Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

Fuzzy Logic

Fuzzy Logic is becoming an essential method of solving problems in all domains. It gives tremendous impact on the design of autonomous intelligent systems. The purpose of this book is to introduce Hybrid Algorithms, Techniques, and Implementations of Fuzzy Logic. The book consists of thirteen chapters highlighting models and principles of fuzzy logic and issues on its techniques and implementations. The intended readers of this book are engineers, researchers, and graduate students interested in fuzzy logic systems

Feasible Form Parameter Design of Complex Ship Hull Form Geometry

This thesis introduces a new methodology for robust form parameter design of complex hull form geometry via constraint programming, automatic differentiation, interval arithmetic, and truncated hierarchical B- splines. To date, there has been no clearly stated methodology for assuring consistency of general (equality and inequality) constraints across an entire geometric form parameter ship hull design space. In contrast, the method to be given here can be used to produce guaranteed narrowing of the design space, such that infeasible portions are eliminated. Furthermore, we can guarantee that any set of form parameters generated by our method will be self consistent. It is for this reason that we use the title feasible form parameter design. In form parameter design, a design space is represented by a tuple of design parameters which are extended in each design space dimension. In this representation, a single feasible design is a consistent set of real valued parameters, one for every component of the design space tuple. Using the methodology to be given here, we pick out designs which consist of consistent parameters, narrowed to any desired precision up to that of the machine, even for equality constraints. Furthermore, the method is developed to enable the generation of complex hull forms using an extension of the basic rules idea to allow for automated generation of rules networks, plus the use of the truncated hierarchical B-splines, a wavelet-adaptive extension of standard B-splines and hierarchical B-splines. The adaptive resolution methods are employed in order to allow an automated program the freedom to generate complex B-spline representations of the geometry in a robust manner across multiple levels of detail. Thus two complementary objectives are pursued: ensuring feasible starting sets of form parameters, and enabling the generation of complex hull form geometry