We define and construct Independent Zero-Knowledge Sets (ZKS) protocols. In a ZKS protocols, a Prover commits to a set S, and for any x, proves non-interactively to a Verifier if x ∈ S or x / ∈ S without revealing any other information about S. In the independent ZKS protocols we introduce, the adversary is prevented from successfully correlate her set to the one of a honest prover. Our notion ...

an object $x$ of a category $mathbf{c}$ with finite limits is called exponentiable if the functor $-times x:mathbf{c}rightarrow mathbf{c}$ has a right adjoint. there are many characterizations of the exponentiable  spaces in the category $mathbf{top}$ of topological spaces. here, we study the exponentiable objects in the category $mathbf{stop}$ of soft topological spaces which is a generalizati...

We study the relationship between the σ-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of closed measure zero sets is equal to the additivity of the ideal of meager sets.

We show how a polynomial-time prover can commit to an arbitrary finite set S of strings so that, later on, he can, for any string x, reveal with a proof whether x 2 S or x 62 S, without revealing any knowledge beyond the verity of these membership assertions. Our method is non interactive. Given a public random string, the prover commits to a set by simply posting a short and easily computable ...

Given a positive integer M and real number q∈(1,M+1], an expansion of x∈[0,M/(q−1)] over the alphabet A={0,1,…,M} is sequence (ci)∈AN such that x=∑i=1∞ciq−i. Generalizing many earlier results, we investigate in this paper topological properties set Uq consisting numbers x having unique form, combinatorial Uq′ their corresponding expansions. We also provide shorter proofs main results Baker [3] ...

The notion of a simplicial set originated in algebraic topology and has also been utilized extensively category theory, but until relatively recently was not used outside those fields. However, with the increasing prominence higher categorical methods wide range applications, it is important for researchers fields to have good working knowledge them. This paper intended as an introduction sets,...

the paper sets forth in detail categorically-algebraic or catalg foundations for the operations of taking the image and preimage of (fuzzy) sets called forward and backward powerset operators. motivated by an open question of s. e. rodabaugh, we construct a monad on the category of sets, the algebras of which generate the fixed-basis forward powerset operator of l. a. zadeh. on the next step, w...