Observed that the above expression is a conjunction of four disjunctions. We will give a protocol which allows the prover to prove a single disjunction. The prover can then prove the entire expression by separately proving that each of the four disjunctions is true. First we recall from last lecture the Chaum­Pedersen honest zero knowledge protocol for proving that two El Gamal ciphertexts, C1 ...

We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and deletes vertices randomly. At time t, with probability α1 > 0 we add a new vertex ut and m random edges incident with ut. The neighbours of ut are chosen with probability proportional to degree. With probability α − α1 ≥ 0 we add m random edges to existing vertices where the endpoints ar...

|α|=2 2! α! aσ(α)(x)∂ α +~b(x) · ∇+ c(x), ∂t = ∂/∂t, ∂j = ∂/∂xj , ∂ = ∇ = (∂1, ∂2), p(ξ) = ξ1ξ2(ξ1 + ξ2), α = (α1, α2) is a multi-index, |α| = α1 + α2, α! = α1!α2!, σ(α) = (α1 − α2)/2, ~b(x) = (b1(x), b2(x)), and aσ(α), bj(x), c(x) are real-valued smooth functions on T2. Such operators arise in the study of gravity wave of deep water. See [1], [3], [4] and the references therein. The purpose of...

In this study the effect of adrenergic receptor agonists and antagonists on physostigmine induced yawning was investigated. Intraperitoneal injection of different doses of physostigmine (0.03, 0.05, 0.1 and 0.2 mg/kg) caused yawning in white rats. The greatest response was seen at a dose of 0.2 mg/kg physostigmine. Phenylephrine, an α1 agonist, and clonidine, an α2 agonist, led to a decrease in...

The involvement of supraspinal alpha-adrenergic receptors in tonic pain was assessed in formalin-induced pain in rats. The alpha2 adrenoceptor agonist clonidine, along with yohimbine and prazosin, α2 and α1 receptor antagonists, were introduced intracerebroventrically (icv) and/or systemically in different doses. The data show that 1) clonidine exerts an alpha adrenergic analgesic effect, ...

Let C(n, p) be the set of p-compositions of an integer n, i.e., the set of ptuples α = (α1, . . . , αp) of nonnegative integers such that α1+ · · ·+αp = n, and x = (x1, . . . , xp) a vector of indeterminates. For α and β two p-compositions of n, define (x + α) = (x1 + α1) β1 · · · (xp + αp) βp . In this paper we prove an explicit formula for the determinant detα,β∈C(n,p)((x + α) ). In the case ...

Let β1, . . . , βn be linearly independent vectors in a vector space. For all j with 0 ≤ j ≤ n and all vectors α1, . . . , αk, if β1, . . . , βn are in the span of β1, . . . , βj, α1, . . . , αk, then j + k ≥ n. The proof of the claim is by induction on k. For k = 0, the claim is obvious since β1, . . . , βn are linearly independent. Suppose the claim is true for k−1, and suppose that β1, . . ....

Let φ : R → R be a continuously differentiable function on an interval J ⊂ R and let α = (α1, α2) be a point with algebraically conjugate coordinates such that the minimal polynomial P of α1, α2 is of degree ≤ n and height ≤ Q. Denote by Mn φ (Q, γ, J) the set of such points α such that |φ(α1)− α2| ≤ c1Q −γ . We show that for a real 0 < γ < 1 and any sufficiently large Q there exist positive va...

Making use a linear operator, which is defined here by means of the Hadamard product (or convolution) involving the generalized hypergeometric function, we introduce two novel subclasses Qp,q,s(α1;A,B,λ) and Qp,q,s(α1;A,B,λ) of meromorphically multivalent functions of order λ (0 λ < p) in the punctured disc U∗ . In this paper we investigate the various important properties and characteristics o...