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
Transcendental Functions
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)
Transcendental Functions
Blackbox Approximations
Transcendental / Special Functions
Polynomials/Rational Functions
CAS Calculations Integer Arithmetic
Rational Values vs Irrational Values
Floating Point Calculation
Question
Given a function f on an interval [a, b], what does it take to show that f is non-negative on [a, b]?
(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
Given a function f on an interval [a, b], what does it take to show that f is non-negative on [a, b]?
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
A. Sturm Sequence Arguments B. Linearity / Monotonicity Arguments C. Special Function Estimates D. Grid Estimates
Iceberg-Type Problems
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=
Iceberg-Type Problems
Extremal Configuration Symmetrization Polarization Variational Methods Boundary Conditions
Iceberg-Type Problems
Iceberg-Type Problems
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
A. Sturm Sequence Arguments B. Linearity / Monotonicity Arguments C. Special Function Estimates D. Grid Estimates
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 Sequence Arguments
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 Sequence Arguments
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
Sturm Sequence Arguments
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
Sturm Sequence Arguments
Polynomial
Sturm Sequence Arguments
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
Iceberg-Type Problems
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.
Iceberg-Type Problems
From the construction we explicitly found
where
Iceberg-Type Problems
Iceberg-Type Problems
where
Iceberg-Type Problems
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
Iceberg-Type Problems
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
A. Sturm Sequence Arguments B. Linearity / Monotonicity Arguments C. Special Function Estimates D. Grid Estimates
Notation & Definitions
{ : | | 1}z z D
Notation & Definitions
{ : | | 1}z z D
2
2 | |( ) | |
1 | |
dzz dz
z
hyperbolic metric
Notation & Definitions
Hyberbolic Geodesics
{ : | | 1}z z D
Notation & Definitions
Hyberbolic Geodesics
Hyberbolically Convex Set
{ : | | 1}z z D
Notation & Definitions
Hyberbolic Geodesics
Hyberbolically Convex Set
Hyberbolically Convex Function
{ : | | 1}z z D
Notation & Definitions
Hyberbolic Geodesics
Hyberbolically Convex Set
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
Extremal Problems for
Euclidean Convexity Nehari (1976):
Spherical Convexity Mejía, Pommerenke (2000):
( ) convex || || 2ff S DD
( ) convex || || 2ff S DD
|| ||fS D
Extremal Problems for
Euclidean Convexity Nehari (1976):
Spherical Convexity Mejía, Pommerenke (2000):
Hyperbolic Convexity Mejía, Pommerenke Conjecture (2000):
( ) convex || || 2ff S DD
( ) convex || || 2ff S DD
( ) 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
Special Function Estimates
Upper bound
Special Function Estimates
Upper bound
Partial Sums
Special Function Estimates
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
Given a function f on an interval [a, b], what does it take to show that f is non-negative on [a, b]?
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