The goal of this paper is to classify linear operators with octonionic coefficients and octonionic variables. While building up to the octonions we also classify linear operators over the quaternions and show how to relate the linear operators over the quaternions and octonions to matrices. We also construct a basis...
Statistical mechanics studies the probability that a system is in a certain state given one or more constraints which are usually fixed conserved quantities. It is a particularly useful and powerful approach for problems with a large number of degrees of freedom where a complete knowledge of the system is...
The Alexander polynomial is a well understood classical knot invariant with interesting symmetry properties and recent applications in knot Floer homology. There are many different ways to compute the Alexander polynomial, some involving algebraic techniques and others more geometric or combinatorial approaches. This is an interesting example of how different...
In probability and statistics, Simpson’s paradox is an apparent paradox in which a trend is present in different groups, but is reversed when the groups are combined. Joel Cohen (1986) has shown that continuously distributed lifetimes can never have a Simpson’s paradox. We investigate the same question for discrete random...
The paper reviews percolation and some of its important properties, particularly on the 2-D square lattice. A bilevel lattice is introduced, with a percolation model representing the spread of a forest fire according to characteristics of the forest. It is proven that the value of the laddering probability may determine...
We consider numerical methods for finding approximate solutions to Ordinary Differential Equations (ODEs) with parameters distributed with some probability by the Generalized Polynomial Chaos (GPC) approach. In particular, we consider those with forcing functions that have a random parameter in both the scalar and vector case. We then consider linear...