Domain%20Theoretic%20Models%20of%20Geometry%20and%20Differential%20Calculus

1
Domain Theoretic Models of Geometry and
Differential Calculus
Abbas Edalat Imperial College http//www.doc.i
c.ac.uk/ae
Joint work with Andre Lieutier
Dassault Systemes
With contributions from Ali Khanban, Marko
Krznaric
ISDT 2001 Chengdu, China
2
Computational Model for Classical Spaces

• A research project since 1993
• Reconstruct basic mathematics!
• Embed classical spaces into the set of maximal
  elements of suitable domains

3
Computational Model for Classical Spaces

• Previous Applications
• Fractal Geometry
• New algorithms for computing fractal objects

• Measure Integration Theory
• The Generalized Riemann Integral

• Exact Real Arithmetic
• C-library based on linear fractional
  transformations for Computation of Elementary
  Functions

4
Part 1 A Domain-Theoretic Model of Geometry

• To develop a Computable model for Geometry and
  Solid Modelling, so that

• the model is mathematically sound, realistic

• the basic building blocks are computable

• it bridges theory and practice.

5
Why do we need a data type for solids?

• Answer To develop robust algorithms!
• Lack of a proper data type and use of real RAM
  in which comparison of real numbers is decidable
  give unreliable programs in practice!

6
The Intersection of two lines

• With floating point arithmetic, find the point P
  of the intersection L1 ? L2. Then
  min_dist(P, L1) gt 0, min_dist(P,
  L2) gt 0.

7
The Convex Hull Algorithm
With floating point we can get
8
The Convex Hull Algorithm
A, B C nearly collinear
With floating point we can get (i) AC,
or
9
The Convex Hull Algorithm
A, B C nearly collinear
With floating point we can get (i) AC,
or (ii) just AB, or
10
The Convex Hull Algorithm
A, B C nearly collinear
With floating point we can get (i) AC,
or (ii) just AB, or (iii) just BC, or
11
The Convex Hull Algorithm
A, B C nearly collinear
With floating point we can get (i) AC,
or (ii) just AB, or (iii) just BC, or (iv) none
of them.
The quest for robust algorithms is the most
fundamental unresolved problem in solid modelling
and computational geometry.
12
A Fundamental Problem in Topology and Geometry

• Subset A ? X topological space.Membership
  predicate ?A X ? tt, ff
• is continuous iff A is both open and closed.

• In particular, for A ? Rn, A ? ?, A ? Rn ?A
  Rn ? tt, ff is not continuous.
• Most engineering is done, however, in Rn.

13
Non-computability of the Membership Predicate

• There is discontinuity at the boundary of the
  set.

False
True
14
Non-computable Operations in Classical CG SM

• ?A Rn ? tt, ff not continuous means it is not
  computable, even for simple objects like
  A0,1n.
• x ? A is not decidable even for simple objects
  for A 0,?) ? R, we just have the
  undecidability of x ? 0.
• The Boolean operation ? is not continuous, hence
  noncomputable, wrt the natural notion of topology
  on subsets? C(Rn) ? C(Rn) ? C(Rn), where
  C(Rn) is compact subsets with the Hausdorff
  metric.

15
Intersection of two 3D cubes
16
Intersection of two 3D cubes
17
Intersection of two 3D cubes
18
This is Really Ironical!

• Topology and geometry have been developed to
  study continuous functions and transformations on
  spaces.
• The membership predicate and the binary operation
  for ? are the fundamental building blocks of
  topology and geometry.
• Yet, these fundamental functions are not
  continuous in classical topology and geometry.

19
Elements of a Computable Topology/Geometry

• The membership predicate ?A X ? tt, ff fails
  to be continuous on ?A, the boundary of A.
• For any open or closed set A, the predicate x ?
  ?A is non-observable, like x 0.

• ?A is now a continuous function.

20
Elements of a Computable Topology/Geometry

• Note that ?A?B iff int Aint B int
  Acint Bc, i.e. sets with the same
  interior and exterior have the same membership
  predicate.
• We now change our view In analogy with classical
  set theory where every set is completely
  determined by its membership predicate, we define
  a (partial) solid object to be given by any
  continuous map
• f X ? tt, ff ?
• Thenf 1tt is open its called the interior
  of the object. f 1ff is open its called
  the exterior of the object.

21
Partial Solid Objects

• We have now introduced partial solid objects,
  since X \ (f 1tt ? f
  1ff)
  may have non-empty interior.
• We partially order the continuous functionsf, g
  X ? tt, ff ? f ? g ? ?x ? X . f(x) ?
  g(x)
• f ? g ? f 1tt ? g 1tt f 1ff
  ? g 1ffTherefore, f ? g means g has more
  information about an idealized real solid object.

22
The Geometric (Solid) Domain of X

• The geometric (solid) domain S (X) of X is the
  poset (X ? tt, ff ?, ? )
• S(X) is isomorphic to the poset SO(X) of pairs of
  disjoint open sets (O1,O2) ordered componentwise
  by inclusion

23
Properties of the Geometric (Solid) Domain

• Theorem For a second countable locally compact
  Hausdorff space X (e.g. Rn), S(X) is bounded
  complete and ?continuous with (U1, U2) ltlt (V1,
  V2) iff the closures of U1 and U2 are compact
  subsets of V1 and V2 respectively.

24
Examples

• A x?R2 ? x 1 ? 1, 2represented in the
  model by(int A, int Ac) ( x ? x lt 1, R2 \
  A )is a classical (but non-regular) solid
  object.

25
Boolean operations and predicates

• Theorem All these operations are Scott
  continuous and preserve classical solid objects.

26
Subset Inclusion

• Subset inclusion is Scott continuous.

27
General Minkowski operator

• For smoothing out sharp corners of objects.
• SbRn (A, B) ? SRn Bc is bounded ?(Ø,Ø).
• All real solids are represented in SbRn.
• Define _?_ SRn ? SbRn ? SRn
  ((A,B) , (C,D)) ? (A ? C , (Bc ? Dc)c) where
  A ? C ac a? A, c? C
• Theorem _?_ is Scott continuous.

28
An effectively given solid domain

• The geometric domain SX can be given effective
  structure for any locally compact second
  countable Hausdorff space, e.g. Rn, Sn, Tn,
  0,1n.
• Consider XRn. The set of pairs of disjoint open
  rational polyhedra of the form K (L1 , L2) ,
  with L1 ? L2 ?, gives a basis for SX.
• Let Kn (p1 ( K n ) , p2 ( K n) ) be an
  enumeration of this basis.

• (A, B) is a computable partial solid object if
  there exists a total recursive function ßN?N
  such that

(A , B) ( ?n p1 ( K ß(n) ) , ?n p2 (
K ß(n) ) )
29
Computing a Solid Object

• In this model, a solid object is represented by
  its interior and exterior.

• The interior and the exterior
• are approximated by two
• nested sequence of rational polyhedra.

30
Computable Operations on the Solid Domain

• F (SX)n ? SX or F (SX)n ? tt,
  ff ?
• is computable if it takes computable sequences
  of partial solid objects to computable sequences.
• Theorem All the basic Boolean operations and
  predicates are computable wrt any effective
  enumeration of either the partial rational
  polyhedra or the partial dyadic voxel sets.

31
Quantative Measure of Convergence

• In our present model for computable solids,
  there is no quantitative measure for the
  convergence of the basis elements to a computable
  solid.
• Example The minimum distance from say a
  rational point to a computable solid is
  semi-computable, but not computable.
• We will enrich the notion of domain-theoretic
  computability to include a quantitative measure
  of convergence.

32
Hausdorff Computability

• We strengthen the notion of a computable solid by
  using the Hausdorff distance d between compact
  sets in Rn.
• d(C,D) min r C ? Dr D ? Cr
  where Dr x ? y ? D.
  x-y ? r

33
Hausdorff computability

• Two solid objects which have a small Hausdorff
  distance from each other are visually close.
• The Hausdorff distance gives a natural
  quantitative measure for approximation of solid
  objects.
• However, the intersection or union of two
  Hausdorff computable solid objects may fail to be
  Hausdorff computable.
• Examples of such failure are nontrivial to
  construct.

34
Boolean Intersection is not Hausdorff computable
is Hausdorff computable.
However Q?(0,1 ? 0) r,1 ? 0 ? R2is
not Hausdorffcomputable.
35
Lebesgue Computability

• (A , B) ? S k, kd is Lebesgue computable iff
  there exists an effective chain K ß(n) of basis
  elements with ß N?N a total recursive
  functions such that
• (A , B) ( ?n p1 ( K ß(n) ) , ?n
  p2 ( K ß(n) ) )
• µ(A) - µ(p1 ( K ß(n) ) ) lt 1/2 n µ(B)
  - µ(p2 ( K ß(n) ) ) lt 1/2 n
• A function is Lebesgue computable if it
  preserves Lebesgue computable sequences.
• Theorem Boolean operations are Lebesgue
  computable.

36
Hausdorff and Lebesgue computability

• Hausdorff computable ? Lebesgue
  computableComplement of a Cantor set with
  Lebesgue measure 1 r with r lim rn left
  computable but non-computable real.

• start with

• stage 1

• stage 2

• At stage n remove 2n open mid-intervals of length
  sn/2n.

37
Hausdorff and Lebesgue computability

• Lebesgue computable ? Hausdorff computable
• Let 0 lt rn ? Q with rn ? r, left
  computable, non-computable 0 lt r lt 1.

38
Hausdorff and Lebesgue Computable Objects

• Hausdorff computable ? Lebesgue computable
• Lebesgue computable ? Hausdorff computable
• Theorem A regular solid object is computable
  iff it is Hausdorff computable.
• However A computable regular solid object may
  not be Lebesgue computable.

39
Conclusion

• Our model satisfies
• A well-defined notion of computability
• Reflects the observable properties of geometric
  objects
• Is closed under basic operations
• Captures regular and non-regular sets
• Supports a methodology for designing robust
  algorithms

40
Part 2 Data-types for Computational Geometry
and Systems of Linear Equations

• The Convex Hull

• Voronoi Diagram or the Post Office problem

• The Partial Circle through three partial points

• Partial Lines and Projective Geometry
  
  the intersection of hyper-planes,
  the solution of systems of
  linear equations

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The Outer Convex Hull Algorithm
42
The Inner Convex Hull Algorithm
43
The Convex Hull Algorithm
44
The Convex Hull Algorithm
45
The Convex Hull map

• Let Hm (R2)m ? C(R2) be the classical convex
  Hull map, with C(R2) the set of compact subsets
  of R2, with the Hausdorff metric.
• Let (IR2, ? ) be the domain of rectangles in R2.

• For x(T1,T2,,Tm)?(IR2)m, define
• Cm (IR2)m ? SR2,Cm(x)(Im(x),Em(x))
  with
• Em(x)?Hm(y) y?(R2)m, yi?Ti, 1 ? I ? m
• Im(x) ?Hm(y) y?(R2)m, yi?Ti, 1 ? I ? m

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The Convex Hull is Computable!

• Proposition Em(x)(H4m((Ti1,Ti2,Ti3,Ti4))1?i?m)c
  Im(x)Int(?Hm((Tin))1?i?m)
  n1,2,3,4).
• Theorem The map Cm (IR2)m ? SR2 is Scott
  continuous, Hausdorff and Lebesgue computable.
• Complexity
• Em(x) is O(m log m).
• Im(x) is also O(m log m).
• We have precisely the complexity of the
  classical convex hull algorithm in R2 and R3.

47
Voronoi Diagrams

• We are given a finite number of points in the
  plane.

• Divide the plane into components closest to
  these points.

• This problem is equivalent to the Delaunay
  triangulation of the points
• (1) Triangulate the set of given points so
  that the interior of the circumference circles do
  not contain any of the given points.

(2) Draw the perpendicular bisectors of
the edges of the triangles.
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Voronoi Diagram Partial Circles

• The centre of the circle through the three
  vertices of a triangle is the intersection of the
  perpendicular bisectors of the three edges of the
  triangle.

• The partial circle of three partial points in the
  plane is obtained by considering the Partial
  Perpendicular Bisector of two partial points in
  the plane.

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Partial Perpendicular Bisector of Two Partial
Points
50
PPBs for Three Partial Points
51
Partial Circles
Each partial circle is defined by its interior
and exterior. The exterior (interior) consists of
all those points of the plane which are outside
(inside) all circles passing through any three
points in the three rectangles.
The exterior is the union of the interiors of the
three red circles.
The Interior is the intersection of the interiors
of the three blue circles.
52
Partial Circles
With more exact partial points, the boundaries of
the interior and exterior of the partial circle
get closer to each other.
53
Partial Circles

• The limit of the area between the interior and
  exterior of the partial circle, and the Hausdorff
  distance between their boundaries, is zero.

• We get a Scott continuous map C (IR2)3?SR2
• We obtain a robust Voronoi algorithm which is m
  log m on average.

54
Lines and hyperplanes

• One can define a Scott continuous function which,
  given any two partial points, gives the partial
  line through them as a partial solid object

L (IR2)2 ? SR2 (x,y) ? (?,A)

• However, if x and y intersect then A ?, i.e. no
  information is provided.
• Moreover, the intersection of two partial lines
  will not be a partial point as one would desire.
• We would like to model a partial line as a set of
  lines rather than a set of points.

55
Projective Lines and hyperplanes

• By going to the projective real space we can take
  lines and hyper-planes as primitives on an equal
  footing with points.
• By going further to a domain model of the
  projective real space we get a computational
  model for projective geometry and also a
  data-type for systems of linear equation.
• The d-dimensional real projective space Pd is
• 1. the set of one dimensional linear
  subspaces of Rd1, or equivalently,
• 2. the quotient of Rd1-0 under the
  colinearity relation X
  Y if ??. X ?Y, or equivalently,
• 3. the quotient of the sphere Sd by
  identifying anti-podal points X Y iff XY or X
  -Y.
• The one dimensional linear subspaces of Rd1
  intersect the hyperplane Xd1 1 at an affine
  subspace or, when the intersection is empty, at a
  point at infinity.

56
The Real Projective Space

• P1

X
X
o
-8
8
-X
57
The Orthogonality Relation Projective Duality

• The orthogonality of vectors in Rd1 induces an
  orthogonality relation on Pd we write p ? q
• O(p) q ?Pd p ? q is a d-1 dimensional
  projective subspace of Pd .
• Conversely, for any d-1 dimensional projective
  subspace S, there exists p ?Pd with SO(p).
• Therefore the set of points and the set of d-1
  dimensional projective subspace of Pd are in
  one-to-one correspondence with each other.
• We have the projective duality
• q ? O(p) iff p ? q iff p ? O(q)

58

• The set ? O(pi) i1,2,,d is the
  intersection of d hyper-planes It
  can be a single point or a hyper-plane, but, in
  contrast to classical geometry, it cannot be the
  empty.

• But O(p) is, by duality, also the set of
  hyper-planes containing p

• Therefore, ? O(pi) i1,2,,d is, by
  duality, also the non-empty set of hyper-planes
  containing all pi (i1,2,,d ).

59
A Domain-theoretic Model for Projective Geometry

• Proposition. For any comact subset A? Pd the
  set
  O(A) q ?Pd ?p ? A. p ? q ? Pd is
  compact.
• Let UPd be the upper space of Pd, i.e.the set of
  non-empty subsets of Pd ordered by reverse
  inclusion.
• Proposition. The map O UPd ?Upd is Scott
  continuous.
• Theorem. (Data Type for Linear Systems) L
  (UPd)d ? UPd defined by L(A1,A2,.,Ad)
  ? O(Ak) k1,,d is Scott
  continuous and computable.
• L computes the intersection of d hyper-planes.
• L also computes, by duality, the hyper-plane
  through d points.
• Remark. The corresponding classical operation is
  not continuous
• The intersection of two lines in R2 can be a
  single point, a line or the empty set.
• The intersection of two projective lines P2 can
  be a projective point or a projective line.

60
Part 3 A Domain-Theoretic Model for
Differential Calculus

• Interval derivative of a function
• Fundamental Theorem of Calculus
• Domain of C1 functions
• Higher interval derivatives Domains for Ck and
  C8
• Picards Theorem
  A data-type for differential
  equations

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Non-smooth Mathematics
Smooth Mathematics

• Geometry
• Differential Topology
• Manifolds
• Dynamical Systems
• Mathematical Physics
• .
• .
• All based on differential
  calculus

• Set Theory
• Logic
• Algebra
• Point-set Topology
• Graph Theory
• Model Theory
• .
• .

62
The Domain of Intervals

• (I0,1, ?) is a bounded complete dcpo ?i?I ai
  ?i?I ai
• a b ? ao ? b
• (I0,1, ?) is ?-continuousBasis q, q q lt
  q q, q ? Q
• Scott topology has basis ?a b ? I0,1 a
  b, a ? I0,1
• x ? x 0,1 ? I0,1Topological embedding

?
x
x
0,1
I 0,1
63
Continuous Functions

• f 0,1 ? R, f ? Co0,1,has continuous
  extensionIf I0,1 ? IR a ? f(a)
  f(x) x ? a
• Scott continuous maps I0,1 ? IR withf ? g ?
  ?x ? I0,1 . f(x) ? g(x)is another bounded
  complete ?-continuous dcpo.
• ? Co0,1 ? (I0,1 ? IR), with f ? Ifis a
  topological embedding.
• Image (?) ? Max(I0,1 ? IR) a proper subset.

64
Step Functions

• We will write a?b as (a, b) on this page.
• Finite lubs of consistent step functions (simple
  functions) ?1?i?n(ai,bi), with ai, bi rational
  intervals, give a basis for I0,1 ? IR
• The finite family A ? I0,1 ? IRis consistent
  iff?B ? A . Con(?1B) ? Con?(?2B)

65
Step Functions-An Example
R
b3
a1
a3
b1
b2
a2
0
1
66
Refining the Step Functions
R
b3
a1
a3
b1
b2
a2
0
1
67
Operations in Interval Arithmetic
68
Interval Derivative
iff ?x1,x2 a. b(x1 x2) ? If (x1) If
(x2)
iff ?x1,x2 a. ?y1,y2 ? IR, x1?y1 ? If
x2?y2 ? If
b(x1 x2) ? y1 y2
69
Interval Derivative
y1
If(x2)
y2
If(x1)
0
1
x2
x1
a
70
Definition of Interval Derivative

• f I0,1 ? IR has an interval derivativeb ? IR
  in a ? I0,1 if ?x1, x2 a, ? y1,y2 ? IR x1?y1,
  x2?y2 ? f b(x1 x2) ? y1 y2.
• The collection of all such f is called the tie of
  a with b, denoted ?(a,b).

71
Proposition.

• Note that
• f ? g ? (f ? ?(a,b) ? g ? ?(a,b)) i.e. ?(a,b)
  ?upper I0,1 ? IR
• If f g c, c ? R, thenf ? ?(a,b) ? g ?
  ?(a,b)

72
Proposition.

• ?(a1,b1) ? ?(a2,b2) iff a2 ? a1 b1 ? b2
• ?ni1 ?(ai,bi) ? ? iff ? ai?bi?ni1 is
  consistent.
• ?i?I ?(ai,bi) ? ? iff ?J?f I . ?i?J ?(ai,bi) ?
  ?(non trivial)
• In fact ?(a,b) behaves exactly like a?b.

73
The Interval Derivative
74
Examples
75
The Interval Derivative Operator

• (I0,1 ? IR) ? (I0,1 ? IR)is
  monotone but not continuous. Note that the
  classical operator is not continuous either.
• (a?b) ?x . ?
• is not linear! For f x ? x I0,1
  ? IR g x ?
  x I0,1 ? IR
• (fg) ? ?x . (1 1) ?x
  . 0

76
The Differential and Integral Operators

• We construct a domain for C1 functions.
• Let L1 ? P(I0,1 ? IR), ( P the power set
  functor) with x ? L1 iff x
  is the non-empty intersection of a family of
  ties x ?i?I ?(ai,bi) ? ?
  
  i.e. x is the collection of functions
  satisfying a given set of differential
  properties.
• Theorem. (L1, ?) is a dcpo and the differential
  operator
• D (L1, ?) ? (I0,1 ? IR)
  x ? ??(a,b)?x a?bis an isomorphism
  with the integral operator as its inverse
• ? (I0,1 ? IR) ? (L1, ?)
• f ? ?a?b?f ?(a,b)

77
Fundamental Theorem of Calculus

• For f, g ? C10,1, define fg if f g c, for
  some c ? R.
• There is a unique map which makes the following
  two diagrams commute

78
Fundamental Theorem of Calculus
79
The Domain of C1 Functions with C1 Norm

• Define the consistency relationCon ? (I0,1 ?
  IR) ? (I0,1 ? IR) with(f,g) ? Con if
  (?f) ? (? g) ? ?
• (?i ai?bi, ?j cj?dj) ? Con iff we have an open
  channel

• (f,g) ? Con iff L ?G . It is decidable on the
  basis.

• The Update is Up(f,g) (fg , g) where
  fg ? (?f) ? (? g)

80
The Domain of C1 Functions with C1 Norm

• Lemma. Con ? (I0,1 ? IR)2 isScott closed.
• Theorem.D1 (f,g) ? (I0,1?IR)2 (f,g) ?
  Conis a bounded complete ?continuous dcpo.

• Theorem.??? C10,1 ? C00,1 ? (I0,1 ?
  IR)2restricts to give a topological embedding
  D1c ? D1 (with C1 norm)
  (with Scott topology)

81
Consistency Test for (f,g)

• Also define L(x) supy?O?Dom(f)(f (y)
  d(x,y)) and G(x)
  infy?O?Dom(f)(f (y) d(x,y))

82
Consistency Test

• Theorem. (f, g) ? Con iff ?x ? O. L(x) ? G(x).
  For (f, g) (?1?i?n ai?bi, ?1?j?m cj?dj)
• the rational endpoints of ai and cj induce a
  partition X x0 lt x1 lt x2 lt lt xk of O.
• Proposition. For arbitrary x ? O, there isp,
  where 0 ? p ? k, with L(x)
  f (xp) d(x,xp).
• Similarly for G(x).

83
Higher Interval Derivative
84
Higher Interval Derivative

• For f I0,1 ? IR we define I0,1 ?
  IR by
• Then ?f ? ?2(a,b) a?b
• Define L20,1 ? P(I0,1 ? IR) by x ? L20,1
  iff x ? ?2(ai,bi) ? ?
• Then L20,1 ? L10,1 is an isomorphism.
  ?i?I ?2(ai,bi) ? ? i?I ?1(ai,bi)

D
85
Domains of C 2and C k functions

• D2 (f0,f1,f2) ? (I0,1?IR)3 ?f0 ? ?f1 ?
  ?? f2? ?

• Theorem. ????? restricts to give a topological
  embedding D2c ? D2

86
Picards Theorem
87
Picards Solution Reformulated
88
Strong Consistency

• Define (f,g) to be strongly consistent,
  (f,g)?SCon, if
• ?h?g . (f,g)?Con
• We define s(x) supy?O?Dom(f) (f (y)
  d(x,y)) and t(x)
  infy?O?Dom(f) (f (y) d(x,y))
• Theorem. (f,g) is strongly consistent iff
• ?x ? O. s(x), t(x) ? f(x)
• If (f, g) (?1?i?n ai?bi, ?1?j?m cj?dj), then
  there exists z in the partition X withs(x) f
  (xz) d(x,xp). Similarly for t(x).
• On basis elements, strong consistency is
  decidable.

89
A Data Type for Differential Equations

• Let F I0,1 ? IR ? IR and?F (I0,1 ? IR)2
  ? (I0,1 ? IR)2 (f,g) ? (
  f , ?T . F(T,f(T)) )
• Let Up D1 ? D1, with Up(f , g) ? ( fg , g )
  where fg ? ( ?f ? ?g )
• DF (f,g) ? D1 (f,g) ? SCon, (f,g) ?
  ?F(f,g) is a dcpo.
• For any initial (f0,g0) ? DF, the continuous map
  P Up o ?F DF ?
  ?(f0,g0) ? DF ? ?(f0,g0)
  has a least fixed point.
• If FIf for a map f0,1 ? R which satisfies the
  Lipschitz property of Picards theorem, then in
  the limit the domain-theoretic solution tends to
  the classical solution.

90
Picards Theorem (Example)
F is approximated by a sequence of step
functions, F1, F2,
.
t
We take gi0 Fi and a sequence of rectangles
ai?bi, representing initial condition (1/2,9/8)
?i ai?bi,
F
t
91
Picards Theorem (Example)
hi fixed point of P wrt Fi with initial
condition (fi0,gi0)
.
At each stage we find L and G with
hi(x)L(x),G(x)
92
Picards Theorem (Example)
hi fixed point of P wrt Fi with initial
condition (fi0,gi0)
.
At each stage we find L and G with
hi(x)L(x),G(x)
93
Picards Theorem (Example)
hi fixed point of P wrt Fi with initial
condition (fi0,gi0)
.
At each stage we find L and G with
hi(x)L(x),G(x)
L and G tend to the exact solutionx ? x2/2 1
94
Further Work

• Differential Calculus with Several Variables
• Analytic Functions
• Implicit and Inverse function Theorems
• Reconstruct Geometry and Smooth Mathematics with
  Domain Theory

95
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