The Algebraic Structure of Quantity Calculus

Abstract

In this paper, the selected results from testing of optimized CMOS friendly signaling method for high-speed communications over cables and printed circuit boards (PCBs) are presented and discussed. The proposed signaling scheme uses modified concept of pulse width modulated (PWM) signal which enables to better equalize significant channel losses during data high-speed transmission. Thus, the very effective signaling method to overcome losses in transmission channels with higher order transfer function, typical for long cables and multilayer PCBs, is clearly analyzed in the time and frequency domain. Experimental results of the measurements include the performance comparison of conventional PWM scheme and clearly show the great potential of the modified signaling method for use in low power CMOS friendly equalization circuits, commonly considered in modern communication standards as PCI-Express, SATA or in Multigigabit SerDes interconnects.

The Algebraic Structure of Quantity Calculus II: Dimensional Analysis and Differential and Integral Calculus

“…In a previous paper [1] the author has produced a new algebraic structure, axiomatically defined, called algebraic space of quantities, which resembles accurately the algebra structure of quantity calculus. In that paper there is a review of past attemps of giving a mathematical framework to quantity calculus and of the motivation to introduce yet another, different, algebraic structure which is now summarized here.…”

“…, r n are real numbers. In this equation α is the numerical value of q with respect to this system of units, while u r 1 1 • • • u r n n is the unit in which q is measured. The set of all units, that is {u r 1 1 • • • u r n n : r i ∈ R, i = 1, .…”

“…It works, of course, since real numbers contain integer numbers, but it is not the algebraic structure which best fits the quantity calculus. The paper [1] addresses these two issues and gives a way out by defining, in axiomatic form, a new algebraic structure which resembles, more accurately, the calculus with quantities (i.e., only integer exponents allowed) and is completely focused on quantities and dimensions. It starts with the definition of the group of dimensions D as a finitely generated free Abelian group.…”

“…In the definition of the algebraic structure, the role of real numbers is taken by an abstract field F. Nevertheless, this abstract field is going to get more specific along the sections of this paper, first as an ordered field in section 3, then as the real numbers in sections 4 and 5. With these elements, a summary of the definition given in [1] is as follows. A space of quantities Q with group of dimensions D over the field F is a set Q, whose elements are called quantities, together with a surjective map dim : Q → D and three operations.…”

“…The system is coherent if σ is a group homomorphism, and it is nonzero if σ (A) is not a zero for any dimension A. In [1] the conditions for its existence are established.…”