Beyond Infinity: Augustine and Cantor

To Infinity and Beyond

We prove that if a group generated by a bireversible Mealy automaton contains an element of infinite order, its growth blows up and is necessarily exponential. As a direct consequence, no infinite virtually nilpotent group can be generated by a bireversible Mealy automaton. The study on how (semi)groups grow has been highlighted since Milnor’s question on the existence of groups of intermediate...

To Infinity and Beyond

Computer algebra systems (CAS) are sometimes expected to compute with infinities symbolically. Since operations on such objects do not always conform to a sufficient and consistent set of rules for ordinary arithmetic, some compromises are necessary. We discuss a few options and implementations.

To infinity, and beyond!

Fourier Asymptotics of Cantor Type Measures at Infinity

where q ≥ 3 is an integer, then lim t→∞φ(t) > 0. Moreover, the average 1 2T ∫ T −T |φ(t)|dt = O(T log 2 log q ), as T → ∞. Note that φ is the characteristic function of the random variable X = ∑∞ n=1 ρ εn, where {εn}n=1 is a sequence of i.i.d. Bernoulli random variables, and the corresponding distribution is a Cantor type measure. This measure is the most basic model in fractal theory; it is ge...

GALILEO'S EMPIRICISM -- AND BEYOND [with an Addendum on Saint Augustine]

Galileo was very much a man of the Renaissance. He was, like that portentous era, a watershed in which the streams of past and present, coming from heterogeneous sources, mixed and mingled. The current that emerged had certain clearly discernible characteristics that made of Galileo a man much more of the future than of the medieval past. Because of this, he is a symbol of the new age of modern...