 0 | | 1}z z D=


Extremal Configuration Symmetrization Polarization Variational Methods Boundary Conditions



We obtained explicit formulas for A = A(r)

and h = h(r). To show that we could write

A = A(h), we needed to show that h = h(r) was monotone.

Practical Methods


Sturm Sequence Arguments

General theorem for counting the number of distinct roots of a polynomial f on an interval (a, b)

N. Jacobson, Basic Algebra. Vol. I., pp. 311-315,W. H. Freeman and Co., New York, 1974.

H. Weber, Lehrbuch der Algebra, Vol. I., pp. 301-313, Friedrich Vieweg und Sohn, Braunschweig, 1898


Sturm’s Theorem. Let f be a non-constant polynomial with rational coefficients and let a < b be rational numbers. Let

be the standard sequence for f . Suppose that

Then, the number of distinct roots of f on (a, b) is where denotes the number of sign changes of

0 1{ , , , }f sS f f f

( ) 0, ( ) 0.f a f b a bV V cV

0 1( ) { ( ), ( ), , ( )}f sS c f c f c f c


Sturm’s Theorem (Generalization). Let f be a non-constant polynomial with rational coefficients and let a < b be rational numbers. Let

be the standard sequence for f . Then, the number of distinct roots of f on (a, b] is where denotes the number of sign changes of

0 1{ , , , }f sS f f f Suppose that ( ) 0, ( ) 0.f a f b

a bV V

cV

0 1( ) { ( ), ( ), , ( )}f sS c f c f c f c


For a given f, the standard sequence is constructed as:

fS

0

1

2 0 1 1 2

3 1 2 2 3

:

:

f f

f f

f f f q f

f f f q f


Polynomial


Polynomial

Linearity / Monotonicity

Consider

where

Let

Then,

0 1( , ) ( ) ( )f x Z c x c x Z

Z

0 1

0 1

( ) ( , ) ,

( ) ( , )Z

Z

f x f x Z c c

f x f x Z c c

( , ) ( , )min { ( ), ( )} ( , ) max{ ( ), ( )}x a b x a b

f x f x f x Z f x f x


We obtained explicit formulas for A = A(r)

and h = h(r). To show that we could write

A = A(h), we needed to show that h = h(r) was monotone.


From the construction we explicitly found

where



where


It remained to show

was non-negative. In a separate lemma, we showed 0 < Q < 1. Hence, using the linearity ofQ in g, we needed to show

were non-negative

0 1 0 1( ) ( ) ( )g g r c c P d d P Q

0 0 1 0 1

1 0 1 0 1

( ) ( ) 0

( ) ( ) 1

g c c P d d P

g c c P d d P


In a second lemma, we showed s < P < t where

Let

Each is a polynomial with rational coefficients for which a Sturm sequence argument show that it is non-negative.

0, 0 0, 0 1, 1 1, 1, , , .s t s tP s P t P s P tg g g g g g g g

0, 0, 1, 1,, , ,s t s tg g g g

Practical Methods


Notation & Definitions

{ : | | 1}z z D


{ : | | 1}z z D

2

2 | |( ) | |

1 | |

dzz dz

z

hyperbolic metric


Hyberbolic Geodesics

{ : | | 1}z z D



Hyberbolically Convex Set

{ : | | 1}z z D




Hyberbolically Convex Function

{ : | | 1}z z D




Hyberbolically Convex Function

Hyberbolic Polygono Proper Sides

{ : | | 1}z z D

Examples

2 2

2( )

(1 ) (1 ) 4

zk z

z z z

k

Examples

12 4 2

0

( ) tan (1 2 cos2 )

2where , 0 2(cos )

z

f z d

K

f

Schwarz Norm

For let

and

where

( )f A D

21

2f

f fS

f f

2|| || sup{ ( ) | ( ) |: }f fS z S z z D D D

2

1( )

1 | |z

z

D

|| ||fS D

Extremal Problems for

Euclidean Convexity Nehari (1976):

( ) convex || || 2ff S DD

|| ||fS D



Spherical Convexity Mejía, Pommerenke (2000):



|| ||fS D



Spherical Convexity Mejía, Pommerenke (2000):

Hyperbolic Convexity Mejía, Pommerenke Conjecture (2000):



( ) convex || || 2.3836ff S DD

|| ||fS D

Verification of M/P Conjecture

"A Sharp Bound on the Schwarzian Derivatives of Hyperbolically Convex Functions," Proceeding of the London Mathematical Society 93 (2006), 395_417, with Roger W. Barnard, Leah Cole and G. Brock Williams.

"The Verification of an Inequality," Proceedings of the International Conference on Geometric Function Theory, Special Functions and Applications (ICGFT) (accepted) with Roger W. Barnard.

Special Function Estimates

Parameter / 2

where cos( )

yK y


Upper bound


Upper bound

Partial Sums


Verification

where/ 2

cos 2 , , 1 1(cos )

c xK

Verification

Straightforward to show that

In make a change of variable

3 0c

0p

q

22 1c y mq

Verification

Obtain a lower bound for by estimating via an upper bound

Sturm sequence argument showsis non-negative

3 0c

mq

8

*m m p

q q

Grid Estimates

Grid Estimates

Given A) grid step size h B) global bound M for maximum of

Theorem Let f be defined on [a, b]. Let

Let and suppose that N is choosen so that . Let L be the lattice . Let

If then f is non-negative on [a, b].

| ( ) |f x

[ , ]max | ( ) | .x a b

M f x

0 ( ) /h b a N

{ : 0 }L a jh j N min ( )x L

m f x

,2

m M

Grid Estimates

Maximum descent argument

Grid Estimates

Two-Dimensional Version

Grid Estimates

Maximum descent argument

Verification

where/ 2

cos 2 , , 1 1(cos )

c xK

Verification

The problem was that the coefficient was not globally positive, specifically, it was not positive for

We showed that by showing that

where

0 < t < 1/4.

2 ( , )c x

2c

4 ( ) 0p t

( ) 0q t

23 2 1( ) ( , ) ( , ) ( , )q t c x t c x t c t

041 , .5 2x

Verification

Used Lemma 3.3 to show that the endpoints

and are non-negative. We partition the parameter space into subregions:

2c

*

0 0(0) ( , )q e y w *

01( )4q

Verification

Application of Lemma 3.3 to

After another change of variable, we needed to show that where

for 0 < w < 1, 0 < m < 1

2c

*

01( )4q

0r

Verification 2c

Verification

Quarter Square [0,1/2]x[0,1/2]

Grid 50 x 50

2c

max{ , } 35

21, 35w m

w m

M M M

M M

0.350M ( , )

min ( , ) 0.400j k

j kw m L

m r w m

[0,1] [0,1] [0,1/ 2] [0,1/ 2] [0,1/ 2] [1/ 2,1]

[1/ 2,1] [0,1/ 2] [1/ 2,1] [1/ 2,1]

Question


Conclusions

There are “proof by picture” hazards There is a role for CAS in analysis

CAS numerical computations are rational number calculations

CAS “special function” numerical calculations are inherently finite approximations

There are various useful, practical strategies for rigorously establishing analytic inequalities

Documents

Use of Computer Technology for Insight and Proof