In this note, we calculate all the basic invariants of the number field K = Q(3 √ 5, ω), where ω = (−1 + √ −3)/2 is a primitive cube root of unity. Here is the notation for the fields and Galois groups to be used. Let

The quintuple identity has a long history and, as Berndt [5] points out, it is difficult to assign priority to it. It seems that a proof of the identity was first published in H. A. Schwartz’s book in 1893 [19]. Watson gave a proof in 1929 in his work on the RogersRamanujan continued fractions [20]. Since then, various proofs have appeared. To name a few, Carlitz and Subbarao gave a simple proo...

We study block decoupling of linear multivariable systems under unity-feedback configuration. We consider plants which have unstable pole-zero coincidences and give necessary and sufficient conditions for the existence of the block decoupling controllers. The conditions are particularly simple for plants with only simple unstable pole-zero coincidences. An illustrative example is given.

Enemy observers, such as cameras and guards, are common elements that provide challenge to many stealth and combat games. Defining the exact placement and movement of such entities, however, is a non-trivial process, requiring a designer balance level-difficulty, coverage, and representation of realistic behaviours. In this work we explore systems for procedurally generating both camera and gua...

Hall-Littlewood functions indexed by rectangular partitions, specialized at primitive roots of unity, can be expressed as plethysms. We propose a combinatorial proof of this formula using Schilling’s bijection between ribbon tableaux and ribbon rigged configurations.

This paper shows how to translate the problem of deciding trace refinement between two data independent (DI) CSP processes to an unreachability problem in a DI Unity program. We cover here the straightforward but practically useful case when the specification satisfies a normality condition, Norm, meaning that we do not have to worry about hidden or unrecorded data variables. This allows us to ...

We present some techniques to obtain smooth derivations of concurrent programs that address both safety and progress in a formal manner. Our techniques form an extension to the calculational method of Feijen and van Gasteren using a UNITY style progress logic. We stress the role of stable guards, and we illustrate the derivation techniques on some examples in which progress plays an essential r...

We must mention that the assumption that we are dealing with a Cartan matrix of finite type and a root of unity appears only at the very end (see Chapter 4). We need these assumptions in order to compare our representations with the conventional definition of the category C. All previous results are valid in more general assumptions. In particular a Cartan matrix could be arbitrary and a deform...

Unity: Unity is a widely-used cross-platform game develop system. It consists of a game engine and an integrated development environment (IDE). It can be used to develop games for many different platforms, including PCs, consoles, mobile devices, and deployment on the Web. In this lecture, we will present the basic elements of Unity. However, this is a complex system, and we will not have time ...