In standard problems of digital circuit design, a switching function (two-valued Boolean function) is specified declaratively as a (usually incomplete) asserted relation R(X,Z), or equivalently as an equation R(X,Z) = 1, where X and Z are inputs and outputs, respectively. To obtain such a function constructively, one might use Boolean-function synthesis (which enlarges propositional logic to first-order predicate logic), or use a ‘big’ Boolean algebra (which acts as an enlargement of switching algebra). This paper explores the utility of Boolean-equation solving in handling the hard or intractable problem of integer factorization by constructing a hardware circuit that achieves this purpose in real time (at least for reasonably large bit sizes). The feasibility of the proposed scheme is verified via the manual solution of the smallest possible problem. However, the results obtained are really encouraging, as they can be automated in a straightforward fashion. A sequel forthcoming paper will treat the scaling, complexity, and automation issues, and will, in particular, determine the upper limit on the bit size that can be treated by the current technique.
Diagnostic testing concerning categorical or dichotomized variables is ubiquitous in many fields including, in particular, the field of clinical or epidemiological testing. Typically, results are aggregated in two-by-two contingency-table format, from which a surprisingly huge number of indicators or measures are obtained. In this paper, we study the eight most prominent such measures, using their medical context. Each of these measures is given as a conditional probability as well as a quotient of certain natural frequencies. Despite its fundamental theoretical importance, the conditional-probability interpretation does not seem to be appealing to medical students and practitioners. This paper attempts a partial remedy of this situation by visually representing conditional probability formulas first in terms of two-variable Karnaugh maps and later in terms of simplified acyclic (Mason) Signal Flow Graph (SFGs), resembling those used in digital communications or DNA replication. These graphs can be used, among other things, as parallels to trinomial graphs that function as a generative model for the ternary problems of conditional probabilities, which were earlier envisioned by Pedro Huerta and coworkers. The arithmetic or algebraic reading or solving of a typical conditional-probability problem is facilitated and guided by embedding the problem on the SFG that parallels a trinomial graph. Potential extensions of this work include utilization of more powerful features of SFGs, interrelations with Bayesian Networks, and reformulation via Boolean-based probability methods.
GSM network design requires efficient interference management technique, which offers significant capacity enhancement and improves cell edge coverage with low complexity of implementation. This is done by assigning different frequencies to adjacent cells to avoid interference or cross talk. Random assignment of these frequencies is quite herculean and inefficient for huge number of cells. This paper proposed a formula for assigning frequencies for uniform cell range and extends it to non-uniform cell range in cell planning. Also, we obtain a functional relationship between the apothem and the circumradius as well as the inner and outer angle and deduce that hexagonal tessellation offers the best radius and angular relationship in GSM cell planning.
Aims/ Objectives: In this paper, based on motivations coming from various physical applications, we consider a coupled system of the wave in a one-dimensional bounded domain with nonlinear localized damping acting in their equations. We also discuss the well-posedness and smoothness of solutions using the nonlinear semigroup theory. Then, we give the asymptotic stability and rates decay to the coupled system, based on solution of an ordinary differential equation, since the feedback functions and the localized functions satisfy some properties widely treated in obtaining uniform decay rates for solutions of semilinear wave equation. Furthermore, the result requires the obtaining of the internal observability inequality for the conservative system.
This review paper deals with Lie algebras, with some concentration on root systems, which help in classification and many applications of symmetric spaces. We deal with the basic concept of a root system. First, its origins in the theory of Lie algebras are exposed, then an axiomatic definition is provided. Bases, Weyl groups, and the transitive action of the latter on the former are explained. Finally, the Cartan matrix and Dynkin diagram are exposed to suggest the multiple applications of root systems to other fields of study and their classification.