F - Duality Science

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CDGO 2007: 2nd International
Conference
on Complementarity,

Duality and Global Optimization

in
Science and Engineering




February 28
-
March 2, 2007


Industrial and Systems Engineering
Department





A Category
-
Theoretic Approach to Duality



Sabah E. Karam, Information Specialist

Morgan State University

Planning & Information Technology

Baltimore, MD 21251



tel
:
443
-
885
-
4597

email
:
Sabah.Karam
@ morgan.edu

CDGO


2007


Historical notes


Categories were first introduced by
S.Eilenberg

and
S. MacLane

during
the years 1942
-
1945, in connection with
algebraic topology
, a branch of
mathematics in which tools from
abstract algebra

are used to study
topological spaces
.



Category theory has come to occupy a central position in
pure
mathematics

and
theoretical computer science
.



Categories are algebraic structures with many
complementary natures
,
e.g.,
geometric
,
logical
,
computational
,
combinatorial
.



Category Theory is an alternative to classical
set theory

as a
foundation
for mathematics
. The primitive, set
-
theoretic concept of "element" or
"membership" is replaced by that of "
function
."

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2007


Applications of CT


Mathematics and Computer Science


Quantum Physics and Tensor CT


Genomes and Computational Biology


Information Systems (databases, OOT)


Unified Modeling Language and Software Engineering


Compiler Optimization


Logic and Philosophy


Natural Transformation Models in Molecular Biology


Neural Network Analysis and Design


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2007


Reasons to use CT


it is a
unifying language

for discussing different
mathematical models and other logic
-
based structures,


it reveals
common structures

in seemingly unrelated
systems and a framework for comparing them,


it reveals
invertible structures
, i.e. for every
categorical construct there is a dual formed by
reversing all the

transformations,


it consolidates the description of similar operations
such as
'
products
'

found in set theory, group theory,
linear algebra, and topology, and


it produces
graphical models

which are
intuitive
,
formal
,
declarative
, and
subject to further analysis
.

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2007


A category consists of 3 entities:
objects, morphisms, and compositions


a class of
objects

(A, B, C, …)



a class of
morphisms

between objects symbolized by


¶)RUHDFKPRUSKLVPRQHREMHFW$LVWKH
domain

of f and another object, B, is the codomain
,
f: A

B.



a binary operation called
composition
. For each pair
of morphisms f: A

B and g: B

C, a composite
morphism, g


f: A

C is defined.


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2007


Morphisms have two properties



Associativity
: If
f

:
A


B
,
g

:
B


C

and
h

:
C


D

then
h



(
g



f
) = (
h



g
)


f
,
and



Identity

: For every object
A
, there exists a
morphism 1
A

:
A


A

called the identity
morphism for A, such that for every morphism
f

:
A


B
, we have 1
A



f

=
f

=
f



1
A
.

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2007


SOME MORE TERMINOLOGY


Every morphism has a
source

object, called the
domain
, and a
target

object, called the
codomain
. If f
is a morphism with X as its source and Y as its target,
we write f: X → Y.




We write
Hom
(X,Y) for the set of morphisms from X to
Y. In traditional set theory morphisms are nothing
more than the set of
functions
from X to Y.



Hom( ) is short for
Homology
.

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2007


What is a homology?


A
correspondence

or
structural parallel
.



In
biology
, two or more structures are said to be
homologous if they are alike because of shared ancestry.
This could be
evolutionary ancestry
, e.g. the wings of bats
and the arms of humans, or
developmental ancestry
, e.g.
the ovaries of female humans and the testicles of males.



Scientists use
physical structures

to reconstruct
evolutionary history.


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2007


What is a homology (cond’t)?


In
mathematics
, especially algebraic topology and
abstract algebra, homology is a certain general procedure
to associate a sequence of
abelian

groups or modules to a
given mathematical object (such as a topological space or
a group).



In
anthropology

and
archaeology
, homology refers to a
type of analogy whereby two human beliefs, practices or
arte
-
facts are separated by time but share similarities due
to genetic or historical connections.

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2007


Elementary Example


Any partial ordering, sequencing, or
arrangement of the elements of a set


(a) Objects are the elements of the partial
order; numbers, sets, points in a plane,
integers, people in a genealogy relationship,


(b) Morphisms:


,

,

divisibility relationship.


(c) Composition works because of transitivity.

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2007


Mathematical Categories


Set
= sets with linear transfomations


Vect

= vector spaces with linear transfomations


Poset

= partially ordered sets with monotone functions


Grp

= groups with group homomorphisms


Top
= topological spaces with continuous functions


Diff

= smooth manifolds with smooth maps


Ring

= rings with ring homomorphisms


Met

= metric spaces with contraction maps


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2007


Functors, Natural Transformations,
and Adjoints


Saunders MacLane, one of the founders of category theory,
remarked, "I didn't invent categories to study functors; I invented
them to study
natural transformations
." Also called
natural
equivalence

or
isomorphism of functors.



The context of Mac Lane's remark was the axiomatic theory of
homology
. With the language of natural transformations he could
easily express: (i) how homology groups are compatible with
morphisms between objects and (ii) how two equivalent homology
theories not only have the same homology groups but also the same
morphisms between those groups.

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2007


Definition of Functor


Let
C

and
D

be categories.


A
functor

F

from
C

to
D

is a
mapping

that


(a)

associates with each object X
ε

C an object
F
(X)
ε

D, and


(b)

associates with each morphism
f: X

Y a
morphism




F
(
f):
F
(X)

F
(Y)

such that the following two properties hold:




(i)
F
(
1
A
) =
1
F
(A)

for every object, and




(ii)
F
(
g



f
) =
F
(
g
)


F

(
f )



That is to say, functors
preserve

identity morphisms and
composition of morphisms.


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2007


Example of a functor


Given

a set S = {a, b, c, 1, 2, 3, @, # ,$)



Objects
:


List
(S) = {a, b2, c$, 3#a3, ...}, L = [s
1
, s
2
, s
3
, s
4
, …]




Morphisms
:

f: S


S’ (e.g. a sort routine)



Identity
: we also need to define an associative binary concatenation operator,


call it *, and an identity operator, call it [ ], such that [ ] * L = L = l * [ ].



Functor
:

F(f):
List
(S)


List
(S’)




List
(f)( L) = [ f(s
1
), f(s
2
), f(s
3
), f(s
4
), …]



Equivalent to the java class
mapList.
It can be used to create a dictionary by
reading a collection of words and definitions.



CDGO


2007


Object
-
Oriented (OO) Technology


Objects are the principle building blocks of object
-
oriented
programs. Each object is a programming unit consisting of
data

(instance variables) and
functionality

(instance methods).


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2007


Customer_Order

CustomerID

customerName

dateShipped

dateReceived

datePayed

checkInventory( )

contactCustomer( )

Ship( )

refund( )

calculateSale( )

Definition of Natural Transformation


Let X and Z be two categories and let F and G be two functors F:
X


Z, and G: X


Z. Let f: A


B


η

is a

NT from F to G, written
η
:F


G, if the diagram commutes.

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2007



η
B

F(f)

F(A)

F(B)

F(B)

η
A

G(f)

F(A)

Examples of Natural Transformations
(NT)


NT’s are structure preserving mappings from one functor to
another functor.


Two decks of playing cards, all analog wrist watches, and all tie
shoes with the same number of holes are isomorphic.


Consider
f
(
x

+
y
) =
f
(
x
) +
f
(
y
). Then
f
(
x
) = 4
x

is one such
preserving map, since
f
(
x

+
y
) = 4(
x

+
y
) = 4
x

+ 4
y

=
f
(
x
) +
f
(
y
).



Consider f
(
a

+
b
) =
f
(
a
) *
f
(
b
), Then
f
(
x
) = e
x

satisfies this
condition

since 5 + 7 = 12 translates into e
5

* e
7

= e
12
.


In group theory, every group is naturally isomorphic to its opposite
group in which the preserving map,
F(a*b) = b*a
, inverts the
binary operation.



The dual of dual is a “
natural transformation
” in category theory

CDGO


2007


Analogous electric and mechanical systems


Electrical and mechanical system have differential equations of
the same form and can be considered isomorphic.


Electrical
:


e = iR


e = voltage, i = current, R = resistance


L = inductance, C = capacitance, Q = charge

LQ'' + RQ' + Q/C = 0




Mechanical
:
f = vB


v = velocity, f = force, B = friction, M = mass






spring
-
mass differential eq.

mx'' + bx' + kx = 0











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2007


Functional programming language


Objects
: Int, Real, Bool, Char, Ref


Morphisms
: isZero: Int


Bool

(test for zero)




not:

Bool


Bool

(negation)




succ
Int
: Int


Int


(successor)




toReal: Int


Real

(conversion)


Constants
: zero (Int) , true/false (Bool)


Composition
: false = not


true

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2007


Contravariant functors & Dual Spaces


There are many constructions in mathematics which
would be
functors

but for the fact that they "turn
morphisms around" and reverse the direction of
composition
F
(
g



f
) =
F
(
f
)


F

(
g ).



Dual vector spaces,
maps which assign to every
vector space

its
dual space

reflect, in an abstract way,
the relationship between
row

vectors and
column

vectors.

The
dual

or
transpose

is a
contravariant
functor

from the category of all vector spaces over a
fixed field to itself.

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2007


Primal/Dual & Minimax Theorem


A
dual

mathematical program has the property that its
objective is always a bound on the original
mathematical program, called the
primal



Minimax theorem proven by von Neumann in 1928, it is
a cornerstone of
duality

and of game theory



Let
X

and
Y

be mixed strategies for players A and B.
Let A be the payoff matrix. Then




max min
X
T
A
Y

= min max

X
T
A
Y




X

Y

Y X

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2007


Commutative diagrams and LP


min{max
F
(x,y): y in Y}: x in X}



= max{min
F
(x,y): x in X}: y in Y}


F
: X*Y


删慮搠R⁡湤 夠慲攠湯n
-
敭灴pⰠ捯湶數Ⱐ捯浰c捴c獥瑳t

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2007


min

min

max

max

LOSS

GAIN

Optimal
solution

Original
problem

Primal/dual commutative diagram



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transpose

matrix A

new obj.
function

The original

objective function

Optimal
solution


max c
T
x

subject to Ax
≤ b


min b
T
y

subject to A
T
x
≥ b

Commutative diagrams as Proofs


Commutative diagrams play the role in category theory that
equations

play in algebra.


Commutative diagrams can be used to assert the validity of program transformations.
Diagram chasing is a method of mathematical proof especially in
homological algebra

and
computer science
.

CDGO


2007


succ
Real

toReal

Int

Int

Real

Real

succ
Int

toReal

Relationship between M
-
theory and
Type IIA supergravity/string theory


In the strong coupling limit type IIA string theory approaches an 11 dimensional Lorentz
invariant theory.


Commutative diagrams are used to assert the validity of relationships between theories.

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2007


S1 compactification

low energy

limit

M theory

Type IIA string theory

D=11
supergravity

Type IIA supergravity

S1 compactification

low energy
limit

Classification of Duals


Dorn's dual


primal/dual are convex quadratic programs


Fenchel's Conjugate Dual



Generalized penalty
-
function/surrogate Dual



Geometric dual



Inference dual



Lagrangian Dual



LP Dual
. This is the cornerstone of duality. In canonical
form:

Primal
:
Min
{cx: x >= 0, Ax >= b}.




Dual
:
Max
{yb: y >= 0, yA <= c}.


CDGO


2007


Duals (cond’t)


Self Dual

when a dual is equivalent to its primal
-

LP problems



Semi
-
infinite Dual



Superadditive Dual


Surrogate Dual



Symmetric Dual


Wolfe's Dual



Hooker
: A
relaxation dual

in which there is a
finite algorithm

for
solving the relaxation is an inference dual. An
inference dual

in which
the
proofs are parameterized

is a relaxation dual.


CDGO


2007


Natural Transformation Models in
Molecular Biology



Molecular models in terms of
categories
,
functors

and
natural transformations

are introduced for:


(a)
unimolecular chemical transformations
,


(b)
multi
-
molecular chemical
, and


(c)
biochemical transformations



Several applications of such
natural transformations

are then presented to analyze


(a)
protein biosynthesis
,


(b)
embryogenesis

and


(c)
nuclear transplant experiments

CDGO


2007


Complex graph matching problems as
combinatorial optimization



Phletora of confusing terms
: computational
complexity; neural networks; linear programming;
weighted graph matching; quadratic optimization;
simplex
-
based algorithm; Hungarian method;
eigendecomposition; pattern recognition;
symmetric polynomial transform; genetic
algorithms; probabilistic relaxation; clustering
techniques.

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2007


Duality, polarity, complementarity


Electronics
:

two devices or two circuits having mathematical
descriptions that are identical except that voltages in one formula
correspond to currents in the other formula.


Chemistry
: Conjugate Acid/Base pairs


Electromagnetic theory
:

electric fields are dual to magnetic fields.


Meterology
: precipitation/evaporation


Mathematics
:
projective geometry, category theory, Morgans laws
(logic), set theory, operations (+/
-
, x/
÷
,

/D
x
)
.


Biology
:
dualism is the theory that blood cells have two origins,
from the lymphatic system and from the bone marrow.


Physics
:
particle/wave nature of light,

electrical
-
mechanical
duality of the differential equations.



CDGO


2007


Duality, polarity, complementarity


Genetics
:
DNA base
-
pairing

(A
-
T, C
-
G)


Philosophy
:
yin
-
yang basis of

Chinese medicine


Endocrinology
:
metabolic processes that assemble/ disassemble
(anabolic/catabolic) molecules in the body = a hormonal process


Theology
:
Koranic verses describe created pairs


Molecular biology
:
code
-
duality (analog/digital)


Language
:
structure from which meaning is derived


Psychology
:
referent and probe in judgment


General theory of relativity
:
4 elementary forces


Quantum field theory
:

fermion
-
boson duality

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2007


Establishing a curriculum based on
a Duality Principle


Dr. Glenda Prime
, Coordinator,


Doctoral Programs in

Mathematics and Science Education




Glenda.Prime@morgan.edu



The
School of Education and Urban Studies
, through the
Doctoral Programs in Mathematics Education and Science
Education seeks to enhance the quality of science and
mathematics education by preparing a cadre of highly
qualified mathematics and science educators, supervisors
and
curriculum specialists
.



CDGO


2007


References


S. Eilenberg and S. MacLane
, "Natural Isomorphisms in Group Theory,"
Proceedings of the National Academy of Sciences
,
28
, (1942), pp. 537
-
543


S. Mac Lane
,

“Categories for the Working Mathematician” 2
nd

edition,
Springer (2000)


J.N. Hooker, Duality in Optimization and Constraint Satisfaction, Carnegie
Mellon Univ., Pittsburgh, PA (2006)
http://wpweb2.tepper.cmu.edu/jnh/duals.pdf



M. Barr and C. Wells
, “Category Theory for Computing Science” 3
rd

edition,
CRM (1999)


Proceedings

SIAM & Society for Mathematical Biology Meeting

N/A(3),
pages pp.

230
-
232, Colorado, 1983.


http://glossary.computing.society.informs.org/second.php?page=duals.html

CDGO


2007


End of Presentation


We would like to thank


Organizers:
Panos M. Pardalos & Altannar Chinchuluun
,
Univ. of Florida

Advisory Committee:
David Y. Gao & Hanif D. Sherali
,

Virginia Tech Univ.


And to everyone who attended this session

CDGO


2007