The decision-making process is a key issue for most manufacturing and services organizations, and forms an essential part of the product design process. Indeed, designers are always confronted with the dilemma of choosing between different alternatives with different parameters, often expressed in fairly vague terms. For this reason, many tools have been proposed offering more and better ways t...

In this paper, we introduce the notions of product vague graph, balanced product vague graph, irregularity and total irregularity of any irregular vague graphs and some results are presented. Also, density and balanced irregular vague graphs are discussed and some of their properties are established. Finally we give an application of vague digraphs.

A novel phased array antenna with wide bandwidth and wide scan angle is presented. The radiating aperture of the phased array consists of periodically and closely spaced octagonal ring elements. Tight capacitive coupling between adjacent elements is realized by interdigitating the end portions of the ring elements. To improve the impedance matching of the individual antenna elements over wide f...

Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension of this vector space from 7n to 3n (where n is the complexity of the underlyi...

A formal approach to feasible numbers, as well as to middle and small numbers, is introduced, based on ideas of Parikh (1971) and improving his formalization. The \vague" set F of feasible numbers intuitively satisses the axioms 0 2 F , F + 1 F and 2 1000 6 2 F , where the latter is stronger than a condition considered by Parikh, and seems to be treated rigorously here for the rst time. Our tec...