The Markoff equation is x+y+z = 3xyz, and all of the positive integer solutions of this equation occur on one tree generated from (1, 1, 1), which is called the Markoff tree. In this paper, we consider trees of solutions to equations of the form x + y + z = xyz + A. We say a tree of solutions satisfies the unicity condition if the maximum element of an ordered triple in the tree uniquely determ...

This paper aims to show that, in order to capture a quite relevant feature such as the recursiveness of abstract data types, Model Theory works better than Category Theory. First, various eategorial notions such as "initiality", "finality", "monoinitiality", "epifinality", "weak monoinitiality" and "weak epifinality' are analyzed, from a model theoretic point of view, as regards the "abstractne...

We give a sheaf theoretic interpretation of Potts models with external magnetic field, in terms of constructible sheaves and their Euler characteristics. We show that the polynomial countability question for the hypersurfaces defined by the vanishing of the partition function is affected by changes in the magnetic field: elementary examples suffice to see non-polynomially countable cases that b...

1. Real Numbers 4 1.1. Approximation of √ 2 4 1.2. Reals, Equality of Reals 5 1.3. The Archimedian Axiom 6 1.4. Nonnegative and Positive Reals 6 1.5. Arithmetical Functions 7 1.6. Comparison of Reals 8 1.7. Non-Countability 11 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 12 2.1. Completeness 12 2.2. Limits and Inequalities 13 2.3. Series 14 2.4. Redundant Dyadic Representat...

The main objective of this study was to examine the role and existence internal control accountability in NGOs Buea Sub-division Cameroon. fol-lowed these research questions: What are concerns governance Cameroon? Among identified items, which ones more important? is implication each item after analyzing? Two theories consti-tuted theoretical framework current study, including agency theory sta...

A widely used theorem of analysis asserts that a uniformly bounded, equicontinuous family of functions has a compact closure in the space of continuous functions. This lemma, variously attributed to Arzela, Escoli, Montel, Vitali, and so on, is of importance in the theory of integral equations, conformal mapping, calculus of variations, and so on. In recent years the lemma has been generalized ...

We pose the problem of whether every FS-domain is a retract of abifinite domain purely in terms of quasi-uniform spaces. 6.1 The problem and its historyEver since domains were introduced by Dana Scott [Sco70] and Yuri Er-shov [Ers75], a question in the centre of interest was to find suitable cartesianclosed categories of domains and the quest for cartesian closed categories ...

1. In a recent paper [18], A. Kleppner proved that every Haar measurable homomorphism 4 : G ~ H between locally compact groups G, H is continuous. This result also follows from a measure theoretic result by D. H. Fremlin [8], (see also [9, 19]), which provides a technique to prove measurability of uncountable unions of measurable sets. The purpose of this note is to present a different and new ...

This paper deals with families of conformal iterated function systems (CIFS). The space CIFS(X, I) of all CIFS, with common seed space X and alphabet I, is successively endowed with the topology of pointwise convergence and the so-called λ-topology. We show just how bad the topology of pointwise convergence is: although the Hausdorff dimension function is continuous on a dense Gδ-set, it is als...

We show that there exists a universal subshift having only a finite number of minimal subsystems, refuting a conjecture in [Delvenne, Kůrka, Blondel, ’05]. We then introduce the still smaller class of quasiminimal subshifts, having finitely many subsystems in total. With N-actions, their theory essentially reduces to the theory of minimal systems, but with Zactions, the class is much larger. We...