The operator Aε = D1g1(x1/ε, x2)D1 + D2g2(x1/ε, x2)D2 is considered in L2(R ), where gj(x1, x2), j = 1, 2, are periodic in x1 with period 1, bounded and positive definite. Let function Q(x1, x2) be bounded, positive definite and periodic in x1 with period 1. Let Q(x1, x2) = Q(x1/ε, x2). The behavior of the operator (Aε + Q ) as ε → 0 is studied. It is proved that the operator (Aε +Q ) tends to ...

Solution. We suppose that we are given g, g ∈ G, and our task is to find a ∈ Zp. We first check if g is the identity element. If so, we know that a = 0. Otherwise, we set u = g and v = g, where we choose b ∈ Zp randomly (though we make sure that b 6= 0). We then call our hash attacker to find distinct pairs (x1, y1), (x2, y2) ∈ Zp × Zp such that u1 · v1 = u2 · v2 . We can rewrite this as: ux1−x...

Let X1, X2 be mixing connected algebraic dynamical systems with the descending chain condition. We show that every equivariant continuous map X1 → X2 is affine (that is, X2 is topologically rigid) if and only if the system X2 has finite topological entropy.

For a real number x, let ||x|| denote the distance from x to the nearest integer. Suppose x1 < x2 < x3 are positive integers with gcd(x1, x2, x3) = 1. This paper proves the following: if (x1, x2, x3) 6= (1, 2, 3s) for an integer s and x3 6= x1+x2, or x3 = x1+x2 but x1 ≡ x2 (mod 3), then there is a real number t such that ||txi|| ≥ 1/3 (for i = 1, 2, 3). If (x1, x2, x3) = (1, 2, 3s) or x3 = x1 +...