On the group of isometries of planar IFS fractals*

For any iterated function system (IFS) on R 2 , let K be the attractor. Consider the group of all isometries on K. If K is a self-similar or self-affine set, it is proven that the group must be finite. If K is a bi-Lipschitz IFS fractal, the necessary and sufficient conditions for the infiniteness (or finiteness) of the group are given. For the finite case, the computation of the size of the group is also discussed.

From Binary Groups to Terminal Rings

Binary groups are a meaningful step up from non-associative rings and nearrings. It makes sense to study them in terms of their nearrings of zero-fixing polynomial maps. As this involves algebras of a more specialized nature these are looked into in sections three and four. One of the main theorems of this paper occurs in section five where it is shown that a binary group V is a P0(V) ring module if, and only if, it is a rather restricted form of non-associative ring. Properties of these non-associative rings (called terminal rings) are investigated in sections six and seven. The finite case is of special interest since here terminal rings of odd order really are quite restricted. Sections eight to thirteen are taken up with the study of terminal rings of order pn (p an odd prime and n ≥ 1 an integer ≤ 7).

Relatively free dimonoids and bar-units

This paper is devoted to the study of the problem of adjoining a set of bar-units to dimonoids. We give necessary and sufficient conditions for adjoining a set of bar-units to the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), and prove that it is impossible to adjoin a set of bar-units to the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid. As consequences, we establish that it is impossible to extend by a set of bar-units the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid to a generalized digroup. We also count the cardinalities of the free left [Formula: see text]-dinilpotent dimonoid and the free [Formula: see text]-dimonoid for a finite case.

Technical Perspective

The paper entitled "Probabilistic Data with Continuous Distributions" overviews recent work on the foundations of infinite probabilistic databases [3, 2]. Prior work on probabilistic databases (PDBs) focused almost exclusively on the finite case: A finite PDB represents a discrete probability distribution over a finite set of possible worlds [4]. In contrast, an infinite PDB models a continuous probability distribution over an infinite set of possible worlds. In both cases, each world is a finite relational database instance. Continuous distributions are essential and commonplace tools for reasoning under uncertainty in practice. Accommodating them in the framework of probabilistic databases brings us closer to applications that naturally rely on both continuous distributions and relational databases.

On Ramsey-Minimal Infinite Graphs

For fixed finite graphs $G$, $H$, a common problem in Ramsey theory is to study graphs $F$ such that $F \to (G,H)$, i.e. every red-blue coloring of the edges of $F$ produces either a red $G$ or a blue $H$. We generalize this study to infinite graphs $G$, $H$; in particular, we want to determine if there is a minimal such $F$. This problem has strong connections to the study of self-embeddable graphs: infinite graphs which properly contain a copy of themselves. We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair $(G,H)$ to have a Ramsey-minimal graph. We use these to prove, for example, that if $G=S_\infty$ is an infinite star and $H=nK_2$, $n \geqslant 1$ is a matching, then the pair $(S_\infty,nK_2)$ admits no Ramsey-minimal graphs.

The Maximum Entropy of a Metric Space

We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of similarity between points). These generalize the Shannon and Rényi entropies of information theory. We prove that on any space X, there is a single probability measure maximizing all these entropies simultaneously. Moreover, all the entropies have the same maximum value: the maximum entropy of X. As X is scaled up, the maximum entropy grows, and its asymptotics determine geometric information about X, including the volume and dimension. And the large-scale limit of the maximizing measure itself provides an answer to the question: what is the canonical measure on a metric space? Primarily, we work not with entropy itself but its exponential, which in its finite form is already in use as a measure of biodiversity. Our main theorem was first proved in the finite case by Leinster and Meckes.

m-Polar Generalization of Fuzzy T-Ordering Relations: An Approach to Group Decision Making

Recently, T-orderings, defined based on a t-norm T and infimum operator (for infinite case) or minimum operator (for finite case), have been applied as a generalization of the notion of crisp orderings to fuzzy setting. When this concept is extending to m-polar fuzzy data, it is questioned whether the generalized definition can be expanded for any aggregation function, not necessarily the minimum operator, or not. To answer this question, the present study focuses on constructing m-polar T-orderings based on aggregation functions A, in particular, m-polar T-preorderings (which are reflexive and transitive m-polar fuzzy relations w.r.t T and A) and m-polar T-equivalences (which are symmetric m-polar T-preorderings). Moreover, the construction results for generating crisp preference relations based on m-polar T-orderings are obtained. Two algorithms for solving ranking problem in decision-making are proposed and validated by an illustrative example.

An Innovative Approach to the Finite Sequences of Prime Numbers

An innovative approach that treats prime numbers as raw experimental data making use of experimental/computational mathematics and the approximation methods is presented in order to get advanced and more exact formulations of the canonical form = being the prime value and its counter. The use of many different functions - such as the inverse of the modified chi-square function with its three parameters , and , the function with the ad-hoc values being , the function , the function , the harmonic series and its approximation by Euler and so on - as fit functions of finite sets i.e. sequences of prime numbers leads to induction algorithms and to new relationships of the kind though within the approximations of the calculations with all the estimations better than that of the standard formulation . In such a manner, refined formulations with higher precisions are got showing that there are many ways to treat the finite sequences of prime numbers. Comparisons among the various methods are made in order to find the best formulation of a new and more refined relationship in a closed form that can be valid to find the most approximate value of a prime starting from its counter in the finite case.

Explicit Kummer theory for the rational numbers

Let [Formula: see text] be a finitely generated multiplicative subgroup of [Formula: see text] having rank [Formula: see text]. The ratio between [Formula: see text] and the Kummer degree [Formula: see text], where [Formula: see text] divides [Formula: see text], is bounded independently of [Formula: see text] and [Formula: see text]. We prove that there exist integers [Formula: see text] such that the above ratio depends only on [Formula: see text], [Formula: see text], and [Formula: see text]. Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction).