♣ Rings . A ring is a non-empty set R with two binary operations ( + , · ) , called addition and multiplication, respectively satisfying : Axiom 1. Closure ( + ) : ∀x, y ∈ R , x + y ∈ R . Axiom 2. Commutative ( + ) : For every x, y ∈ R , x + y = y + x . Axiom 3. Associative ( + ) : ∀x, y, z ∈ R , x + (y + z) = (x + y) + z . Axiom 4. Neutral ( + ) : ∃ θ ∈ R , such that ∀x ∈ R, x + θ = θ + x = x ...

The following notion of forcing was introduced by Grigorieff [2]: Let I ⊂ ω be an ideal, then P is the set of all functions p : ω → 2 such that dom(p) ∈ I. The usual Cohen forcing corresponds to the case when I is the ideal of finite subsets of ω. In [2] Grigorieff proves that if I is the dual of a p-point ultrafilter, then ω1 is preserved in the generic extension. Later, when Shelah introduced...

A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of zfc has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion v=hod that...

Development of new algorithms in the area machine learning, especially clustering, comparative studies such as well testing according to software engineering principles requires availability labeled data sets. While standard benchmarks are made available, a broader range sets is necessary order avoid problem overfitting. In this context, theoretical works on axiomatization clustering algorithms...

In this paper, the Urysohn and completely Hausdorff axioms in general topology are generalized to L-topological spaces so as to be compatible with pointwise metrics. Some properties and characterizations are also derived

At-speed testing is becoming crucial for modern very-large-scale-integration systems, which operate at clock speeds of hundreds of megahertz. In a scan-based test methodology, it is common to use a transition delay fault model for at-speed testing. The launching of the transition can be done either in the last cycle of scan shift [launch-off-shift (LOS)], or in a functional launch cycle that fo...

The starting point of the paper is the (well-known)observation that the “classical” Rough Set Theory as introduced by PAWLAK is equivalent to the S5 Propositional Modal Logic where the reachability relation is an equivalence relation. By replacing this equivalence relation by an arbitrary binary relation (satisfying certain properties, for instance, reflexivity and transitivity) we shall obtain...

Amartya Sen has correctly sought to correct some shortcomings within neoclassical economic theory. Nonetheless, there still exists a tension in his work. His overall frame of analysis is still congruent with much of the neoclassical tradition. However, his critiques seem to imply that a sharp break with neoclassical theory is necessary. This conflict is examined in light of the connections betw...