We begin with some reminders and remarks on non-commutative rings. First, recall that if R is a not necessarily commutative ring, an element x ∈ R is invertible if it has both a left (multiplicative) inverse and a right (multiplicative) inverse; it then follows that the two are necessarily equal. Note however, that unlike the example of n× n matrices over a field, in a general noncommutative ri...

Dickson bases have recently been introduced in [1] for finite field arithmetic. Such a basis exists for any extension field and under certain conditions it represents a permutation of type II optimal normal bases. In this paper Dickson bases are developed using simpler mathematical terms and their properties are discussed. An algorithm/architecture based on the model presented in [1] is develop...

silica-bonded n-propyltriethylene-tetramine (sbnptt) was found as an efficient solid base for the synthesis of 2-amino-5-oxo-5,6,7,8-tetrahydro-4h-chromenes. chromene derivatives were obtained via the three-component condensation reaction of aromatic aldehydes, malononitrile, and dimedone at 80 oc under solvent-free conditions. the heterogeneous solid base showed much the same efficiency when u...

The β-numeration, born with the works of Rényi and Parry, provides a generalization of the notions of integers, decimal numbers and rational numbers by expanding real numbers in base β, where β > 1 is not an integer. One of the main differences with the case of numeration in integral base is that the sets which play the role of integers, decimal numbers and rational numbers in base β are not st...

We introduce a class of Gabor-type matrices and show that the product of two Gabor-type matrices is still a Gabor-type matrix possessing the same type. The key point is based on the fact that the multiplication of Gabor-type matrices can be converted into the \multiplication" of small block matrices. We propose an eecient algorithm, the specially devised block-multiplication. As an interesting ...

An all-digital phase-locked loop (PLL) circuit in which resolution in the phase detector and digitally controlled oscillator (DCO) exactly matches the gate-delay time is presented. The pulse delay circuit is connected in a ring shape with 32 inverters (2 inverters). With the inverter gate-delay time as the time base, the pulse phase difference is detected simultaneously with the generation of t...

In a pairwise interaction, an individual who uses costly punishment must pay a cost in order that the opponent incurs a cost. It has been argued that individuals will behave more cooperatively if they know that their opponent has the option of using costly punishment. We examined this hypothesis by conducting two repeated two-player Prisoner's Dilemma experiments, that differed in their payoffs...