Cubical complexes are defined in a manner analogous to that for simplicial complexes, the chief difference being that cubical complexes are unions of cubes rather than of simplices. A very natural cubical complex to consider is the complex C(k_1,...,k_n) where k_1,...,k_n are nonnegative integers. This complex has as its underlying space [0,k_1]x...x[0,k_n] subset of R^n with vertices at all points having integer coordinates and higher dimensional cubes formed by the vertices in the natural way. The genus of a cubical complex is defined to be the maximum genus of all surfaces that are subcomplexes of the cubical complex. A formula is given for determining the genus of the cubical complex C(k_1,...,k_n) when at least three of the k_i are odd integers. For the remaining cases a general solution is not known. When k_1=...=k_n=1 the genus of C(k_1,...,k_n) is shown to be (n-4)2^{n-3}+1 which is equivalent to the genus of the graph of the n-cube. Indeed, the genus of the complex and the genus of the graph of the 1-skeleton of the complex, are shown to be equal when at least three of the k_i are odd, but not equal in general.

On étudie divers problèmes concernant le n-cube. On décrit les exemples connus de (0,2) graphes (bipartis de diamètre 2 ou 3). On présente des constructions de (0,2) graphes. On étudie les (0,2) graphes avec des triangles. On montre comment construire certains des (0,2) graphes comme graphes de Cayley de groupes. On étudie les invariants immédiats du n-cube.

Computer modeling and simulation is a practical way to design and test a system without actually having to build it. Simulation has many benefits which apply to many different domains: it reduces costs creating different prototypes for mechanical engineers, increases the safety of chemical engineers exposed to dangerous chemicals, speeds up the time to model physical reactions, and trains soldiers to prepare for battle. The motivation behind this work is to build a common software framework that can be used to create new networking simulators on top of an HLA-based federation for distributed simulation. The goals are to model and simulate networking architectures and protocols by developing a common underlying simulation infrastructure and to reduce the time a developer has to learn the semantics of message passing and time management to free more time for experimentation and data collection and reporting. This is accomplished by evolving the simulation engine through three different applications that model three different types of network protocols. Computer networking is a good candidate for simulation because of the Internet's rapid growth that has spawned off the need for new protocols and algorithms and the desire for a common infrastructure to model these protocols and algorithms. One simulation, the 3DInterconnect simulator, simulates data transmitting through a hardware k-array n-cube network interconnect. Performance results show that k-array n-cube topologies can sustain higher traffic load than the currently used interconnects. The second simulator, Cluster Leader Logic Algorithm Simulator, simulates an ad-hoc wireless routing protocol that uses a data distribution methodology based on the GPS-QHRA routing protocol. CLL algorithm can realize a maximum of 45% power savings and maximum 25% reduced queuing delay compared to GPS-QHRA. The third simulator simulates a grid resource discovery protocol for helping Virtual Organizations to find resource on a grid network to compute or store data on. Results show that worst-case 99.43% of the discovery messages are able to find a resource provider to use for computation. The simulation engine was then built to perform basic HLA operations. Results show successful HLA functions including creating, joining, and resigning from a federation, time management, and event publication and subscription.