Smoothness of the truncated display functor

We show that to every p-divisible group over a p-adic ring one can associate a display by crystalline Dieudonne theory. For an appropriate notion of truncated displays, this induces a functor from truncated Barsotti-Tate groups to truncated displays, which is a smooth morphism of smooth algebraic stacks. As an application we obtain a new proof of the equivalence between infinitesimal p-divisible groups and nilpotent displays over p-adic rings, and a new proof of the equivalence due to Berthelot and Gabber between commutative finite flat group schemes of p-power order and Dieudonne modules over perfect rings.Comment: 38 page

Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees

We study the computational difficulty of the problem of finding fixed points of nonexpansive mappings in uniformly convex Banach spaces. We show that the fixed point sets of computable nonexpansive self-maps of a nonempty, computably weakly closed, convex and bounded subset of a computable real Hilbert space are precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A uniform version of this result allows us to determine the Weihrauch degree of the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is equivalent to a closed choice principle, which receives as input a closed, convex and bounded set via negative information in the weak topology and outputs a point in the set, represented in the strong topology. While in finite dimensional uniformly convex Banach spaces, computable nonexpansive mappings always have computable fixed points, on the unit ball in infinite-dimensional separable Hilbert space the Browder-Goehde-Kirk theorem becomes Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive mappings may not have any computable fixed points in infinite dimension. We also study the computational difficulty of the problem of finding rates of convergence for a large class of fixed point iterations, which generalise both Halpern- and Mann-iterations, and prove that the problem of finding rates of convergence already on the unit interval is equivalent to the limit operator.Comment: 44 page

Frames and finite group schemes over complete regular local rings

Let p be an odd prime. We show that the classification of p-divisible groups by Breuil windows and the classification of finite flat group schemes of p-power order by Breuil modules hold over any complete regular local ring with perfect residue field of characteristic p. We use a formalism of frames and windows with an abstract deformation theory that applies to Breuil windows.Comment: 22 page

Playing Smart - Artificial Intelligence in Computer Games

Abstract: With this document we will present an overview of artificial intelligence in general and artificial intelligence in the context of its use in modern computer games in particular. To this end we will firstly provide an introduction to the terminology of artificial intelligence, followed by a brief history of this field of computer science and finally we will discuss the impact which this science has had on the development of computer games. This will be further illustrated by a number of case studies, looking at how artificially intelligent behaviour has been achieved in selected games

Real-Time Character Animation for Computer Games

The importance of real-time character animation in computer games has increased considerably over the past decade. Due to advances in computer hardware and the achievement of great increases in computational speed, the demand for more realism in computer games is continuously growing. This paper will present and discuss various methods of 3D character animation and prospects of their real-time application, ranging from the animation of simple articulated objects to real-time deformable object meshes

A Classification of Scripting Systems for Entertainment and Serious Computer Games

The technology base for modern computer games is usually provided by a game engine. Many game engines have built-in dedicated scripting languages that allow the development of complete games that are built using those engines, as well as extensive modification of existing games through scripting alone. While some of these game engines implement proprietary languages, others use existing scripting systems that have been modified according to the game engine's requirements. Scripting languages generally provide a very high level of abstraction method for syntactically controlling the behaviour of their host applications and different types of scripting system allow different types of modification of their underlying host application. In this paper we propose a simple classification for scripting systems used in computer games for entertainment and serious purposes

A NPC Behaviour Definition System for Use by Programmers and Designers

In this paper we describe ZBL/0, a scripting system for defining NPC (Non Player Character) behaviour in FPS (First Person Shooter) games. ZBL/0 has been used to illustrate the use of scripting systems in computer games in general and the scripting of NPC behaviour in particular in the context of a book on game development. Many novice game designers have clear ideas about how the computer game they imagine should work but have little knowledge – if any – about how their ideas can be implemented. This is why books on game creation (design, programming etc.), as well as all-in-one game creation systems – especially designed for ease of use and intended for an amateur audience – enjoy great popularity. A large proportion of these books however merely present solutions in the form of descriptions and explanations of specific implementations with inadequate explanations of principles. While this may benefit rapid application development it often does not lead to a deeper understanding of the underlying concepts. The understanding of rule-based behaviour definition through simple scripting in computer games and the development of such scripts by programmers and designers is what we aim to address with the ZBL/0 system