linear algebra and others

[Chris Landauer]

glen got it almost right about linear algebra and exactly right about the topic 
-
i am also talking much more about the evolution in time of multi-agent systems
(regardless of what it is called)

details of some mathematical subject area definitions

in english language mathematics, algebra in general
(sometimes called "abstract" algebra to distinguish it from grade school 
algebra,
which is mostly about polynomials over the reals)
is the study of particular systems defined by sets and operations on those sets
that satisfy certain axioms

"universal algebra" is the abstract study of such systems,
and particular examples include groups, rings, vectors spaces and such
(posets are a special case when you allow generalization of the concept to sets
and relations on the sets)

"linear algebra" is the study of vector spaces,
which have an additive group of "vectors" and a field of "coefficients"
(usually but not always assumed to be commutative) -
there is no notion of multiplication of vectors except by these coefficients -
the fundamental notion of linear algebra is (linear) independence

orthogonality and independence

the notion of orthogonality is appropriate for one kind of linear algebra,
typically called an inner product space (or sometimes dot product space),
which is a space with a certain two-variable function to the coefficient field
that satisfies certain axioms -
examples include the real and complex vector spaces we usually see
in a college linear algebra or calculus course -
many of the linear independence concepts that are important for linear spaces
become simpler to define in an inner product space

orthogonality in general is probably not the right name anyway
for the property glen wants -
it is much more like a kind of independence,
which has been studied in many different guises,
perhaps the most general of which is the area of combinatorics called
"matroid" theory (more details on request)

the difficulty with basing a theory of agents on independence
is that agents aren't independent -
there is some kind of locality of interaction
(this is another of the "qualitites" i am looking for),
which means that agents further away tend to have a weaker impact
(on a given agent) -

message passing

using a notion of message passing as the basis for formalization
is interesting and important,
because there is no adequate model for object-and-message-based
systems at all

since some messages cause new objects to be created or destroyed,
this has remained a difficult problem -
so far, the only formal models i have seen do no more than
operationalize the definition of the message passing process -
this is not very useful because there is no analytical power to it
(which is the main point of formalization) -
that means you can't learn anything about the system without "running" it -
it is for precisely this reason that we believe need some new mathematics -
it certainly does not exist now

more later,
cal

Dr. Christopher Landauer
National Systems Group, The Aerospace Corporation
The Hallmark Building, Suite 187
13873 Park Center Road, Herndon, Virginia 20171
e-mail: address@hidden
Phone: (703) 318-1666, FAX: (703) 318-5409


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