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Operator calculus on graphs : theory and applications in computer science
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Description
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Title
Operator
calculus
on
graphs
:
theory
and
applications
in
computer
science
Creator
Schott, Rene.
Contributors
Staples, G. Stacey.
World Scientific (Firm)
DescriptionAbstract
This
pioneering
book
presents
a
study
of the
interrelationships
among
operator
calculus
,
graph
theory
, and
quantum
probability
in a
unified
manner
, with
significant
emphasis
on
symbolic
computations
and an
eye
toward
applications
in
computer
science
.
Presented
in this
book
are
new
methods
,
built
on the
algebraic
framework
of
Clifford
algebras
, for
tackling
important
real
world
problems
related
, but not
limited
to,
wireless
communications
,
neural
networks
,
electrical
circuits
,
transportation
, and the
world
wide
web
.
Examples
are
put
forward
in
Mathematica
throughout
the
book
,
together
with
packages
for
performing
symbolic
computations
.
DescriptionTable Of Contents
1
.
Introduction
.
1.1
.
Notational
preliminaries

2
.
Combinatorial
algebra
.
2.1
.
Six
group
and
semigroup
algebras
.
2.2
.
Clifford
and
Grassmann
algebras
.
2.3
. The
symmetric
Clifford
algebra
Cl[symbol]
.
2.4
. The
idempotentgenerated
algebra
Cl[symbol]
.
2.5
. The
nparticle
zeon
algebra
Cl[symbol]
.
2.6
.
Generalized
zeon
algebras

3
.
Norm
inequalities
on
Clifford
algebras
.
3.1
.
Norms
on
Cl[symbol]
.
3.2
.
Generating
functions
.
3.3
.
Clifford
matrices
and the
CliffordFrobenius
norm
.
3.4
.
Powers
of
Clifford
matrices

4
.
Specialized
adjacency
matrices
.
4.1
.
Essential
graph
theory
.
4.2
.
Clifford
adjacency
matrices
.
4.3
.
Nilpotent
adjacency
matrices

5
.
random
graphs
5.1
.
Preliminaries
.
5.2
.
Cycles
in
random
graphs
.
5.3
.
Convergence
of
moments

6
.
Graph
theory
and
quantum
probability
.
6.1
.
Concepts
.
6.2
. From
graphs
to
quantum
random
variables
.
6.3
.
Connected
components
in
graph
processes

7
.
Geometric
graph
processes
.
7.1
.
Preliminaries
.
7.2
.
Dynamic
graph
processes
.
7.3
.
Timehomogeneous
walks
on
random
geometric
graphs

8
.
Timehomogeneous
random
walks
.
8.1
.
Cl[symbol]
and
random
walks
on
hypercubes
.
8.2
.
Multiplicative
walks
on
Cl[symbol]
.
8.3
.
Induced
additive
walks
on
Cl[symbol]

9
.
Dynamic
walks
in
Clifford
algebras
.
9.1
.
Preliminaries
.
9.2
.
Expectation
.
9.3
.
Limit
theorems

10
.
Iterated
stochastic
integrals
.
10.1
.
Preliminaries
.
10.2
.
Stochastic
integrals
in
Cl[symbol]
.
10.3
.
Graphtheoretic
iterated
stochastic
integrals

11
.
Partitiondependent
stochastic
measures
.
11.1
.
Preliminaries
.
11.2
.
Cycle
covers
,
independent
sets
, and
partitions
.
11.3
.
Computations
on
lattices
of
partitions
.
11.4
.
Free
cumulants

12
.
Appell
systems
in
Clifford
algebras
.
12.1
.
Essential
background
.
12.2
.
Operator
calculus
on
Clifford
algebras
.
12.3
.
Generalized
raising
and
lowering
operators
.
12.4
.
Clifford
Appell
systems
.
12.5
.
Fermion
algebras
and the
fermion
field

13
.
Operator
homology
and
cohomology
.
13.1
.
Introduction
.
13.2
.
Clifford
homology
and
cohomology
.
13.3
.
Homology
and
lowering
operators
.
13.4
.
Cohomology
and
raising
operators
.
13.5
.
Matrix
representations
of
lowering
and
raising
operators
.
13.6
.
Graphs
of
raising
and
lowering
operators
.
13.7
.
Operators
as
quantum
random
variables

14
.
Multivectorlevel
complexity
.
14.1
.
Preliminaries
.
14.2
.
Graph
problems
.
14.3
. A
matrixfree
approach
to
representing
graphs
.
14.4
.
Other
combinatorial
applications

15
.
Bladelevel
complexity
.
15.1
.
Blade
operations
.
15.2
.
Counting
cycles
.
15.3
.
Further
remarks
on
complexity

16
.
Operator
calculus
approach
to
minimal
path
problems
.
16.1
.
Pathidentfying
nilpotent
adjacency
matrices
.
16.2
.
Operator
calculus
approach
to
multiconstrained
paths
.
16.3
.
Minimal
path
algorithms
.
16.4
.
Application
:
precomputed
routing
in a
storeandforward
satellite
constellation

17
.
Symbolic
computations
with
Mathematica
.
17.1
.
CliffMath
:
Computations
in
Clifford
Algebras
of
Arbitrary
Signature
.
17.2
.
CliffSymNil
: A
Companion
Package
.
17.3
.
CliffOC
:
Operator
Calculus
on
Clifford
Algebras
.
17.4
. "
Fast
Zeon
"
implementation
.
Publisher
Imperial College Press
Distributed by World Scientific Pub. Co.
Subject
Calculus, Operational.
Graph theory.
Quantum statistics.
Computer science.
Identifier
9781848168770
(electronic
bk.)
;
1848168764
;
9781848168763
;
http://www.worldscientific.com/worldscibooks/10.1142/P843#t=toc
Language
eng
Type
Electronic books.
FormatExtent
xv, 411 p. : ill. (some col.)
Date
c2012
.
OCLC number
874498579
CONTENTdm number
241
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