This paper presents a discussion of complex shift dynamics of some elementary cellular automaton rules (rules 30, 41, and 110). Equations that show some degree of self-similarity are obtained. It is demonstrated that rules 30, 41, and 110 exhibit Bernoulli shifts and are topologically mixing on one of their own closed invariant subsystems. Furthermore, many complex Bernoulli shifts are explored...

Supersymmetric quantum mechanics is discussed in the spaces of one and three dimensions. The structure of the continuum portion of the spectrum of the hamiltonian is investigated by making use of Callias' and Weinberg's generalized index theorems. The difference of the densities of bosonic and fermionic states at a fixed energy is calculated and it turns out to be topologically invariant and no...

A positive real interval, [a, b] can be partitioned into sub-intervals such that sub-interval widths divided by sub-interval ”‘average”’ values remains constant. That both Arithmetic Mean and Geometric Mean ”‘average”’ values produce constant ratios for the same log scale is the stated invariance proved in this short note. The continuous analog is briefly considered and shown to have similar pr...

In this paper, we study the behavior of $\Upsilon_K(t)$ under cabling operation, where is knot concordance invariant defined by Ozsv\'ath, Stipsicz, and Szab\'o, associated to a $K\subset S^3$. The main result an inequality relating $\Upsilon_{K_{p,q}}(t)$, which generalizes inequalities Hedden Van Cott on Ozsv\'ath-Szab\'o $\tau$-invariant. As applications, give computation $\Upsilon_{(T_{2,-3...

The topological order of a (2+1)D phase matter is characterized by its chiral central charge and unitary modular tensor category that describes the universal fusion braiding properties anyonic quasiparticles. I discuss topologically invariant quantities associated with these identify ones are useful for determining order. propose variety physical experiments probe detail relation measured data ...