Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

Universal coalgebra: a theory of systems

In the semantics of programming, finite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with infinite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (1988) on a theory of non-wellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages (Milner, 1980; Park, 1981). Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras (Aczel and Mendler, 1989). Thus the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to: coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken as the basic ingredients of a theory called universal coalgebra. Some standard results from universal algebra are reformulated (using the afore mentioned correspondence) and proved for a large class of coalgebras, leading to a series of results on, e.g., the lattices of subcoalgebras and bisimulations, simple coalgebras and coinduction, and a covariety theorem for coalgebras similar to Birkhoff's variety theorem.

An introduction to timetabling

1. Programs. 2. Functions and Categories. 3. Applications. 4. Relationships and Allegories. 5. Datatypes in Allegories. 6. Optimisation Problems. 7. Thinning Algorithms. 8. Dynamic Programming. 9. Greedy Algorithms. Appendices.

Algebra of programming

Relational methods can be found at various places in computer science, notably in data base theory, relational semantics on concurrency, relational type theory, analysis of rewriting systems, and modern programming language design. In addition, they appear in algorithms analysis and in the bulk of discrete mathematics taught to computer scientists. This book is devoted to the background of these methods. It is the first to explain how to use relational and graph-theoretic methods systematically in computer science. A formal framework of relational algebra is developed with respect to applications to a diverse range of problem areas. Results are first motivated by practical examples, often visualized by both Boolean O-1 matrices and graphs, and then derived algebraically.

Relations and Graphs: Discrete Mathematics for Computer Scientists

I. Introduction.- 1 Background Material.- II. Algebras.- 2 Relation Algebras.- 3 Heterogeneous Relation.- 4 Fork Algebras.- III. Logics.- 5 Relation Algebra and Modal Logics.- 6 Relational Formalisation of Nonclassical Logics.- 7 Linear Logic.- IV. Programs.- 8 Relational Semantics of Functional Programs.- 9 Algorithms from Relational Specifications.- 10 Programs and Datatypes.- 11 Refinement and Demonic Semantics.- 12 Tabular Representations in Relational Documents.- V. Other Application Areas.- 13 Databases.- 14 Logic, Language, and Information.- 15 Natural Language.- Bibliography (compiled by Wolfram Kahl, Thomas Stroehlein).- Symbol Table.- Addresses of Contributors.

Relational Methods in Computer Science

Relation Algebras We have deduced Propositions 2.3.6 and 2.3.7 from Propositions 2.3.2, 2.3.3, and 2.3.4 by arguments involving no element relationship (x,y) E R. In the exercises we require similar proofs for Dedekind's formula 2.3.5. Proofs not involving quantifiers are usually shorter and more transparent, hence also easier to check and to memorize. Moreover, this type of proof is capable of being performed by a machine. We define an (abstract) relation algebra to be an algebraic structure that satisfies the laws of an atomic, complete, Boolean lattice, the properties of multiplication (2.3.2), the Tarski rule (2.3.3), and the SchrOder equivalences (2.3.4). By contrast, when we consider a given set V and relations on it, we refer to concrete relations. We call a proof relation-algebraic, if it involves only the laws of a relation algebra, and their consequences, such as (2.3.6), but makes no use of the matrix-like structure of ReV x V. Results proved in such a way hold true in every relation algebra, hence also in those constructed by the procedure of (7.5.14).

Relations and graphs

Inhaltsubersicht: Mengen.- Homogene Relationen.- Transitivitat.- Heterogene Relationen.- Graphen: Assoziierte, Inzidenz und Adjazenz.- Erreichbarkeit.- Strukturfragen.- Kerne und Spiele.- Zuordnungen und Uberdeckungen.- Programme: Korrektheit und Verifikation.- Anhang.- Allgemeine Literaturhinweise.- Symbolverzeichnis.- Sachverzeichnis.

Relationen und Graphen

Relational mathematics is to operations research and informatics what numerical mathematics is to engineering: it is intended to help modelling, reasoning, and computing. Its applications are therefore diverse, ranging from psychology, linguistics, decision aid, and ranking to machine learning and spatial reasoning. Although many developments have been made in recent years, they have rarely been shared amongst this broad community of researchers. This first comprehensive overview begins with an easy introduction to the topic, assuming a minimum of prerequisites; but it is nevertheless theoretically sound and up to date. It is suitable for applied scientists, explaining all the necessary mathematics from scratch using a multitude of visualised examples, via matrices and graphs. It ends with tangible results on the research level. The author illustrates the theory and demonstrates practical tasks in operations research, social sciences and the humanities.

Gunther Schmidt

Papers

Relations and Graphs: Discrete Mathematics for Computer Scientists

Relational Methods in Computer Science

Relations and graphs

Relationen und Graphen

Relational Mathematics