Fermat's Quotient and Related Arithmetic Functions

Fibers, Surjective Functions, and Quotient Groups

Spectral disjointness of dynamical systems related to some arithmetic functions

We present a new and general approach to prove the spectral disjointness of dynamical systems related to digital expansions of natural numbers and Gaussian integers. The main tools are ideas from automata theory and rigid time in ergodic theory. This extends earlier work of T. Kamae and M. Queffélec.

The Rayleigh Quotient of Bent Functions

The Rayleigh quotient of a bent function is an invariant under the action of the orthogonal group, and it measures the distance of the function to its dual. An efficient algorithm is derived that generates all bent functions of given Rayleigh quotient. The Rayleigh quotient of some bent functions obtained by primary (Maiorana McFarland, Dillon) or secondary (direct and indirect sum) constructio...

On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions

We begin by introducing an interesting class of functions, known as the Schemmel totient functions, that generalizes the Euler totient function. For each Schemmel totient function Lm, we define two new functions, denoted Rm and Hm, that arise from iterating Lm. Roughly speaking, Rm counts the number of iterations of Lm needed to reach either 0 or 1, and Hm takes the value (either 0 or 1) that t...

Neural Computation of Arithmetic Functions

The basic processing unit of a neural network i s a linear threshold element. I t has been known that neural networks can be much more powerful than traditional logic circuits, assuming that each threshold element can be built at a cost comparable to that o f AND, OR, Nor logic elements. Whereas any logic circuit o f polynomial size (in n) that computes the product of two n-bit numbers requires...