In multiwinner approval voting, the goal is to select k-member committees based on voters' ballots. A well-studied concept of proportionality in this context justified representation (JR) axiom, which demands that no large cohesive group voters remains unrepresented. However, JR axiom may conflict with other desiderata, such as coverage (maximizing number who approve at least one committee memb...

‎let ${x_{alpha}:alphainlambda}$ be a collection of topological spaces‎, ‎and $mathcal {g}_{alpha}$ be a grill on $x_{alpha}$ for each $alphainlambda$‎. ‎we consider tychonoffrq{}s type theorem for $x=prod_{alphainlambda}x_{alpha}$ via the above grills and a natural grill on $x$ ‎related to these grills, and present a simple proof to this theorem‎. ‎this immediately yields the classical theorem...

Two-person bargaining problem is considered as to allocate a number of goods between two players. This paper suggests that any non-trivial division of goods cause a non-zero change on the solution of bargaining. So, a axiom of sharing division is presented, as an alternative axiom to Nash’s axiom of independence of irrelevant alternatives and Kalai-Smorodinsky’s axiom of monotonicity. This solu...

We prove criteria for axiom (b1) of [Cluckers, Loeser, b-minimality, http://www.dma.ens.fr/∼cluckers/]. We establish criteria for b-minimality when the auxiliary sorts are small. These criteria show that in many naturally occurring situations, for example in the situation of local fields, axiom (b1) is the only important axiom of b-minimality, in the sense that the other axioms of b-minimality ...

In this note, we give a survey on recent developments pertaining to forcing axioms with emphasis on their characterizations and consequences connected to the Continuum Problem. Nonspecialists in set theory and/or students in mathematics are supposed to be the readers, and thus we tried to make this as self-contained as possible. Due to the limitation on the extent of the article, however, many ...

We show that bounded forcing axioms (for instance, the Bounded Proper Forcing Axiom and the Bounded Semiproper Forcing Axiom) are consistent with the existence of (!2; !2)-gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen’s combinatorial principles for L at the level !2, and therefore with the existence of an !2-Suslin tree. We also show that the axiom we ...