Some remarks on modular arithmetic and parallel computation

Some remarks on the arithmetic-geometric index

Using an identity for effective resistances, we find a relationship between the arithmetic-geometric index and the global ciclicity index. Also, with the help of majorization, we find tight upper and lower bounds for the arithmetic-geometric index.

Some remarks on arithmetic physics *

There have been some recent speculations on connections between quantum theory and modern number theory. At the boldest level these suggest that there are two ways of viewing the quantum world, the usual and the arithmetic, which are in some sense complementary. At a more conservative level they suggest that there is much mathematical interest in examining structures which are important in quan...

Some remarks on cardinal arithmetic without choice

One important consequence of the Axiom of Choice is the absorption law of cardinal arithmetic. It states that for any cardinals m and n, if m 6 n and n is infinite, then m+ n = n and if m 6= 0, m · n = n. In this paper, we investigate some conditions that make this property hold as well as an instance when such a property cannot be proved in the absence of the Axiom of Choice. We further find s...

Some Notes on Parallel Quantum Computation

We exhibit some simple gadgets useful in designing shallow parallel circuits for quantum algorithms We prove that any quantum circuit composed entirely of controlled not gates or of diagonal gates can be parallelized to logarithmic depth while circuits composed of both cannot Finally while we note the Quantum Fourier Transform can be parallelized to linear depth we exhibit a simple quantum circ...

DNA-based Parallel Computation of Simple Arithmetic