The author uses irrationality and linear independence measures for certain algebraic numbers to derive explicit upper bounds for the solutions of related norm form equations. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm is then utilized to show that the integer solutions to NK/Q(x 4 √ N4 − 1 + y 4 √ N4 + 1 + z) = ±1 (where K = Q( 4 √ N4 − 1, 4 √ N4 + 1)) are given by (x, y, z) =...

In this paper, we present a method for color reduction of Persian carpet cartoons that increases both speed and accuracy of editing. Carpet cartoons are in two categories: machine-printed and hand-drawn. Hand-drawn cartoons are divided into two groups: before and after discretization. The purpose of this study is color reduction of hand-drawn cartoons before discretization. The proposed algorit...

In this paper, we study the (geodesic) hull number of graphs. For any two vertices u, v ∈ V of a connected undirected graph G = (V,E), the closed interval I[u, v] of u and v is the set of vertices that belong to some shortest (u, v)-path. For any S ⊆ V , let I[S] = ⋃ u,v∈S I[u, v]. A subset S ⊆ V is (geodesically) convex if I[S] = S. Given a subset S ⊆ V , the convex hull Ih[S] of S is the smal...

Sequences generated by an F2-linear recursion have wide applications, in particular, pseudorandom number generation. The dimension of equidistribution with v-bit accuracy is a most important criterion for the uniformity of the generated sequence. The fastest known method for computing these dimensions is proposed by Couture and L’Ecuyer, based on Lenstra’s lattice basis reduction and the dual l...

The reduction number r(G) of a graph G is the maximum integer m ≤ |E(G)| such that the graphs G − E,E ⊆ E(G), |E| ≤ m, are mutually non-isomorphic, i.e., each graph is unique as a subgraph of G. We prove that r(G) ≤ |V (G)| 2 +1 and show by probabilistic methods that r(G) can come close to this bound for large orders. By direct construction, we exhibit graphs with large reduction number, althou...

The Turing degree of a real number is defined as the Turing degree of its binary expansion. In this note we apply the double witnesses technique recently developed by Downey, Wu and Zheng [2] and show that there exists a ∆2-Turing degree which contains no divergence bounded computable real numbers. This extends the result of [2] that not every ∆2-Turing degree contains a d-c.e. real.

Maurer’s test is nowadays a basic statistical tool for testing physical random number generators in cryptographic applications. Based on a statistical analysis of this test we propose simple and effective methods for its improvement. These methods are related to the m spacing technique common in goodness-of-fit problems and the L leave out method used for a noise reduction in the final Maurer t...