Embeddings Of Linear Arrays, Rings And 2-D meshes On Extended lucas Cube Network

A Fibonacci string is a length ii binary string containing no two consecutive 1 s. Fibonacci cubes (FC), Extended Fibonacci cubes (ELC) and Lucas cubes (LC) are subgraphs of hvpercube defined in terms of Fibonacci strings. All these cubes were introduced in the last ten years as models for interconnection networks and shown that their network topology posseses many interesting properties that are important in parallel processor network design and parallel applications. In this paper, we propose a new family of Fibonacci-like cube, namely Extended Lucas Cube (ELC). We address the following network simulation problem : Given a linear array, a ring or a two-dimensional mesh; how can its nodes be assigned to ELC nodes so as to keep their adjacent nodes near each other in ELC ?. We first show a simple fact that there is a Hamiltonian path and cycle in any ELC. We prove that any linear array and ring network can be embedded into its corresponding optimum ELC (the smallest ELC with at least the number of nodes in the ring) with dilation 1, which is optimum for most cases. Then, we describe dilation 1 embeddings of a class of meshes into their corresponding optimum ELC. Keywords: (Extended) Fibonacci cube, Extended Lucas cube, Fibonacci number, Hamiltonian path, Hamiltonian cycle, linear array, ring , mesh, networ

Semantic Agreement Maintenance

A Fibonacci string is a length ii binary string containing no two consecutive 1 s. Fibonacci cubes (FC), Extended Fibonacci cubes (ELC) and Lucas cubes (LC) are subgraphs of hvpercube defined in terms of Fibonacci strings. All these cubes were introduced in the last ten years as models for interconnection networks and shown that their network topology posseses many interesting properties that are important in parallel processor network design and parallel applications. In this paper, we propose a new family of Fibonacci-like cube, namely Extended Lucas Cube (ELC). We address the following network simulation problem : Given a linear array, a ring or a two-dimensional mesh; how can its nodes be assigned to ELC nodes so as to keep their adjacent nodes near each other in ELC ?. We first show a simple fact that there is a Hamiltonian path and cycle in any ELC. We prove that any linear array and ring network can be embedded into its corresponding optimum ELC (the smallest ELC with at least the number of nodes in the ring) with dilation 1, which is optimum for most cases. Then, we describe dilation 1 embeddings of a class of meshes into their corresponding optimum ELC

pp-adic quotient sets

For $A \subseteq \mathbb{N}$, the question of when $R(A) = \{a/a' : a, a' \in A\}$ is dense in the positive real numbers $\mathbb{R}_+$ has been examined by many authors over the years. In contrast, the $p$-adic setting is largely unexplored. We investigate conditions under which $R(A)$ is dense in the $p$-adic numbers. Techniques from elementary, algebraic, and analytic number theory are employed in this endeavor. We also pose many open questions that should be of general interest.Comment: 24 page