ZK proofs. Zero-knowledge proofs involve a prover P trying to prove a statement to a verifier V without revealing any knowledge beyond the fact that the statement is true. For example, consider the problem of proving membership in an NP language L, (e.g., graph Hamiltonicity, 3-coloring, etc.). A ZK proof protects against a cheating prover, in the sense that if a prover tries to give a proof fo...

We provide a characterization of strictly stationary solutions to the stochastic recurrence equation zk = c(εk−1)zk−1 + g(εk−1) with Borel–measurable functions c and g, and independent, identically distributed random variables {εk}. Strictly stationary solutions that are functions of the past, respectively, of the future exist if and only if the expected value E log |c(ε0)| is negative, respect...

The purpose of this study is to examine the properties dust-acoustic solitary waves in a complex magnetoplasma made up negatively charged moving dust grains lower ionospheric region and inertialess electrons ions obeying Maxwell kappa distributions, respectively. In context, reductive perturbation technique carried out obtain Zakharov–Kuznetsov (ZK) equation within given framework. obtained evo...

Although the main role of 1alpha,25-dihydroxyvitamin D(3) [1,25-(OH)(2)D(3)] is to regulate calcium homeostasis, the valuable therapeutic applications of this compound have led to the search of new 1,25-(OH)(2)D(3)-vitamin D receptor (VDR) ligands with less side effects. In this work we have characterized seven 1,25-(OH)(2)D(3) derivatives (ZK136607, ZK161422, ZK157202, ZK159222, ZK168492, ZK19...

Let f (z1, . . . ,zk)∈ C(Ik,I) be a given function, where I is (bounded or unbounded) subinterval of R, and k ∈ N. Assume that f (y1, y2, . . . , yk) ≥ f (y2, . . . , yk, y1) if y1 ≥ max{y2, . . . , yk}, f (y1, y2, . . . , yk) ≤ f (y2, . . . , yk, y1) if y1 ≤ min{y2, . . . , yk}, and f is nondecreasing in the last variable zk. We then prove that every bounded solution of an autonomous differenc...