Diierent requirements for the representation of the factual knowledge for the use in mechanised reasoning systems are represented. In particular, we distinguish diier-ent types of knowledge: axioms, deenitions (for introducing concepts like \set" or \group") and theorems (for relating the concepts) for formulating concepts and their relationships, as well as proofs, examples and counterexamples...

This note contains a Stone-style representation theorem for compact Haus-dorff spaces. The note is very much inspired by some existing representation theorems and is expository in nature. The first representation theorem is by Jung and Sünderhauf in [JS] and there is also a version of it for compact Hausdorff spaces noted by D. Moshier [Mo], the ideas of which were described to us by A. Jung. T...

Kusuoka (2001) has obtained explicit representation theorems for comonotone risk measures and, more generally, for law invariant risk measures. These theorems pertain, like most of the previous literature, to the case of scalar-valued risks. Jouini-Meddeb-Touzi (2004) and Burgert-Rüschendorf (2006) extended the notion of risk measures to the vector-valued case. Recently Ekeland-Galichon-Henry (...

Our goal in this paper is to illustrate how the representation theorems for finite interval orders and semiorders can be seen as special instances of existence results for potentials in digraphs. This viewpoint yields short proofs of the representation theorems and provides a framework for certain types of additional constraints on the intervals. We also use it to obtain a minimax theorem for t...

. . . . . . . . . . . . . . . . . . . . . . ix Possible lecture schedules . . . . . . . . . . . . . . x Acknowledgments . . . . . . . . . . . . . . . . . . xi Chapter

Diierent requirements for the representation of the factual knowledge for the use in mechanised reasoning systems are represented. In particular, we distinguish diier-ent types of knowledge: axioms, deenitions (for introducing concepts like \set" or \group") and theorems (for relating the concepts) for formulating concepts and their relationships, as well as proofs, examples and counterexamples...

We give a relatively short and self contained proof of Ratner’s theorem in the special case of SL(2, R)-invariant measures.

Generalized random processes by various types of continuity are considered and classified as generalized random processes (GRPs) of type (I) and (II). Structure theorems for Hilbert space valued generalized random processes are obtained: Series expansion theorems for GRPs (I) considered as elements of the spaces L(A, S(H)−1) are derived, and structure representation theorems for GRPs (II) on K{...