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

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

What is modal logic

Real and Complex Analysis

Preface- Acknowledgments- Introduction- I Background- Preliminaries- Computability Theory- Kolmogorov Complexity of Finite Strings- Relating Plain and Prefix-Free Complexity- Effective Reals- II Randomness of Sets- Martin-Lof Randomness- Other Notions of Effective Randomness- Algorithmic Randomness and Turing Reducibility- III Relative Randomness- Measures of Relative Randomness- The Quantity of K- and Other Degrees- Randomness-Theoretic Weakness- Lowness for Other Randomness Notions- Effective Hausdorff Dimension- IV Further Topics- Omega as an Operator- Complexity of CE Sets- References- Index

Algorithmic Randomness and Complexity

It is time for a new paper about open questions in the currently very active area of randomness and computability. Ambos-Spies and Kucera presented such a paper in 1999 [1]. All the question in it have been solved, except for one: is KL-randomness different from Martin-Lof randomness? This question is discussed in Section 6. Not all the questions are necessarily hard—some simply have not been tried seriously. When we think a question is a major one, and therefore likely to be hard, we indicate this by the symbol ▶, the criterion being that it is of considerable interest and has been tried by a number of researchers. Some questions are close contenders here; these are marked by ▷. With few exceptions, the questions are precise. They mostly have a yes/no answer. However, there are often more general questions of an intuitive or even philosophical nature behind. We give an outline, indicating the more general questions. All sets will be sets of natural numbers, unless otherwise stated. These sets are identified with infinite strings over {0, 1}. Other terms used in the literature are sequence and real.

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Randomness and computability: Open questions

We define a new class of hybrid automata for which reachability is decidable--a proper superclass of the initialized rectangular hybrid automata--by taking parallel compositions of simple components. Attempting to generalize, we encounter timed automata with algebraic constants. We show that reachability is undecidable for these algebraic timed automata by simulating two-counter Minsky machines. Modifying the construction to apply to parametric timed automata, we reprove the undecidability of the emptiness problem, and then distinguish the dense and discrete-time cases with a new result. The algorithmic complexity-- both classical and parametric--of one-clock parametric timed automata is also examined. We finish with a table of computability-theoretic complexity results, including that the existence of a Zeno run is Σ11 -complete for semi-linear hybrid automata; it is too complex to be expressed in first-order arithmetic.

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Decidability and Complexity Results for Timed Automata and Semi-linear Hybrid Automata

One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define X ≤K Y to mean that (Vn) K(X | n) < K(Y | n) + 0(1). The equivalence classes under this relation are the K-degrees. We prove that if X ⊕ Y is 1-random, then X and Y have no upper bound in the K-degrees (hence, no join). We also prove that n-randomness is closed upward in the K-degrees. Our main tool is another structure intended to measure the degree of randomness of real numbers: the vL-degrees. Unlike the K-degrees, many basic properties of the vL-degrees are easy to prove. We show that X ≤K Y implies X ≤vL Y, so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for ≤c, the analogue of ≤K for plain Kolmogorov complexity. Two other interesting results are included. First, we prove that for any Z ∈ 2", a 1-random real computable from a 1-Z-random real is automatically 1-Z-random. Second, we give a plain Kolmogorov complexity characterization of 1-randomness. This characterization is related to our proof that X ≤C Y implies X ≤vL Y.

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On initial segment complexity and degrees of randomness

In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that for three fragments of the logic of iterated relativization and transitive closure, the satisfiability problems are fi1 Σ11–complete. Two of these fragments do not include transitive closure. We also show that the question of whether a sentence in these fragments has a finite (tree) model is fi0 Σ01–complete. These results go via reduction to problems concerning domino systems.

The Undecidability of Iterated Modal Relativization

We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees non-total. Several fundamental results are proved: a continuous function with non-total degree has no least degree representation, settling a question asked by Pour-El and Lempp; every non-computable f ∈ [0,1] computes a non-computable subset of ℕ there is a non-total degree between Turing degrees a <Tb iff b is a PA degree relative to a; ⊆ 2ℕ is a Scott set iff it is the collection of f-computable subsets of ℕ for some f ∈ [0,1] of non-total degree; and there are computably incomparable f, g ∈ [0,1] which compute exactly the same subsets of ℕ. Proofs draw from classical analysis and constructive analysis as well as from computability theory.

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Joseph S. Miller

Papers

Randomness and computability: Open questions

Decidability and Complexity Results for Timed Automata and Semi-linear Hybrid Automata

On initial segment complexity and degrees of randomness

The Undecidability of Iterated Modal Relativization

Degrees of unsolvability of continuous functions