RESEARCH PAPERS IN PREPARATION
Paul
Sundheim, Ph.D.
TOPIC: Clifford Algebras
Properties of the Pascal Determinant An
n-matrix of order m will be an m×m×m×…×m (n factors)
matrix. A determinant for 3-matrices of order 2 was first discovered by Blaise E. Pascal. It was later generalized to n-matrices of
order m and properties of the determinant were established. This paper provides
new proofs of these properties in English that greatly simplifies the previous
approach, demonstrates the determinant as an extension of the classical
determinant and highlights the fact that the determinant, while having all of
the essential properties in the even dimensions, is well defined only up to
sign in the odd dimensions.
Status:
Submitted to a journal on linear algebra. Under consideration
by the journal.
An Associative
Multiplication for An
associative multiplication for multidimensional matrices of order 2 is Multidimensional Matrices of Order 2 defined
and properties of the multiplication and the
resulting system of rings are
discussed. One result of this multiplication is the existence of an infinite
associative system of hypercomplex numbers.
Status:
Submitted to a journal on linear algebra. Under consideration
by the journal.
An Associative
Multiplication for An
associative multiplication for multidimensional matrices is defined and Multidimensional Matrices properties
of the multiplication and the resulting system of rings are
discussed. Also, properties of a
related determinant are presented.
Status:
Submitted to a journal on linear algebra. Under consideration
by the journal.
Status:
Submitted to a journal on linear algebra. Under consideration
by the journal.
An
Infinite System of Associative A
communative and associative multiplication of
multidimensional
Hypercomplex Numbers matrices gives a unique insight into
an extension of the complex
numbers and hypercomplex
numbers in dimension four to all dimensions that are powers of two. The purpose
of this research grant would be to explore this view of hypercomplex
numbers in order to better understand the geometrical and analytical
consequences of the system.
Status:
In preparation for submission.
Invariants of Links in Three I have done an initial
investigation into the possibility of a polynomial invariant for links in
Dimensional
Manifolds three
manifolds. It looks extremely promising
for an infinite class of these spaces.
Once the polynomial is proven to be an invariant, it will be necessary to
discover the properties of the invariant.
These would include which infinite sub-collections of three manifolds
have polynomial invariants and possibly (and much harder) which do not. Other
properties would include; how good at identifying differing links the
polynomial is, what the properties of
the links the polynomial identifies when evaluated at specific values and
perhaps other topics that might arise while pursuing the above. For example, in R3 there is an addition that one can perform on knots
and typically the polynomial for the knot sum is the product of the polynomials
of the knots for each term in the sum.
Is there such an addition for knots in three manifolds and, if so, how
does the polynomial treat this sum? If discovered, the knot polynomial would be
the first found in these spaces.
Status: In preparation
for submission.